voluntary coercion

VOLUNTARY COERCION

Collective Action and the Social Contract _________________________________________________________

Magnus Jiborn

Department of Philosophy Lund University

 Magnus Jiborn ISBN 91-628-3913-6 Printed by Universitetstryckeriet Lund 1999

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To the memory of Rolf Gottfries

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ACKNOWLEDGEMENTS ______________________________________________________________________

A number of people have been important to me in writing this book. First and foremost, I am deeply indebted to Wlodek Rabinowicz. who has been my supervisor during the major part of the time I have worked on it. Without the combination of support, inspiration, knowledge and generosity – but also sharp and rigorous criticism – that has characterized Wlodek’s supervisorship, this book would not have been written. I remember, for example, a day in August 1996, when I had finally – after weeks of gloomy meditation over the impossibility of writing a philosophical dissertation - decided to quit philosophy and find myself some other job. I went to Wlodek’s office to inform him of my irrevocable decision. A good hour later, I left his office in high spirits, filled with thoughts and ideas, with a clearer understanding of the objects of my project and determined to complete it. I owe thanks also to Ingmar Persson who guided my first staggering steps in the world of philosophical ideas, and to Göran Hermerén who inspired and encouraged me to go there. Clas Pihlström has been invaluable as a conversation partner on issues of evolutionary game theory. Clas has also devoted a large number of hours to helping me understand the dynamics of iterated n-player games by computer simulations. A few of these simulations appear in this work. The simulation program was written by Johan Lindén. I have benefited a lot from comments and criticisms raised by participants at the philosophical seminars in Lund where various drafts of different chapters in this work have been presented and discussed. In particular, I want to thank Lena Halldenius, Björn Petersson, Johan Brännmark and Mats Johansson for many useful comments. For the ideas behind chapter 5, I have benefited much from conversations with Jan Morén at LUCS. I also want to thank Johannes Persson who took the time and trouble to read and comment on the entire manuscript at the end of the process.

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John Broome read and gave valuable comments to an early version of chapter 2. This chapter was also presented and discussed at a seminar at SCASSS in Uppsala 1998. Just a few days before the manuscript was completed, Edward McClennen and Philippe Mongin gave a number of very useful comments to chapter 6. Although I have clearly not been able to do justice to all these insightful remarks and suggestions in the final text, I am most grateful for their help. The final parts of this work were completed thanks to financial support from Riksbankens Jubileumsfond. Many friends outside the philosophical sphere have influenced and supported me during the writing of this book. I want to thank Dan-Erik Andersson, Anders Folkesson, Jonas Jiborn and many others for the inspiration, ideas and support that they have provided me with. Finally, I want to thank my family for the encouragement, loyalty and support that they have given me during these years. My wife, Maria, has had a hard time taking care of the family when I have been busy completing this work, and still had patience enough to listen to endless expositions of my philosophical worries. Maria and I are also extremely privileged to have parents who have helped and supported us in every possible way during these years. Without their constant support, I do not know if this work could have been completed. This book is dedicated to the memory of Rolf Gottfries, a dear friend and colleague, who’s presence at the Philosophical Department in Lund made my first time as a PhD-student much more pleasant and interesting than it would otherwise have been. Lund, November 1, 1999 Magnus Jiborn

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CONTENTS

CHAPTER 1 INTRODUCTION ............................................................................................................................9 1.1 THE HOBBESIAN JUSTIFICATION OF THE STATE............................................................................................9 1.2 STRUCTURE OF THE BOOK . ............................................................................................................................14 CHAPTER 2 GAME THEORY AND CULTURAL EVOLUTION ......................................................... 19 2.1 SOME BASIC CONCEPTS OF NON-COOPERATIVE GAME THEORY...............................................................20 2.2 EVOLUTIONARY GAME THEORY....................................................................................................................27 2.3 CULTURAL SELECTION ...................................................................................................................................34 2.4 IMITATION AND UTILITY................................................................................................................................38 2.5 SUMMARY........................................................................................................................................................43 CHAPTER 3 COLLECTIVE ACTION ............................................................................................................ 45 3.1 THE TRAGEDY OF THE COMMONS.................................................................................................................46 3.2 COLLECTIVE GOODS. ......................................................................................................................................50 3.3. THE 2-PLAYER PRISONER’S DILEMMA.......................................................................................................54 3.4. THE N-PLAYER PRISONER’S DILEMMA.......................................................................................................61 3.5 A LTERNATIVE GAMES. ...................................................................................................................................67 CHAPTER 4 THE STATE OF NATURE......................................................................................................... 81 4.1 THE WAR OF EVERY MAN AGAINST EVERY MAN . .......................................................................................81 4.2 THE STATE OF NATURE AS A PRISONER’S DILEMMA.................................................................................82 4.3 RESOURCE COMPETITION...............................................................................................................................85 4.4 THE COLD WAR................................................................................................................................................87 4.5 A PARTIAL STATE OF NAT URE .......................................................................................................................98 4.6 EGOISM .............................................................................................................................................................99 CHAPTER 5 COOPERATION IN THE PRISONER'S DILEMMA. ....................................................101 5.1 THE SHADOW OF THE FUTURE .....................................................................................................................102 5.2 BACKWARD INDUCTION...............................................................................................................................105 5.3 THE EVOLUTION OF COOP ERATION.............................................................................................................112 5.4 NETWORK COOPERATION . ...........................................................................................................................115 5.5 CONSTRAINED MAXIMIZAT ION...................................................................................................................115 5.6 THE N-PLAYER PRISONER'S DILEMMA – THE CASE FOR COOPERATION................................................121

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5.7 P ROBLEMS WITH TAYLOR’S ARGUMENT ...................................................................................................124 5.8 THE N-PLAYER PRISONER'S DILEMMA – THE CASE AGAINST COOPERATION.......................................126 CHAPTER 6 SANCTIONS AND COOPERATION ...................................................................................133 6.1 THE POWER OF COORDINATION. .................................................................................................................134 6.2 A N A SSURANCE GAME .................................................................................................................................138 6.3 EVOLUTIONARY ANALYSIS..........................................................................................................................146 A PPENDIX TO CHAPTER 6....................................................................................................................................154 CHAPTER 7 THE CREATION AND MAINTENANCE OF SANCTIONS ........................................157 7.1 INTRODUCTION..............................................................................................................................................157 7.2 COMMONWEALTH BY ACQUISITION ...........................................................................................................159 7.3 COMMONWEALTH BY INSTITUTION............................................................................................................162 7.4 OVERLAPPING RELATIONS...........................................................................................................................165 7.5 M ETA-NORMS AND THE INFINITE REGRESS ...............................................................................................169 7.6 THE PACKAGE -SOLUTION.............................................................................................................................170 7.7 I S GOVERNMENT NECESSARY? ....................................................................................................................173 CHAPTER 8 THE IDEA OF A SOCIAL CONTRACT .............................................................................179 8.1 THE IDEA OF A SOCIAL CONTRACT .............................................................................................................180 8.2 OBJECTIONS AGAINST SOCIAL CONTRACT ARGUMENT S..........................................................................184 8.3 CONTRACT BY CONVENTION.......................................................................................................................188 8.4 COMMONWEALTH BY ACQUISITION...........................................................................................................199 8.5 SOME FURTHER OBJECTIONS AND LIMITATIONS.......................................................................................203 BIBLIOGRAPHY ..................................................................................................................................................209

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CHAPTER 1

INTRODUCTION ______________________________________________________________________

There are matters in which interference of law is required, not to overrule the judgement of individuals respecting their own interest, but to give effect to that judgement; they being unable to give effect to it except by concert, which concert again cannot be effectual unless it receives validity and sanction from the law. John Stuart Mill1

1.1 The Hobbesian justification of the state. According to what we might call the Hobbesian justification of the state, state coercion is often necessary to enable people to achieve mutually beneficial cooperation, and, hence, to realize fundamental common interests. In the absence of coercion, the argument goes, people will fail to cooperate, not because they are stupid, shortsighted or malevolent, but because the structure of individual incentives is such that each agent, even if it is in her interest that cooperation is brought about, has rational reasons not to cooperate herself, regardless of what others do. For Thomas Hobbes the basic problem was that of establishing peace and security in a fictive, pre-social State of Nature. In the absence of government, Hobbes believed, rational individuals, concerned to promote their own self–preservation, would inevitably find themselves trapped in fierce conflict; in a constant ”war of every man against every man”. Although everyone, according to Hobbes, must prefer peace to this miserable and uncertain state of war, without a common power to protect them and “keep them in awe”, each agent must use whatever means available to protect herself, including unlimited force and violence.

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Mill (1965), Book V, Chapter 11, § 12, p 956

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The only way to terminate this devastating “war of all against all”, according to Hobbes, is by everyone agreeing to confer all their powers and rights to one common power, a sovereign, who thereby becomes strong enough to enforce a peace agreement and “tie them by fear of punishment to the performance of their covenants”. 2 Thus, government is thought to arise as the result of an agreement, a social contract, entered into by the prospective subjects in order to satisfy their most fundamental interests of peace and personal security. David Hume explicitly rejects the idea of a social contract as the foundation of political society. Still, there are similarities between Hume’s views on the motives for the institution of government, and the views of Hobbes. The idea that political authority is a remedy to the problem of achieving mutually beneficial cooperation is central to Hume’s discussion of the origins of government. In an oft quoted passage, Hume states that, Two neighbors may agree to drain a meadow, which they possess in common ; because ‘tis easy for them to know each others mind , and each must perceive that the immediate consequence of his failing in his part, is, the abandoning the whole project. But ‘tis very difficult, and indeed impossible, that a thousand persons shou´d agree in any such action ; it being difficult for them to concert so complicated a design, and still more difficult for them to execute it , while each seeks a pretext to free himself of the trouble and expense, and wou´d lay the whole burden on others. Political society easily remedies both these inconveniences.3

The “original motive” for the institution of government, as well as “the source of our obedience to it”, is, according to Hume, “the security and protection which we enjoy in political society, and which we can never attain when perfectly free and independent”. 4 Thus, Hume contends, “interest, therefore, is the immediate sanction of government”. 5 The idea of political authority as a prerequisite for successful cooperation in certain situations is also expressed in John Stuart Mill’s argument for 2

Hobbes (1996), Ch 17, p 111. Hume (1978) Book III, Part II, Sect. VII, p 538. 4 op. cit. Sect. IX, pp 550-1. 5 ibid. 3

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state intervention, quoted above. Whereas Mill is normally a defender of laisser faire, there are cases where he finds government intervention suitable. One of the examples that he gives is of workers who have an interest to coordinate their actions in order to diminish hours of labor. Assuming then that it really would be the interest of each to work only nine hours if he could be assured that all others would do the same, there might be no means of their attaining this object but by converting their supposed mutual agreement into an engagement under penalty, by consenting to have it enforced by law.6

The kind of problem, for which political authority is thought to provide a solution, is today commonly known as the problem of collective action. The characteristic feature of a collective problem is an inherent disharmony between individual incentives and collective interests; if each agent acts rationally according to her own individual interests, the joint result is likely to be bad for all. With a, somewhat polemic, reference to Adam Smith, Russell Hardin speaks of the problem as ”the back of the invisible hand”. Contrary to Adam Smith’s claim that self interested individuals are ”led by an invisible hand” to promote collective interests, Hardin contends that all too often we are less helped by the benevolent invisible hand than we are injured by the malevolent back of that hand; that is, in seeking private interests, we fail to secure greater collective interests. The narrow rationality of self-interest that can benefit us all in market exchange can also prevent us from succeeding in collective endeavours.7

Consider, for example, the problem of pollution. Most people, presumably, desire to live in a decently healthy environment. We all have fundamental common interests in avoiding global warming, stopping the ongoing destruction of the ozone-layer and reducing dumping of carcinogenic substances into the air we breathe and the water we drink. Still, lacking enforced international regulations, we seem to be unable to cooperate efficiently in order to achieve these ends. Everyone might agree (in 6 7

Mill (1965), Book V, Chapter 11, § 12, p 958. Hardin (1982) p 6.

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principle) on the need for universally observed constraints against pollution; still, as individuals, we find little reason (in practice) to observe such constraints ourselves. There are numerous similar problems of social interaction. Destruction of valuable common resources, such as the atmosphere, is one typical example. Another is the case where a group of people, by joint efforts, can produce some collective good, which everyone desires but which, once produced, no one can be excluded from enjoying. Each agent, although she desires that the good be provided, might then prefer to be a free-rider on the (eventual) contributions of others, rather than to contribute herself. But if everyone acts on her inclination to be a free rider, the good is not produced, and everyone is worse off than if everyone had contributed. Consider, for example, the tax-system and the public sector. We pay a considerable part of our income to the political authorities as taxes. In exchange, they provide us with a number of important goods and services; a legal system, national defense, public education, medical service, public roads and other infrastructure etc. Presumably, since there is no serious political opposition to the system as such8, most of us believe that the benefits that we derive from the existence of these public institutions (as long as nearly everyone contributes) outweigh the cost of our contribution. Still, how many of us would contribute voluntarily if taxes were not enforced? The Hobbesian justification of the state is built on the claim that coercion is sometimes necessary to remedy such collective action predicaments. Where the invisible hand fails, it is claimed, people need the visible hand of a coercive power to ensure that individual incentives do not counteract collective interests. Since, by being subject to coercion, people are enabled to improve their situation by successful cooperation, the existence of such a coercive power is taken to be in everyone’s interest.

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There is, of course a considerable amount of quarrelling about the size, effectiveness and precise objects of the public sector.

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In our century, versions of this way of justifying the state have been put forward by economists such as William Baumol (1952), as well as political philosophers, such as e.g. Gregory Kavka (1986), Jean Hampton (1986), (1997) and Russell Hardin (1991). This kind of justification has been criticized by e.g. Michael Taylor (1995). The Hobbesian justification of the state rests on four central claims. (1) Coercion is necessary to solve certain, important problems of collective action. (2) Coercion actually solves these problems, i.e. it changes the structure of incentives of the problem in such a way that cooperation is individually rational. (3) Coercion is mutually advantageous, i.e. general cooperation under a coercive regime is better for all than general non-cooperation in the absence of coercion. (4) The establishment of a coercive regime is possible, given the motivational assumptions behind the statement of the problem of collective action. That is, it does not merely reproduce an identical collective action problem at a new level. The aim of this essay is to examine and evaluate these claims. Initially, the existence of “coercion” will be interpreted rather abstractly as the existence of an external sanction system, without taking position on its particular form. The Hobbesian argument, of course, makes the further claim that a certain kind of sanction system, i.e. a state, is necessary to achieve effective cooperation. I shall direct this issue towards the end of this work, after having first considered the necessity, efficiency and possibility of sanctions more generally. The analysis will be conducted within a game theoretical framework. I shall adhere to the, by now, established tradition of interpreting the basic problem of social interaction in a Hobbesian state of nature in terms of a Prisoner's Dilemma game. The characteristic feature of a Prisoner's Dilemma is that, in a single-shot game, each player prefers non-cooperation to cooperation, regardless of the actions taken by others, but each also

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prefers the outcome where all, including herself, cooperates to the outcome where no-one does. To evaluate the claim that external sanctions are necessary, I shall discuss the prospects for spontaneous cooperation to emerge in Prisoner's Dilemma situations, in the absence of such sanctions. I will then investigate how the introduction of external sanctions might affect the incentive structure, and hence the prospects for successful cooperation. By “external” I mean here that imposing sanctions requires that the underlying structure of the interaction problem - i.e. the structure of the game – is somehow manipulated. That is, sanctions are not immediately available to the agents as strategies in the game for which sanctions are invoked. However, it need not be external in the further sense that sanctions are established, administered and/or financed by an external agency. In most realistic cases, the players who are involved in the collective action situation must invent and maintain their sanction system themselves. What must be assumed, then, is not the existence of some external deus ex machina but the existence of some additional structure of relations that is external to the original game. 1.2 Structure of the book. CHAPTER 2 is methodological. It introduces some of the basic concepts and tools of game theoretic analysis, which will later be used in the analysis of collective action and social contract issues. Traditional, non-cooperative game theory was developed in order to study economic behavior; the pioneering work is John von Neumann’s and Oscar Morgenstern's Theory of Games and Economic Behavior (1944). It works with the idealizing assumption that each agent is perfectly rational, and it aims to identify equilibrium behavior for rational agents. Evolutionary biologists have since then picked up game theoretic analysis as a way of structuring interaction among animals. But evolutionary biologists didn’t just carbon copy the traditional theory, they adjusted and

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developed it in important ways. In particular, rationality assumptions are left to the side, and replaced by the assumption that agents are genetically programmed for certain strategies, and that a process of natural selection operates on the strategy distribution among the entire population. Equilibrium behavior is achieved as more successful strategies increase and less successful strategies decrease. The current trend is that the evolutionary approach is applied to problems in economics and the social sciences. After all humans are not perfectly rational. The evolutionary approach offers a way of understanding equilibrium behavior without postulating unrealistic degrees of rationality and knowledge. In this context, strategies are not thought to be genetically inherited, but transmitted through a process of imitation. I consider the possibilities, but also some difficulties related to such an application of evolutionary thinking. CHAPTER 3 presents the logic behind collective action. It discusses some of the classical discussions, e.g. Garrett Hardin’s “The Tragedy of the Commons”9 and Mancur Olson’s analysis of logic of collective action. 10 This chapter also presents and discusses the Prisoner's Dilemma and some other games that are relevant for collective action situation. The two-player Prisoner's Dilemma is well known and much discussed. The n-player version, however, is less frequently invoked, although it is probably more relevant for collective action problems in general than the special case with only two players. Other games that are considered are Chicken and the Assurance game. Both these games are related to the Prisoner's Dilemma, only a small change in payoff structure distinguishes a Prisoner's Dilemma from a Chicken game or an Assurance game. Still the resulting equilibrium behavior is very different. Both Chicken and Assurance game can occur with n players.

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Hardin (1968). Olson (1965).

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CHAPTER 4 deals with the Hobbesian idea of the state of nature. It asks for the source and structure of conflict in such a state. It is common to interpret conflict in the Hobbesian state of nature as a Prisoner's Dilemma, something that I will do as well. In this chapter I consider how resource competitions in the absence of government might degenerate into a Prisoner's Dilemma. CHAPTER 5 starts from the Prisoner's Dilemma interpretation of conflict in a Hobbesian state of nature, and considers the prospects for spontaneous cooperation to emerge in Prisoner's Dilemma situations. Concerning the two-player Prisoner's Dilemma, there is very much work done, and there are strong arguments to the effect that cooperation is viable provided that the game is indefinitely iterated between the same two players. This kind of argument is based on the possibility to adopt a reciprocal strategy, i.e. to punish defection in the current game by defecting in a future game. Similar arguments have been raised for cooperation based on community enforcement, i.e. defections against one player in the community trigger retaliation from another player in the community.11 For n-player Prisoner's Dilemma, however, the prospects for cooperation seems to be much more gloomy. I consider one argument to the effect that rational agents would be capable of sustaining cooperation in an iterated Prisoner's Dilemma, regardless of the number of players 12, and contrast it with an evolutionary analysis showing the cooperative equilibrium to be very fragile and unlikely to prevail. The only evolutionary stable state is a rather “shabby” mix of a cooperators and non-cooperators, such that everyone earns a payoff barely above what they would earn if everyone defected.13 CHAPTER 6 develops a model of a limited sanction system, and shows that when it is applied to a group involved in an ongoing n-player Prisoner's Dilemma, the structure of the game changes radically. The modified game turns out to be an Assurance game. I demonstrate that, in contrast to the 11

Kandori (1992). Taylor (1995). 13 Molander (1992). 12

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Prisoner's Dilemma, cooperation in this game is to be expected, although near general defection remains an equilibrium. Moreover, I demonstrate that a state of general cooperation under threat of sanctions might be better for all than the shabby equilibrium with no sanctions, even if the players have to pay to maintain the sanction system. CHAPTER 7 considers the question whether the establishment and maintenance of sanctions is itself a Prisoner’s Dilemma. If this is the case, then it offers no remedy to the problem; to escape the original Prisoner's Dilemma we must have solved another Prisoner's Dilemma already. I argue, however, that this need not be the case. To establish sanctions it is, of course, necessary to achieve cooperation, but this need not pose a second-order Prisoner's Dilemma. In this chapter I also discuss some different ways that a sanction system could be organized. CHAPTER 8 deals with some ambiguities in the notion of a social contract. A quasi-legalistic interpretation, where the normative force of the argument rests on an alleged duty to do what one has promised to, is contrasted with the notion of contract by convention. The relation between social contract and convention, as defined by David Lewis 14is brought to discussion. I argue that interpreting the social contract as a particular kind of convention, allows us to avoid some of the standard objections that are raised against social contract views. However, I also discuss the limitations of such a view. In particular, its normative bite seems to be very limited. Whereas this kind of argument might say something of about the grounds for accepting political authority at all, it says very little about the justification of particular kinds of political regimes.

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Lewis (1986).

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CHAPTER 2

GAME THEORY AND CULTURAL EVOLUTION ______________________________________________________________________

A central assumption of the Hobbesian tradition is that individuals are independent decision makers; each one choosing her actions according to her own preferences and beliefs, each one trying, to the best of her ability, to promote her own individual ends. At the same time, however, they live under conditions characterized by interdependence. The extent to which an individual will succeed in promoting her ends depends, not only on her own actions, but also on the actions of all others, and her behavior, in turn, affects their interests as well her own. Cooperation and conflict, the emergence and continuance of society, must thereby be explained as the result of individual choices in such a complex, strategic environment. Game theory provides a to ol-box for analyzing such strategic interaction formally. We shall here consider two different forms of game theoretical analysis. Classical, non-cooperative game theory operates with the assumption that agents are rational in the sense of having consistent sets of preferences and beliefs, and choosing (what they believe to be) the most effective means to obtain their ends. Evolutionary game theory, on the other hand, operates without assumptions of rationality. Instead, each agent is considered to be pre-programmed for a certain strategy, and the analysis focuses on how strategy profiles of large populations of such pre-programmed agents evolve over time under pressure of natural selection. Although originally developed to model the evolution of genetically transmitted traits, evolutionary game theory has found its way also to issues of economic theory, philosophy and the social sciences. The evolutionary approach is, to use one of its own favorite expressions, "invading" the population of game theorists. Some examples of this development are Robert Axelrod’s The Evolution of Cooperation (1984), Ken Binmore’s

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Game Theory and the Social Contract (1994), and Brian Skyrms’s The Evolution of the Social Contract (1996). This chapter has two aims. First, I will present some of the basic game theoretical tools and concepts that are later to be used in the analysis of collective action and social contract issues. Second, I will discuss some difficulties in the application of evolutionary game theory to traits that are supposed to be culturally, rather than genetically, transmitted. In particular, I will focus on the problem of how to interpret payoffs in this context. As we shall see, whereas both classical, non-cooperative game theory and the biological application of evolutionary game theory work with well-defined payoff concepts, it is somewhat unclear what payoff functions represent in the context of evolutionary games based on cultural transmission, and how these payoffs are related to growth rates of different strategies. Having spelled out these difficulties, I shall suggest a way of understanding the payoff concept, which is applicable to some, but not necessarily all, types of culturally transmitted behavior. 2.1 Some basic concepts of non-cooperative game theory. A non-cooperative game is a situation of strategic decision-making, such that each member of a group chooses individually her course of action and the payoff to each depends on everyone’s choice. Thus, in every game, G, there is a finite set of players, N ={1,...,n} where n ≥ 2. For each player i∈N there is a finite set of alternative pure strategies, Si. A pure strategy, si, is interpreted as a complete specification of player i’s course of action throughout the game. An outcome of the game G is a certain strategy-profile, s = (s1,...,sn), consisting of one pure strategy for each player i∈N. Thus, the set of alternative outcomes of a game is equal to the set S=×i∈NSi, of possible combinations of the players’ pure strategies. Sometimes it is reasonable to assume that agents, instead of having to choose one particular pure strategy, may randomize over the set of

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alternative pure strategies. In such cases, a mixed strategy, xi, is taken to be a certain probability distribution over the pure strategies in Si. We say that ∆i is the set of such probability distributions over Si. Similarly, ∆=×i∈N∆i is the set of all possible combinations of mixed strategies, one for each player i∈N, sometimes called the mixed-strategy space15 of game G. Each player i∈N is supposed to have a consistent - i.e. complete and transitive - preference ordering, ? i , over the set S of pure strategy profiles, or outcomes, of the game. The relation s? i s* means that player i weakly prefers s to s*, i.e. either she prefers s to s*, or she is indifferent among those two alternatives. Strict preference, ? , is defined in terms of weak preference such that iff s? i s* and not s*? i s, then s? i s* . In many cases, however, it is convenient to represent the preference ordering of each player i∈N as a payoff function, ui(s) , assigning a numerical value to each outcome s∈S, such that ui(s) ≥ ui(s*) iff s ? i s*. In standard, noncooperative game theory, these payoff functions are normally interpreted as von Neumann-Morgenstern (vNM) utility functions. A vNM utility function is a cardinal representation of a player’s preferences, i.e. besides information about the ordering of a set of alternatives, it also conveys certain information about utility differences between different alternatives. More precisely, a vNM utility function is thought to represent a player’s preferences over all possible lotteries with the alternative outcomes of G as prizes. Thereby it allows us to calculate and compare the expected utility of different strategies in cases where the outcome is uncertain, for example when one (or more of the others can be expected to use a mixed strategy. However, the information that a vNM utility function conveys is limited in important respects. Most importantly, a vNM utility function is unique only up to a positive linear transformation, i.e. any positive linear transformation of such a function is interchangeable. That is, if ui(s) is a vNM utility function representing i’s preferences over the set S of alternative outcomes, then any function u’i(s) = a + b ui(s), where a and b are constants and 15

Weibull (1995) p. 6.

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b > 0, is an equally good representation of i’s preferences over S. That means that there is no fixed, natural zero point, and no fixed, natural unit of measurement. For any set of outcomes, there is an infinite number of numerical functions, mutually related by positive linear transformations, each of which represents a persons utility equally well. This means that, if payoffs are represented in terms of vNM utility, they do not provide for interpersonal comparisons. For classical, non-cooperative game theory, this does not matter, since the analysis does not invoke interpersonal comparability anyway. For evolutionary game theory, however, matters are different; the evolutionary analysis requires a payoff concept that does provide for interpersonal comparisons. We shall return to this issue in subsequent sections. In a strategic game, or game in normal form, all players are supposed to choose simultaneously, once and for all, complete strategies for the entire game. For some purposes, however, it is useful to consider games that extend over time, with choices made at different points in time. This may be the case, either when choices are not made simultaneously by all players, or when each of the players faces a number of sequential choices during a game, and the temporal order of their choices may affect the outcome. Such a game in extensive form is usually represented as a decision tree (fig. 2.1) At each choice node, one, or more, of the players makes a choice between a set of alternative actions. A pure strategy, si, is here interpreted as a function that, for each choice node where i is among the players who choose, ascribes to player i one of the alternative actions that are available to her at that choice node.

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j

i

(s a, sb)

(s a, s*b) j

(s*a, sb)

(s*a, s*b) fig. 2.2

A type of extensive game that will be of particular interest here, is where the same strategic game is iterated an indefinite number of times between the same players. In an iterated game, provided that players have memory, each player can choose to apply a conditional strategy such that her choice of action at any move of the game depends on how the game has developed up to that point. As was said in the beginning, classical, non-cooperative game theory assumes that players are rational. Rationality here means that their beliefs and preferences satisfy certain consistency requirements, for example that they handle probabilities in a consistent way, and that they act so as to maximize their own expected utility. It is sometimes held that non-cooperative game theory models agents as complete egoists. This is correct only in the sense that a player’s payoff function is taken to reflect everything that player cares about. There are no restrictions on the kinds of motives – egoistic, altruistic, sadistic, moral, aesthetic etc – that could enter into the specification of a player’s utility function. But once her utility function has been settled, once the payoff

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matrix of the game is fixed, each player is assumed to choose her strategy in order to maximize her own payoff, without trying to increase or decrease the payoffs of other players. Other-regarding preferences might affect a player’s evaluation of the set of outcomes, but they do not enter a second time in determining her actions given these evaluations. Further, players are often assumed to be fully informed about all aspects of the game. Thus, each player is supposed to have complete knowledge about the set of alternative strategies available to her, as well as the strategy sets available to her opponent(s). Further, each player is supposed to have complete knowledge, not only of her own utility function over the set of outcomes, but also the utility function(s) of her opponent(s). Finally, all this is often supposed to be common knowledge among them, i.e. each player knows that each of the other players is rational and fully informed, and each player also knows that each of the other players knows this, and each knows that each knows that each knows, and so on. Assuming common knowledge of rationality might be problematic, however. Such common knowledge implies that each player knows that she herself is rational. But if rationality is a feature of choice, then the player knows, before she made her choice, that her choice will be rational. Fredrick Schick argues that assuming players to know that they are rational may lead to contradictions, and suggests a weaker, alternative assumption of mutual belief in rationality.16 That is, each player believes that all other players are rational, each believes that each believes that all others are rational, and each believes that each believes that each believes, and so on. The object of non-cooperative game theory is to identify the best strategy for each player in a given situation. Which strategy is best, however, will often depend on what the other players of the game do. Let s-i = (sj)j∈N \{i} be a strategy vector, consisting of one strategy for every player j∈N except i. Then s*i∈Si is a best response to s-i iff, for all si∈Si, (s*i, s-i) ? i (si, s-i).

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Schick (1999).

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In some cases, however, some strategy is better than others, regardless of which strategies are used by other players. Such strategies are labeled dominant. A strategy s*i∈Si is weakly dominant iff, for all si∈Si and all si∈S -i, (s*i, s-i) ? i (si, s-i) and, for all si ≠s*i and some s’-i∈S-i, (s*i, s’-i) ? i (si, s’-i). A strategy s*i∈Si is strictly dominant iff, for all si∈Si such that si ≠s*i and all s-i∈S-i, (s*i, s-i) ? i (s i, s-i). Similarly, a strategy s*i∈Si is said to be weakly dominated iff there exist some strategy si∈Si such that, for all s-i∈S-i, (si, s-i) ? i (s*i, s-i) and, for some s’-i∈S-i, (si, s’-i) ? i (s*i, s’-i). A strategy si∈Si is said to be strictly dominated iff there exist some strategy si∈Si such that, for all s-i∈S-i, (si, s-i) ? i (s*i, s-i). The central solution concept of noncooperative game theory is the Nashequilibrium. In a Nash-equilibrium, each player’s strategy is a best response to the strategy vector of the other player(s). None of the players can then gain by unilaterally changing her strategy. Thus s* is a Nashequilibrium iff, for all i∈N and all si, (s*i, s-i) ? i (si, s-i). Similarly, s* is said to be a strict Nash-equilibrium iff, for all i∈N and all si, (s*i, s-i) ? i (si, s-i).

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When only pure strategies are considered, some games do not have any Nash-equilibria. When mixed strategies are allowed, however, it can easily be shown that every game has at least one Nash-equilibrium. However, many games have more than one Nash-equilibrium, and it may then be difficult to determine which among a number of possible equilibrium strategies that a player ought rationally to use. If we do not assume that players know beforehand what strategies their opponents will use, only that they are rational and know each other to be rational, the outcome of a game might therefore be indeterminate. A number of refinements of the Nash-equilibrium concept have been suggested in response to this issue, but none of these refinements will play a significant role in the present work. A rather different approach is to try to identify strategies that are rationalizable. A strategy si∈Si is said to be rationalizable if it is a best response to some strategy profile x-i, which is compatible with i’s belief that all other players are rational. That is, every component of x-i must itself be a rationalizable strategy. Rationalizability, thus, is a weaker solution concept than Nash equilibrium. In many games, every strategy for every player is rationalizable. Another set of concepts from economics that will frequently be invoked here is Pareto-optimality and Pareto-superiority. These concepts do not strictly belong to the game theoretical tool-box, but are useful in comparing the relative advantages to the group of different outcomes. An outcome, s*∈S, is Pareto-superior to s’∈S (and s’ Pareto-inferior) iff, for all i∈N, s*? i s’, and for some j∈N s*? js’. The outcome s*∈S, is strongly Pareto-superior to s’∈S (and s’ strongly Pareto-inferior) iff, for all i∈N, s*? i s’.

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An outcome s*∈S, is said to be Pareto-optimal (or Pareto-efficient) iff there is no outcome s’∈S that is Pareto-superior to s*. The assumptions of perfect rationality and information are, of course, extreme idealizations. Real-life people are neither perfectly rational nor well-informed about the logic structure of their interactions. The question naturally arises then, whether game theoretic analysis can say anything about the behavior of imperfect individuals in more realistic situations of social interaction. Another limitation of the classical analysis is its static perspective. It focuses on static equilibria, i.e. states that are such that, when they obtain, none of the players has rational grounds to change her strategy. However, it does not consider the question how different equilibria might arise, and, in the case of multiple equilibria, which equilibria are more likely to obtain. We might therefore want a more dynamic analysis, that allows for changes and experiment, and that allows imperfectly rational agents to adapt their behavior to the environment on basis of previous experience. Evolutionary game theory offers such a dynamic perspective.

2.2 Evolutionary game theory. Evolutionary game theory arose in the 1970s as an application of game theoretical analysis to problems of evolutionary biology. Whereas classical game theory was developed to analyze rational behavior in economics and social sciences, John Maynard Smith and other evolutionary biologists demonstrated that game theoretical analysis also provided powerful tools for explaining various aspects of animal behavior. 17 But evolutionary biologists didn’t just carbon copy the traditional theory, they adjusted and developed it in important ways. As Ken Binmore puts it

17

Maynard Smith & Price (1973), Maynard Smith (1974),(1976),(1982)..Maynard Smith (1982) mentions Lewontin (1961) and Hamilton (1967) as early introductions of game theoretic concepts in evolutionary theory, and claims that the "method of thinking was foreshadowed by" Fisher (1930).

27

After all, insects can hardly be said to think at all, and so rationality cannot be so crucial if game theory somehow manages to predict their behavior under appropriate conditions.18

Instead of assuming individuals to be rational deliberators, the evolutionary version of game theory starts from the assumption that agents, who compete for possibilities to reproduce, are pre-programmed for certain strategies. The theory focuses on how the strategy profiles of large populations of such pre-programmed agents evolve over time, when the same game is played over and over again between randomly drawn agents, and the outcomes of current games determine the frequency of different strategies in the future. Who are the players of this kind of game, and what do game payoffs represent here? In classical game theory, as we have seen, the basic condition is rationality. Any kind of entity to which we can ascribe instrumental rationality and a consistent set of preferences and beliefs could play games in this sense. That may include not only human beings but also entities like firms, organizations, states etc. Payoffs represent the preferences of such rational players. In evolutionary theory the basic and necessary assumptions are replication, variation and heredity. Any kind of entity capable of replicating itself with differential success, where differences in success are related to hereditary variations, will be subject to an evolutionary process. Richard Dawkins 19 has coined the term replicator for such entities. In the biological case the basic unit of replication is, of course, the gene, but there is no a priori reason to count out other types of entities as potential replicators.20

18

Binmore “Foreword” to Weibull (1995), p. x. Dawkins (1989). 20 The other day I received an e-mail urging me to vote against the creation of a nazi discussion-group on internet, and also to forward the message to all I knew with an email address. As it turned out, the vote was over since several years, but the message had continued to "live" and reproduce. A "mutation" had occurred in the message during the transmission process (the final date of the vote was erased from the message) which enhanced the message’s survival value. See http://www.sub-rosa.com/whitepower/ for the history of this electronic "replicator". 19

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One possibility would be to think of evolutionary games as something played directly between replicators. In the biological application, the players would then be genes. The argument for such an interpretation might be that genes are much more stable entities than individuals. Individuals are merely temporary "survival machines" that their gene sets use for short periods of time, and could without much loss be left aside in evolutionary thinking. Pay-offs in such a model would represent changes in a replicatorplayer’s fitness due to the outcome of a game, i.e. in the expected number of future copies that a gene will be able to make of itself. A more natural interpretation, however, is to think of evolutionary games as something that goes on at the level of the phenotype. This is the way Maynard Smith sees it. “Evolutionary game theory,” Maynard Smith contends, is a way of thinking about evolution at the phenotypic level when the fitnesses of particular phenotypes depend on their frequencies in the population.”21

A phenotype here is identical to a strategy, i.e. a complete specification of an individual’s course of action under any contingency. 22 A player is an instance of such a behavioral phenotype, i.e. an individual member of a population. Game payoffs, finally, represent changes in individual fitness due to the outcome of a game. The fitness of an individual is defined as the expected number of (surviving) offspring per time unit. However, evolutionary game theory does not take much interest in the payoff to particular, identifiable individuals. Its interest is how different strategies do on average when games are played repeatedly between individuals who are randomly drawn from a large population. The average payoff to a certain strategy in this setting will depend on the whole environment of different strategies in which it operates. Strategies that do well in a certain environment may be complete disasters in another. An environment here is a population state, i.e. a certain frequency 21 22

Maynard Smith (1982) p 1. Maynard Smith (1982) p 10.

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distribution of different strategies. The average payoff to a strategy in a certain environment will in turn determine its future frequency in the population. Strategies that, on average, earn high payoffs in the current environment are assumed to increase their future frequencies and strategies that, on average, earn lower payoffs are assumed to decrease. Thus, the strategic environment will not be fixed but will evolve as long as there are some strategies that do better and some strategies that do worse. The process where high-performing strategies increase whereas lowperforming strategies decrease, and eventually disappear, is the process of natural selection. The selection process presupposes variations in behavior. In the case of genetic evolution, this variation is provided by random mutations and recombinations of genes. Genes are copied from parent to offspring, but sometimes errors occur in this process; the “copy” is not identical to the original. Such random errors might give rise to new strategy phenotypes. In most cases, copy errors disappear rapidly from the population, but sometimes a new well-adapted strategy occurs. To simplify matters, it is often convenient to restrict attention to games that are symmetric, or uniform, in the sense that each player has the same strategy set and the same payoffs. 23 By saying that players have the same strategy set it is meant that (i) each player has the same number of alternative strategies, and (ii) there is some natural description of the available alternatives that is the same for each player. What does it mean that all players have “the same” payoffs? Of course, it does not mean that all players receive the same payoff in every possible outcome – that would imply that we only consider pure coordination games. Rather, it means something like the following. Suppose that, for every i and j in N, Si and Sj are “identical” in the sense described above. 23

This symmetry assumption is, of course, only an artificial restriction, motivated by considerations of analytical simplification. In biology, games are often played between individuals in radically different positions, such as, for example, games played by predator and prey or between male and female in a game of reproduction. For such cases, we need somewhat more complex models, where individuals who occupy different positions are conceived of as belonging to different populations. See Weibull (1995) pp 163 ff.

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That is, for every strategy s*i∈Si there is a corresponding strategy s*j∈Sj which can be described as the same strategy. Then, payoffs are uniform if, for all i, j∈N, and for all x-i∈∆-i , ui(s*i, x-i) = uj(s*j, x-j). In a symmetric game, thus, we have a finite set of pure strategies, S ={1,…,m}, the same for all players, and a corresponding set ∆ of mixed strategies. A population state is equivalent, then, to a mixed strategy x∈∆. Such a population state x could be interpreted either as a state where all members of the population are programmed for the same mixed strategy x, or as a polymorphic state, where each individual is programmed for some pure strategy such that the proportion of different strategies corresponds to x. In a two-player game, being matched against a randomly drawn individual in population state x is equal to being matched against an individual actually playing the mixed strategy x. Consequently, the average payoff to strategy y in population state x is equal to the expected payoff to strategy y when matched against the mixed strategy x, i.e. u(y, x). The population average in such a case is equal to the expected payoff to the mixed strategy x when matched against itself, i.e. u(x, x). The central solution concept of evolutionary game theory is that of an “evolutionary stable strategy” (ESS), a concept that was first defined by John Maynard Smith and Price24. An evolutionary stable strategy is a strategy that, once it dominates in a population, does strictly better than any possible mutant strategy, and hence cannot be invaded. In a symmetric, two-player game, x is an ESS iff, for all y∈∆ u(x, x) > u(y, x), or, u(x, x) = u(y, x) and u(x, y) > u(y, y). An alternative formulation is the following. x is an ESS iff, for all y≠x, there exist some ε y∈(0,1) such that for all ε∈(0, ε y), u(x, w) > u(y, w),

24

Maynard Smith and Price (1973), Maynard Smith (1982)

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where w=εy + (1-ε)x is the mixed strategy corresponding to the population state that results when population state x is modified by the entrance of a proportion, ε, of strategy y. An evolutionary stable strategy, thus, is immune against invasion by a small proportion of any possible mutant. If a mutant strategy occurs in a proportion that is smaller than that, it does strictly worse than the incumbent strategy, and will hence be wiped out. If a current population state x is evolutionarily stable, then, it will return to x whenever it is exposed to some small disturbance. The concept of evolutionary stability can be generalized to cover strategy sets as well as single strategies. Let X⊂∆ be a closed and nonempty set of strategies. Then, for a symmetric 2-player game, X is an evolutionary stable set (ES set) iff each for each x∈ X there is a neighborhood W ⊂∆, such that, u(x, w) = u(w, w) for all w∈Y, and u(x, w) > u(w, w) if w ∉ X. The neighborhood W of x, consists of all mixed strategies w = εy + (1-ε)x that can be achieved by modifying x by the entrance of some proportion ε∈(0, ε y) of some strategy y∈∆. That is, if w = εy + (1-ε)x belongs to W then, for all ε’= ε, w’ = ε’y + (1-ε’)x does also belong to W. Thus, the following definition is equivalent. X is an ES set iff, for all x∈X and for all y∈∆, there exist some ε y ∈ (0,1) such that, for all ε ∈(0, ε y), Ax(w) = Aw(w) and Ax(w) > Aw(w) if y ∉ X, where w=εy + (1-ε)x is the population state that results when population state x is modified by the entrance of a proportion, ε, of strategy y, In words, an ES set is a set of mixed strategies such that there are no invasion barriers between strategies within the set, but, for every strategy in the set, there is a positive invasion barrier against every strategy that is not a member of the set. A neighborhood of x is an open set which contains x.

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There is an interesting connection here between the solution concepts of evolutionary and classical game theory. An ESS is a strategy that earns higher payoffs than any other strategy when it dominates the population, i.e. it is the best response to itself. Every evolutionary stable state is thus also a Nash-equilibrium. The reverse does not hold, however. The set of ESS in a game is a subset of the set of Nash-equilibria. Thus, the evolutionary analysis shows how a population of less than ideally rational beings might achieve and sustain a Nash-equilibrium outcome. Since the ESS concept is “sharper” than Nash-equilibrium, it also offers a partial solution to the problem of equilibrium selection. It explains why certain equilibria are viable, but others are not. In order to describe more precisely how the distribution of different strategies evolve over time in a population, evolutionary game theorists have developed mathematical models of the selection dynamics, expressing the growth rate of a certain pure strategy j in population state x as a function of the size of j’s average payoff in x relative to the average payoff to other strategies in x. One commonly used such model is the replicator dynamics25, according to which the growth rate of the frequency of a pure strategy j in given population state x equals the difference between the average payoff to j and the population average in x. However, there are other possible kinds of selection dynamics, linking growth rates to payoffs in different ways. Weibull (1995) lists a number of different properties that might characterize an evolutionary selection dynamics. Payoff monotonicity implies that a strategy i∈S has a higher growth rate in population state x than another strategy j∈S if and only if the average payoff to i in x is higher than the average payoff to j in x.

25

Taylor and Jonker (1978).

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Payoff positivity, which is a weaker requirement than payoff monotonicity, implies that a strategy i∈S has a positive growth rate in population state x if and only if the average payoff to i in x is higher than the population average in x. The weakest requirement is weak payoff positivity implying only that, unless every strategy in x earns exactly the same payoff, there is at least one strategy among those earning above average that has a positive growth rate.26 Which of these models yields the most realistic picture of the dynamic process in a certain type of situation depends on the particular type of selection mechanism that is operating. In the case of cultural evolution, which we shall deal with here, the precise properties of the different possible selection mechanisms remain rather unexplored. There is, therefore, some uncertainty about the relevant kind selection dynamics in different cases. However, many interesting results concerning the stability of equilibria are valid for any payoff-positive (or even weakly payoffpositive) dynamics. 2.3 Cultural selection. The basic principle behind the evolutionary approach is simple and general. Using a formulation of Robert Axelrod, it says that "whatever is successful is likely to appear more often in the future".27 In the context of biological evolution this principle works through genetical heredity and differential reproductive success. Successful behavior is behavior that promotes an individual’s chances for survival and reproduction, and genes for successful behavior will thus enjoy greater possibility to be passed forward into the next generation than genes for less successful behavior. The raw material for the selection process, i.e. the variation of competing strategies to which individuals are genetically

26 27

Weibull (1995), pp. 139-150 Axelrod (1984), p 169.

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programmed, is provided by random mutations and recombination of gene sequences. In a non-genetical, or cultural, context it is assumed that a similar selection mechanism could work through processes of imitation and learning. The intuition is that players are able to imitate each other, and that (the strategies of) more successful players will be imitated more frequently than (the strategies of) less successful ones 28. Again, the raw material for the selection process can be thought to be provided by something similar to mutation and recombination (although not necessarily random). New strategies may enter the field of competition either by random mistakes (copy errors) or by purposive innovations.29 Although the intuition may seem natural and reasonable, the extension of the evolutionary approach to non-genetically transmitted strategies runs into some difficulties. First of all, it is unclear what payoffs should be taken to represent in this kind of games. Neither biological fitness nor vNMutility seem to fit very well with the model. A vNM-utility interpretation is problematic since evolutionary models generally presume that payoffs are interpersonally comparable, something which vNM-utility does not provide. In order to speak of the growth rate of a strategy as a function of its average payoff, there must be some natural way of calibrating the utility scales of different players such that we can meaningfully add together and compare the payoffs for different players. Also, if we think of payoffs in terms of preference based utility it must be explained why an individual who satisfies her preferences should thereby also be assumed to promote the expected number of individuals imitating her behavior. In the biological case an individual cannot promote her individual fitness without thereby promoting the expected number of offspring programmed for similar behavior.

28 29

Axelrod (1984) p 169-70, Weibull (1995) pp. 152 ff. , Skyrms (1996) p 11. Cavalli-Sforza and Feldman (1981) p 351.

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Cultural transmission of strategies may occur by a number of different mechanisms. Parents teach their children how to behave, we learn in school, from friends, books, movies, oral traditions etc.30 The selection dynamics connected to different modes of transmissions is likely to vary. Here, I will focus on one type of case only - imitation of successful agents. I shall ignore the effects of other types of transmission, such as that from parents to children in upbringing, in which biological fitness obviously plays an important role . Are more successful agents in general imitated more frequently than others? If so, how can we account for this fact? There is at least one interpretation by which it is trivially true that more successful agents are imitated more frequently than less successful ones. Suppose that payoffs represent increments in "cultural fitness"31 due to the outcome of the game, just as payoffs in biological game theory represent increments in biological fitness. Suppose further that "cultural fitness" is defined as expected number of future imitators, exactly as "biological fitness" is defined as expected number of future offspring. It is obviously (and trivially) true, then, that more "successful" agents are, on average, imitated more frequently. This follows from the definition of success. The attraction of this interpretation is the close analogy to concepts and definitions in biological theory of evolution. Results from the biological context will easily be transferable to evolution by imitation. But does it answer any interesting questions? One objection against this interpretation is that it leaves us with very strange concepts of "success", "self-interest" and similar. Robert Axelrod’s evolutionary analysis, for example, focuses on the problem of collective action which, "occurs when the pursuit of self-interest by each leads to a poor outcome for all" 32. But we do not normally consider having few 30

Cavalli-Sforza and Feldman (1981) gives a survey of different types of cultural transmission. The kind discussed here is belongs to what Cavalli-Sforza and Feldman calls “horizontal transmission”. 31 Cavalli-Sforza and Feldman (1981). 32 Axelrod (1984) p 7.

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people imitating us as a "poor outcome", and we do not normally think of "pursuit of self-interest" as a matter of trying to maximize the expected number of disciples. Another difficulty is how to determine the payoff matrix of a game, before it is played. How do we calculate the effect on "expected number of imitators" of different outcomes? To do that we need to have some background knowledge about statistical relations between imitation frequencies and observable variables in the outcomes. It should be noted that we are interested here in the application of abstract game theoretic models to certain types of problems in human, social interaction. To account for such an application, we must be able to explain how variables in the theoretic models are related to observable variables in real-life situations. Without such correspondence between observable variables and variables in the model, it is difficult to see in what sense the theoretical construct is a “model” of some particular kind of situation. In biological theory, there exist relevant background knowledge about statistical relations between number of offspring and observable variables in the outcomes, which allow us to have a well-founded opinion about the effect on "expected number of offspring per time unit" of different outcomes. These observable variables are such things as energy intake, energy consumption, probabilities for mating etc. In a fight over territory, we can compare different territories concerning variables such as supply of food and water, possibilities to find shelter, risk for predators etc, factors that we know affect the number of surviving offspring. It is, I believe, such background knowledge that allows us to take the step from pure theory to application. For the cultural application of evolutionary game theory, we must account for some similar relations between theoretical concepts and variables in the real world. Suppose that the outcomes of a game are different distributions of some measurable good or combination of goods (such as time, money, food, water, risk of injury, ...). One way of expressing the payoff to an agent is in

37

terms of increments to her bundle of goods. Suppose further that agents have preferences over the set of possible bundles of goods. Now if the cultural fitness value of an outcome is defined as the effect of that outcome on an agent’s "expected number of imitators" then there must be some statistical relationship between imitation frequencies and variations in bundles of goods. Such a relationship might be for example, that agents who earn more money are, on average, imitated more frequently than agents who earn less money. It might, of course, be the case that there is such a relation between variations in bundles of goods and imitation frequency, independently of the agents’ evaluations of different bundles. However, if there is no correlation between imitation and evaluation, I think it is misleading to speak of the process as "imitation of the successful". Imitations are then based on something else than what we normally mean by “success”. I doubt that, if there are such cases, they are best be described in terms of games at all. My interest here, is in cases where the imitation frequency is in some way related to the agents’ preferences over different bundles of goods. That is, we are interested in the assumption that imitation frequencies are related to individual "success" in a more ordinary sense of that word, i.e. where "success” is measured in terms of some values that the agents can plausibly be assumed to care about. 2.4 Imitation and utility. Why should we believe that agents imitate those who are judged more successful? One natural explanation is that each agent wants to maximize her own utility. In a world of perfect information and perfectly rational agents, an agent might try to maximize her own utility by rationally calculating which of the available strategies is her best response to the strategies that she expects other to use. In a world of less than perfect information and rationality, however, such direct maximizing might be difficult to apply, and also a rather unreliable method. She must, then, find some other method.

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One possibility is to learn by experience. Each agent tries a number of different strategies, and then continues with one that yields her a satisfying payoff. By applying relevant learning rules, individuals who are less than perfectly rational and/or informed, can learn to play a Nash-equilibrium. A striking example from the literature is the following. Baldwin and Meese (1979) studied the behavior of a pair of pigs in a Skinner box, arranged so that when a lever at one end of the box was pressed, food was dispensed at the other. They found that in those cases in which the pair developed a stable pattern of behavior, the dominant pig pressed the bar and then rushed over to the food dispenser, while the subordinate pig waited at the dispenser. Such behavior is stable for the following reason. Provided that enough food is dispensed at each press of the bar to ensure that some is left when the dominant pig arrives, the dominant pig is rewarded for bar-pressing; obviously, the subordinate is rewarded for waiting at the dispenser. The reverse pair of behaviors would not be stable; the subordinate would not be rewarded for pressing the bar because the dominant would prevent it from eating. 33

These pigs did not calculate the expected utility of different strategies. Nor were they genetically programmed for bar-pressing respectively waiting at the dispenser. They tried some different strategies, and continued to use the strategies for which they were rewarded. Still, they ended up acting on their Nash-equilibrium strategies. The application of such learning rules does not require interpersonal comparisons. Each agent evaluates only her own payoffs with different strategies according to her learning rule. Pure trial and error is a rather slow process, however, although faster than biological evolution. Each new pig in the box has to begin from scratch and make all mistakes for herself. If the game played is reasonably complex and the number of possible strategies large, then players might never reach an equilibrium. As Maynard Smith notes One lifetime would not be long enough for such an inefficient learning process.34 33 34

Maynard Smith (1982), p 55. Op. cit. p 171.

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If agents were able learn from the experience of others, they would avoid a lot of unnecessary costs, and each newcomer could start off directly from an equilibrium already established by others. Imitation of agents who have been successful so far might, in other words, be an efficient general metastrategy in world of imperfect knowledge and reasoning abilities. Agents do not generally know all logically possible strategies for each situation they find themselves in, and they seldom have any realistic possibility to calculate with any degree of certainty the expected payoff of the strategies that they do know about. By imitating others, an agent is able to pick up a strategy that has proved to work well so far. The background assumption is that "what has worked well for others will probably work well for me too". Consider first a simple model of an imitation process35. Agents in a large population are matched at random and play a game G. Each agent plays the same strategy in every matching, but with some frequency revises her strategy. At such a revision, she samples some other agent from the population, observes the other players strategy and payoffs, and then decides whether to copy the strategy of the other or continue with her current strategy. Such a process of strategy revision may take payoffs into account in three different ways. First, the revision rate for each agent may depend on her average payoff with her current strategy. Second, each revising agent may compare the payoffs of the sampled agent with her own payoff, and change strategy only if the other player’s payoff exceeds her own by some unit. Third, the probability that an agent will be sampled by a revising agent may depend on her payoff. If imitations are random, and only revision rates are thought to vary with payoffs, then we have a version of a trial and error learning process, where imitations simply serve to supply agents with alternatives to try out.

35

See Weibull (1995) Young (1996).

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However, it is easily seen that all these ways of taking payoffs into account presuppose some kind of interpersonal comparison of success. If revision rates are to depend on current payoffs, such that less successful agents are more likely to revise their strategy, then there must be some natural way of calibrating their payoffs. Fredrick Schick has suggested that a calibration of individual utilities might be achieved by equalizing the upper and lower limits of each individual’s utility scale, i.e. the best and the worst outcomes for each player are assigned the same utility value, say 1 respectively 0. This idea is criticized by Amartya Sen, Richard Jeffrey and Peter Hammond 36. In this context, however, there might be a natural, operational interpretation of such a calibration. We might measure the “dissatisfaction” of an individual with her current strategy, as the probability of revising her strategy within a certain time interval, t. If we assume that each agent will be less dissatisfied, the larger her current payoff, we have a natural way of calibrating payoffs between 0 and 1. Suppose that ui(s*i, x-i) > ui(si, x-i). Then, if i changes from si to s*i in population state x, she will be less dissatisfied. Hence, i will be less likely to revise her strategy in x if her current strategy is s*i than she would be if her strategy was si. Now, utility functions might be calibrated in the following way. For each player i, we assign the value 1 to some outcome, o*, that i considers so extremely good that she would never revise her strategy si in x if ui(si, x-i) = ui(o*). Conversely, we assign the value 0 to some outcome, o’, that i considers so terribly bad that she would with absolute certainty revise her strategy si in x if ui(si, x-i) = ui(o’). The following requirements are assumed. For all i, ui(s*i, x-i) = 1 ⇒ ui(s*i, x-i) = ui(si, x-i) for all si∈Si. ui(s*i, x-i) = 0 ⇒ ui(s*i, x-i) = ui(si, x-i) for all si∈Si.

36

Schick (1971), Sen (1970), Jeffrey (1974), Hammond (1991).

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Consider a game that is symmetric in the following sense. The strategy set is the same for all players, and, for all players i,j and for all x∈∆, ui(s*i, x-i) > ui(si, x-i) ⇔ uj(s*j, x-j) > uj(sj, x-j).37 If all strategy revisions are made in accordance with the above, the process of strategy revision in a large population will yield a payoff monotonic selection dynamics. If imitation frequencies are to depend on the relative “success” of the imitated agents, then we must assume that the revising agents are capable of judging the relative success of different agents. These interpersonal comparisons need not be objective, i.e. we need not assume that there is an objectively correct answer to who is the more successful of two agents. But there must be some level of correspondence between the subjective judgments of different players. Without such correspondence, strategy changes will be entirely unpredictable. We will not have a process of selection that can meaningfully be said to favor more efficient strategies over less efficient ones. There are different ways of achieving the required stability in subjective judgments. Suppose, for example, that we accept probability of revision as a measure of a player’s dissatisfaction with her current strategy. Then, players might simply prefer to imitate strategies that are less frequently abandoned. Another possibility is this. Suppose that the outcomes of a game are different distributions of some quantifiable good V. Suppose further that each player has a preference ordering over different quantities v of V, such that she either prefers more of V to less, or she prefers less of V to more. For a reasonably large class of goods it is plausible also to assume that nearly everyone has the same preference ordering over different quantities. That is, for nearly every player i,j, (v+v’) ? i (v) ⇒ (v+v’) ? j (v)

37

This is a somewhat weaker notion of symmetry than the one defined on p 29.

42

Nearly all people prefer ceteris paribus a shorter term in prison to a longer, nearly all in the population of market investors prefer higher interests on invested money to lower and nearly all in the population of athletes prefer being number one in their sport to being number two or number three (or worse). For games where the outcomes are different distributions of V, a good for which nearly everyone has the same preference ordering, payoffs can simply be stated in terms V directly, without making the detour over individual vNM-utilities. Again, given that any agent is more likely to imitate someone who is successful in terms of V than someone who is unsuccessful in terms of V, it is straightforward to define the growth rate of a strategy as a positive function of its average payoff in terms of V. 2.5 Summary. To conclude this chapter, we have seen that the evolutionary approach to game theory offers new insights to classical problems. It offers a partial solution to the problem of equilibrium selection, it explains how an equilibrium can be reached through a dynamic process where strategies of different players are adapted to each other, and it allows us to do without extreme assumptions of rationality and common knowledge; in fact, no rationality or common knowledge assumptions at all are necessary to explain equilibrium behavior. The application of the evolutionary approach to behaviors that are not thought to be genetically inherited, but rather culturally transmitted through a process of imitation, is not entirely unproblematic. Although the theory as such is general, certain concepts and relations must be reinterpreted to fit with the cultural application. I have focused on some unclear points regarding the payoff concept and the relation between individual payoffs and strategy growth rates – problems that have not received much attention in the literature. Neither the payoff concept of classical game theory, i.e. vNM utility, nor that of biological evolutionary game theory, i.e. Darwinian fitness, can be applied in the new context without complications.

43

However, I have also showed that there are ways in which these difficulties can be met. Agents in social interaction are, normally, neither perfectly rational nor brainless robots. Ideally, a theory of social interaction should be able to deal with agents with limited rationality, agents who try to act with foresight, who try to predict the future and choose the best response, but whose horizon and knowledge of the game they are playing is limited, and whose reasoning abilities are far from infallible. Such an integrated theory is still lacking, however. As a second best alternative, I will in this work try to consider the issues from both the rationalistic and the evolutionary point of view.

44

CHAPTER 3

COLLECTIVE ACTION ______________________________________________________________________

Ruin is the destination toward which all men rush, each pursuing his own best interest in a society that believes in the freedom of the commons. Garrett Hardin38

A collective action problem is a situation where a group of individuals is prevented from achieving common interests by the structure of their individual incentives. If each one acts rationally to promote her own private interests, the group is likely to realize an outcome that is worse for all than some possible, alternative outcome. There are several examples of situations where individual incentives seem to run counter to common interests in this way and, hence, individually rational behavior may yield outcomes that are bad for all. Some examples that are frequently mentioned are: Overexploitation of valuable common resources. Pollution. • The problem of providing of collective goods, such as national defense, certain public infrastructure, radio/TV-broadcasting, increased prices due to cartel-activities or union negotiated wage-increases. • The security dilemma involved in international armaments and nuclear deterrence. • The security dilemma faced by individuals in a Hobbesian state of nature. The first part of this chapter (section 3.1 and 3.2) will discuss some of these situations. The second part is devoted to the formal, game theoretical representation of collective action. Thus, section 3.3 presents the twoplayer version of the Prisoner’s Dilemma, section 3.4 deals with the nplayer generalization of the same game, whereas section 3.5 presents some 38

Hardin (1968), p 1244.

45

alternative game structures that are of interest in the study of cooperation and political authority. 3.1 The tragedy of the commons. A typical situation that is likely to raise a collective action problem, is when a valuable common resource is exploited by a number of independent agents. If the resource in question is freely accessible to all, but in limited supply; individually maximizing behavior is likely to result in overexploitation, such that the resource pool is eventually exhausted. The classical formulation of this kind of problem is Garrett Hardin’s “The Tragedy of the Commons”. 39 Hardin asks us to consider a situation where herdsmen put their cattle to graze on common pasture. Each herdsman, according to Hardin, has an interest in keeping as many cattle as possible. However, the number of cattle that the common pasture can sustain is limited. Above that number, overgrazing occurs, and the pasture will be damaged. Ultimately, the pasture may be permanently destroyed. As long as the number of animals remains below the carrying capacity of the land, individually maximizing behavior creates no problem. Each herdsman can add new animals to his herd without costs to the group. As soon as the maximum sustainable level has been reached, however, each herdsman adding an animal to his herd imposes a cost on the group. This cost consists in increased overgrazing, which, in turn, diminishes the productivity of the land. Now, when considering whether to add a new animal or not, each rational herdsman asks, “‘What is the utility to me of adding one more animal to my herd?’ ”40 Whereas the cost of increased overgrazing is dispersed over the entire group of herdsmen, the whole benefit from the additional animal accrues to the individual cattle owner. As long as the benefit produced by one additional animal is larger than the individual herdsman’s share of the cost, adding an animal is the rational thing to do. But, as Hardin says 39 40

Hardin (1968). Op. cit., p. 1244.

46

this is the conclusion reached by each and every rational herdsman sharing a commons. Therein is the tragedy. Each man is locked into a system that compels him to increase his herd without limit – in a world that is limited. 41

The logic behind this conclusion can be brought out by considering an example. Suppose there are n herdsmen sharing a commons, and suppose that the land can support ω cattle without any loss of productivity. Suppose, further, that when the total number of animals is below ω , each animal produces a net benefit of 1 unit. Every additional animal above ω , however, decreases the productivity of the land by an amount that is equal to the production of one animal. Thus, when the (ω +1)th animal is added, total production is reduced by 1 unit, when the (ω +2)th animal is added, total production is further reduced by (ω -1)/ (ω +1), and so on. Let X be the state where there are x animals on the commons. Let π(x) be the total production and let α(x) be the production per head in X. Thus, assuming that production is symmetric - i.e. in every state X, every animal has the same production - we suppose that π(x) and α(x) satisfy the following conditions. (C1) (C2) (C3)

∀x(α(x) = π(x)/x) ∀x((0 ≤ x ≤ ω ) ⇒ π(x) = x) ∀x( x ≥ ω ⇒π(x+1) = π(x) - π(x)/x)

Let λ(x) be the reduction in productivity per head that results from adding one more animal in X. Let ν i(x) be the number of animals in i’s herd at X. Then, i’s share of the cost that results from adding one more animal in X is λ(x) ×ν i(x) Being a rational maximiser, i will increase his herd in X if (C4)

α(x+1) > λ(x) ×ν i(x)

Proposition 3.1 For every state X such that n >2 and ω >3, there must be at least one herdsman for whom condition C4 is satisfied.

41

Ibid.

47

Proof (i) ∀x(λ(x) = α(x) - α(x+1)) By (C1) and (C2), we have, (ii) ∀x((0 ≤ x < ω ) ⇒ α(x) = 1 ∧ α(x+1) =1) By (i) and (ii), it follows that, (iii) ∀x((0 ≤ x < ω ) ⇒ λ(x) = 0) By (ii) and (iii), it follows that, (iv) ∀x∀i((0 ≤ x < ω ) ⇒ (α(x+1) > λ(x) × ν i(x)). (That is, whenever x is smaller than ω , (C4) is satisfied for every i.) By (C1) and (C3), we have, (v) ∀x(x ≥ ω ⇒ α(x+1) = (π(x) – π(x)/x)/ (x+1)) ⇔ ∀x(x ≥ ω ⇒ α(x+1) = π(x)×(1-1/x)/(x+1)) ⇔ ∀x(x ≥ ω ⇒ α(x+1) = π(x)×(x-1)/x(x+1)) By (i) and (v), it follows that, (vi) ∀x(x ≥ ω ) ⇒ λ(x) = π(x)/x -π(x)×(x-1)/x(x+1)) ⇔ ∀x(x ≥ ω ⇒ λ(x) = π(x) ×2/x(x+1)) By (v) and (vi), it follows that, (vii) ∀x(x ≥ ω ⇒ (α(x+1) > λ(x) × ν i(x)) ⇔ (ν i(x) < π(x)×((x-1)/x(x+1)) /π(x)×(2/x(x+1))) ⇔ ∀x(x ≥ ω ⇒ (α(x+1) > λ(x) × ν i(x)) ⇔ (ν i(x) < (x-1)/2)) Thus for all x such that x ≥ ω , C4 is identical to (C5) ν i(x) < (x-1)/2 Now, let ν min(x) denote the smallest number of cattle that anyone owns in X. By necessity, (viii) ν min(x) ≤ x/n It is obvious that (C5) will be satisfied for some i iff ν min(x) < (x-1)/2. Thus, (ix)

48

∀x((x/n < (x-1)/2) ⇒ ∃i(ν i(x) < (x-1)/2)))

It is easily checked that x/n < (x-1)/2 for all (x, n) such that x > 3 and n > 2. Thus, (x)

∀x∀n((x > 3 ∧ n > 2) ⇒ ∃i(ν i(x) < (x-1)/2))

By (ii) and (x), it follows that, (xi)

∀x∀n∀ω ((x ≥ω ∧ ω > 3 ∧ n>2) ⇒ ∃i(α(x+1) > λ(x) × ν i(x))). QED.

For any value of x, then, if ω > 3 and n>2, there is always someone who gains individually by adding one more animal to his herd. This means that, as long as each herdsman is a rational maximiser, herds will grow larger and larger without limit, whereas the production per animal eventually approaches zero. Total production decreases rapidly above ω , and goes towards zero as the number of animals goes to infinity. Fig. 3.1. shows how total production and production per animal develops when ω = 100. With 1000 animals, total production is reduced to less than 10. total prod. α (x)

prod./animal

π ( x) 100

1

0

100

200

300

400

500

600

700

800

900

1000

fig. 3.1 The destructive logic of the commons can be seen at work in many situations; serious global issues as well as more local ones. Garrett Hardin’s article focuses on the problem of overpopulation. A finite world cannot support a human population that grows without limit; hence, Hardin

49

claims, there is no “technical solution” to the problem of a constantly growing humanity. One might doubt, however, that this is really a good example; it depends on whether it is true that individual parents always benefit by putting another child to the world. The propensity to have many children is likely to depend on a number of social factors such as economic opportunities, level of education, systems of social security, the health situation, infant mortality and so on. But there are other, and perhaps better, examples. Destruction of open sea fish populations, extinction of whales and other species of economically valuable animals, the cutting down of virgin forests; these are typical cases. Each fisherman might have an interest in restricting the total amount of fish catched, to some level that will be compensated for by natural reproduction. However, the size of her own catches does not affect future fish populations at large to any perceptible degree. Hence, although each fisherman might support universally observed restrictions, each of them also has reason to continue catching as much as possible, irrespective of what others do. The problem of pollution is very similar. Air and water are, in some sense, common assets that can be used freely by anyone. Up to some level, the atmosphere might handle the wastes that we dispose of, but above that level, pollution becomes destructive to the conditions for life. Driving my own car to work involves both a cost and a benefit. The cost is that it adds to the greenhouse effect and the level of carcinogenic substances in the air. The benefit is that I save half an hour. Just as with Hardin’s villagers, although the total cost outweighs by far the total benefit, I may still benefit individually by driving the car, since the cost is dispersed over everyone, whereas the benefit accrues to me alone. 3.2 Collective goods. The tragedy of the commons is caused by a combination of valuable resources being freely available to anyone, and in limited supply. The property of being freely available to anyone is also called nonexcludability; no one can be excluded from using it.

50

As Russell Hardin notes, in real life, it might be difficult to find very clear and unambiguous cases of perfectly non-excludable goods42. However, it is not strictly necessary that exclusion is really impossible, only that it would be so costly or difficult to achieve as to make it uneconomic or practically unfeasible 43. Russell Hardin argues that de facto unfeasibility of exclusion may even obtain as a result of legal restrictions. The important point is that, under the circumstances at hand, for whatever reason, selective exclusion is not a practically feasible option. Also, a good may be non-excludable only with respect to a certain group. A good is non-excludable with respect to a certain group of individuals if it is the case that, if some group member consumes the good, “it cannot feasibly be withheld from the others in that group”44. Non-group members might still be excluded from benefiting from it. A good that is non-excludable in this sense, with respect to some group of individuals, Olson calls a collective good. In the tragedy of the commons the common resource was also supposed to be in limited supply. However, that a collective good is in limited supply is not a necessary condition for a collective action problem to occur. In fact, according to the classical definition of a public good, the provision of which is often presumed to raise a problem of collective action, a public good is a good that is characterized by non-excludability and jointness of supply (or indivisibility). That a commodity is in joint supply means that any individual can consume any amount of it without reducing the amount available to others. The provision of such a public good may generate a collective action problem if it requires that each of a number of agents makes a positive, and individually costly, contribution. Since, once the good is provided, noncontributors cannot be excluded from benefiting from it, there is a temptation to be a free rider. As in the tragedy of the commons, making a contribution involves a cost and a benefit. In this case, however, the cost accrues entirely to the individual whereas the benefit is dispersed over the 42

Hardin (1993). Olson (1965). 44 Op. cit., p. 14. 43

51

entire group. Hence, even when the total value of a contribution outweigh its costs, the share of the benefit that falls to the individual contributor may still be far less than her cost. However, since it seems that neither jointness of supply, nor limited supply are necessary conditions for the generation of a collective action problem, we will follow Mancur Olson in speaking henceforth of collective goods referring to goods that are non-excludable with respect to some group. There are various types of collective goods. One common distinction is that between continuous and step goods45. A continuous good is such that any additional contribution adds to the value of the good, whereas a step level good is such that it requires some number k of contributions before some amount of the good is contributed. A special case is where the good is either provided entirely, or not at all, i.e. a single-step good. Standard examples of this type of problem is the construction of a bridge or the election of a candidate. A bridge, presumably, has no value at all until it is finished. No one can cross a river on a half a bridge. Another distinction is that between goods that are maximized only at the maximal level of contribution and goods for which there is an optimal level of contributions, such that additional contributions above that level either make no difference or even subtract from the value of the good. The incentive problems behind these different kinds of collective goods provisions are likely to vary. Mancur Olson’s analysis of the logic of collective action proceeds from a cost/benefit calculus for each individual member of a group 46. An individual i is supposed to contribute voluntarily in the production of a collective good, only if there is some amount of the good that can be produced at a total cost that falls below i’s share of the benefit from that amount. When this is the case, it might be in i’s interest to pay the whole cost of producing that amount (unless he has reasons to believe that if he won’t pay, someone else will pay instead.) 45 46

see Hardin (1993) pp 50 ff. Olson (1965).

52

Olson claims that a group is likely to succeed in providing itself with a collective good if and only if the group contains at least one individual who has an interest to pay the entire cost. In such cases, Olson speaks of the group as being privileged. If a group does not contain any such individual, it is said to be latent. Olson’s main conclusions concern the effects of group size on the likelihood that a collective good will be produced. Large groups are likely to be latent whereas small groups are more likely to be privileged. However, Olson also admits the existence of “intermediate” groups for which the result is somewhat indeterminate. The conclusion that larger groups are less likely to succeed in producing a collective good is based on the following arguments. First, the larger the group, the smaller the “fraction” of the total group benefit that an individual receives. Therefore, the larger the group, the less likely it is that any individual’s gain from the collective good will be large enough to outweigh the total cost of its production. Also, the larger the group, the larger the “organization costs” involved in producing a collective good. In large, latent groups, collective goods will, according to Olson, be provided only as the result of “selective incentives”, i.e. rewards or punishments. Since the main object of this essay is to investigate the effects of such selective incentives, specifically punishments, on collective action, this issue will be more thoroughly discussed in later chapters. That intermediate groups are indeterminate, according to Olsson, results from the possibility that one agent’s behavior may influence the behavior of others. If, in a reasonably small organization, a particular person stops paying for the collective good that he enjoys, the costs will rise noticeably for each of the others in the group; accordingly, they may then refuse to continue making their contributions, and the collective good may no longer be provided. However, the first person could realize that this might be the result of his refusal to pay anything for the collective good, and that he

53

would be worse off when the collective good is not provided than when it was provided and he met part of the cost. 47

This argument anticipates to some extent the dynamic analysis of n-player interactions that we shall consider in later chapters. Olson assumes that agents might apply strategies that are conditional on the choices made by others; they may “refuse to continue making their contributions” if someone “stops paying”. Hence, the argument presupposes that there is some dynamic mechanism, like iteration, involved in the situation. The smaller the latent group, according to Olson, the more likely it is that one individual’s contribution will have a noticeable effect on the cost/benefit calculus of other members of the group, and, hence, the more likely it is that one’s behavior will influence choices of others. Olson’s contention that larger groups are less likely to succeed in providing themselves with collective goods is well in line with much of the later results on n-player Prisoner’s Dilemma games. However, his argument is difficult to assess without a proper dynamic analysis of n-player interaction. We will therefore return to the effects of group size in later chapters. 3.3. The 2-player Prisoner’s Dilemma. In the Prisoners Dilemma, each agent has one dominant strategy, i.e. one strategy that she prefers irrespective of which strategy the other agent(s) choose. It is normally assumed that each agent faces a binary choice, i.e. she must choose one of two pure strategies, commonly labeled Cooperation (c) and Defection (d). Each agent is supposed to prefer d regardless of which strategy the other(s) choose. However, each agent is also supposed to prefer the outcome where everyone, including herself, cooperates, to the outcome where everyone, including herself, defects. Thus, the outcome that results when each agent uses her dominant strategy is strictly Pareto-inferior to the outcome that would result if each agent would instead use her dominated strategy.

47

Op. cit., p. 43.

54

In the simplest case, with two players and two alternative pure strategies, the situation can be represented as in fig 3.2. Player 2’s payoffs are given in the upper right corner of each box. Player 2 c d 3

4

c Player 1

3

0 0

1

d 4

1 fig. 3.2

Since each player has one dominant pure strategy, it follows that there is one unique Nash-equilibrium; mutual defection. According to the orthodox view, this means that mutual defection is the inevitable outcome of a single-play Prisoner's Dilemma. One argument to the contrary goes as follows 48. We assume that it is common knowledge among the agents that each of the agents is fully rational and fully informed about the conditions and the payoff matrix of the game they are playing. That is, each knows that each of them is fully rational, each knows that each knows that each of them is fully rational, each knows that each knows that each knows…, and so on, ad infinitum. Since each of them is fully informed about the conditions of the game, they acknowledge that their respective options are symmetric in a certain sense; if c is the rational choice for one of them, c must the rational choice for the other as well. Likewise, if d is the rational choice for one of them, d must be the rational choice for the other as well. They both know that both know this. Hence, since both are rational, each of them can assume that they will both choose exactly the same way. This means that there are only two plausible outcomes; i.e. mutual defection and mutual cooperation. Of these two, 48

Rapoport (1966), Davis (1985a), (1985b), Watkins (1985).

55

mutual cooperation is strictly Pareto-superior. Hence, being rational maximizers, they must both choose cooperation, and each can take this alternative, “secure in the knowledge that the other will take it also.”. 49 The active substance in this argument is symmetry. As Rapoport writes, “because of the symmetry of the game, rationality must prescribe the same choice to both”. However, we should beware not to read too much into the apparent symmetry of a Prisoner's Dilemma game. For example, the fact that the strategies are normally labeled Cooperation and Defection for both players need not imply that each agent actually faces a choice between the same two actions. In an exchange relation, cooperation might for one of the agents involve performing some piece of work (e.g. mowing a lawn), whereas, for the other agent, it involves handing over a certain sum of money. However, such symmetry is not necessary for a game to be a Prisoner's Dilemma. Consider a game with the payoff structure represented in fig. 3.3. This is also a 2-player Prisoner's Dilemma. There are no requirements concerning how the payoffs of different players relate to each other, only concerning the internal structure of each player’s payoff function. Player 2 c d 999

1000

c Player 1

3

0 0

1

d 4

1 fig. 3.3

49

Davis (1985a) p 48.

56

However, even if we grant that the situation is symmetric in the way that the Symmetry Argument suggests, the argument is ill founded. It mixes together two distinct notions; that of an action being evidence for a certain state of affairs obtaining with that of an action being the cause of a certain state of affairs. In the words of Jon Elster, the Symmetry Argument involves “a slide from unexceptional diagnostic thinking to an unjustified causal argument” 50 Knowing that I am a fully rational person, and knowing that the other player is exactly like me, i.e. fully rational, my Cooperating might indicate that the other player will also Cooperate. This is a good sign, of course. However, Cooperating myself does not cause the other player to Cooperate.. What the other player does is causally independent of what I do; that is part of the definition of the problem. This means that, if Defecting is the rational thing to do, Cooperating would not have been rational. If Defecting is rational, my rational opponent will defect. But he would still defect, if I had Cooperated. The symmetry argument also seems to raise questions about what it means to be a fully rational person and what it means to know that one is fully rational. Can one really know that one is a fully rational person, and still deliberate? That is, can one really know that one will choose the rational alternative, prior to knowing which alternative to choose? If one is uncertain about which alternative is the right one, it seems, it should at least be possible that one could make a mistake. Common knowledge that one will do the right thing should not, therefore, be allowed to enter into an argument about what is the right thing to do. The position that will be taken here is the orthodox one. In a single-shot Prisoner's Dilemma, if players are rational and the game is not modified in any way, mutual defection is inevitable. The Pareto-superior cooperative outcome is out of reach for rational agents. Some modifications that might allow for rational cooperation will be considered in Chapter 5. A two player Prisoner's Dilemma can be formally defined as follows. 50

Elster (1985) p 144.

57

Definition 3.1. 2-player Prisoner's Dilemma Let G=〈N, (Si), (ui)〉 be a strategic game such that N={1,2} and, for each player j∈N, Si={c, d}. Then G is a Prisoner's Dilemma iff, for all i∈N, (i) ui(di, c-i) > ui(ci, c-i) > ui(di, d-i) > ui(ci, d-i) (ii) ui(ci, c-i) > (ui(di, c-i) + ui(ci, d-i))/2 Some comments on the definition are appropriate. First, formulations of the Prisoner's Dilemma often contain the assumption that players are unable to communicate with each other. The motivation for this condition seems to be that if players are able to communicate before making their choices, they might promise each other to cooperate. However, since the game is not a coordination game, such promises would be of little value. Each player would still have to decide whether to comply with her promise or not, and this compliance problem would be identical to their original Prisoner's Dilemma. What is required, then, is not that players are unable to communicate, but that they cannot take on causally binding commitments prior to playing. In other words, it is required that the players’ choices are causally independent of each other, i.e. that the game is non-cooperative. Second, consider condition (ii). It ensures that mutual cooperation is Pareto-optimal even when mixed strategies are allowed, i.e. there is no mixed strategy profile that is Pareto-superior to mutual cooperation. When iterated games are considered, this condition ensures that, if the same two players,1 and 2, were to play a series of games, they could not gain by an agreement to oscillate between (d1 , c2) and (c1, d2). In a way, however, condition (ii) might seem redundant, since if there was a Pareto-superior mixed strategy profile, x*, then the cooperative move for each player i would be to choose the mixed strategy, x*i, that x* ascribes to her. There would be no more reason for any player i to apply her mixed strategy, x*i , than there was for her to choose her pure strategy ci, since (i) implies that, for every i∈N, every mixed strategy, xi, is strictly dominated by her pure strategy d. Thus, (i) is sufficient to ensure the existence of a unique, strictly Pareto-inferior Nash-equilibrium, and a strictly Paretosuperior outcome that is inaccessible to rational players.

58

Likewise, in an iterated game, if two players, 1 and 2, could gain by alternating between (d1 , c2) and (c1, d2), then every two consecutive moves of the iterated game could be described as two moves in a sequential Prisoner's Dilemma that satisfies (ii). To see this, consider the game of fig.3.4, which does not satisfy (ii).

Player 2 c d 3

7

c Player 1

3

0 0

1

d 7

1 fig. 3.4

Now, suppose that Player 1 and Player 2 are going to play this game exactly twice, and they make the agreement that Player 1 should cooperate in the first move, and Player 2 in the second. Should they comply with this agreement? Assuming that both players will choose d when this is in accordance with their agreement, the situation can be represented by the decision tree of fig 3.5.

59

Player 2 Player 1

7,7

c d

c

1,8 c

d

8,1

d

2,2

fig 3.5

Reducing the situation to a game in normal form, with the strategies Comply, c, and Defect, d, we get the following matrix (fig. 3.6), which satisfies both (i) and (ii). Player c

2 d 7

8

c Player 1

7

1 1

2

d 8

2 fig. 3.6

This is a Prisoner’s Dilemma that satisfies (ii). This argument, of course, requires that the binary choice assumption is relaxed, such that agents are allowed to choose among a much wider set of strategies. Such an extension of the definition of a Prisoner's Dilemma should require

60

(i) (ii)

That each player has one strictly dominant pure strategy That the dominant strategy equilibrium is strictly Pareto-inferior to at least one (mixed or pure) non-equilibrium outcome.

However, as we shall see in the following section, extending the definition in this way also opens for certain difficulties. When there are mixed strategy profiles that are strictly Pareto-superior to every pure strategy profile, it might be difficult to determine which of a number of possible Pareto-optimal outcomes cooperation should target on. 3.4. The n-player Prisoner’s Dilemma. The 2-player Prisoner’s Dilemma game has attracted very much attention in the analysis of cooperation and conflict ever since it was formulated in the 1950’s51. However, in real life, most interesting collective action situations involve more than two agents. An n-player generalization of the PD-structure was suggested already by Luce and Raiffa. 52 Their account, however, is anecdotal rather than formal. They ask us to consider a number of wheat farmers who sell their production on a common, free market. By restricting production, they could increase the market price for wheat. Each farmer “as an idealization”, is assumed to have a choice between two pure strategies, “restricted production” and “full production”. The strategy of a given farmer, however, does not significantly price level – this is the assumption of a competitive market regardless of the strategies of the other farmers, he is better circumstances with full production. Thus full production restricted production; yet if each acts rationally they all fare poorly.53

affect the – so that off in all dominates

In this game, as in the two-player game considered in the previous section, each player has one strictly dominant, pure strategy (“full production”), but the unique Nash-equilibrium that obtains when each one applies her

51

The game was originally invented by Merrill Flood and Melvin Drescher, and got its name from an illustrative anecdote about two prisoner’s that was formulated by A.W. Tucker. See e.g. Hardin R, (1993) p 24, Skyrms (1996) p 48-49. 52 Luce and Raiffa (1989/1957). 53 Op. cit., p 97.

61

dominant strategy is also strictly Pareto-inferior to the outcome that would obtain if each would instead apply her dominated strategy. It is often convenient to assume also that the strategy sets and payoff functions for all players are uniform. Doing so, allows us to represent graphically, in a way suggested by Thomas Schelling54, the payoff structure of a n-player game with binary choices. Schelling gives the following definition of a uniform n-player Prisoner’s Dilemma. 1. There are n people, each with the same binary choice and the same payoffs. 2. Each has a preferred choice whatever the others do; and the same choice is preferred by everybody. 3. Whichever choice a person makes, he or she is better off, the more there are among the others who choose their unpreferred alternative. 4. There is some number, k, greater than 1, such that if individuals numbering k or more choose their unpreferred alternative and the rest do not, those who do are better off than if they had all chosen their preferred alternatives, but if they number less than k this is not true.55

A game that satisfies these conditions is represented in fig. 3.7. As in the 2player case, the alternative strategies are labeled Cooperation, (c), and Defection, (d). The payoff for each pure strategy is given as a function of the number of others in the group who choose to Cooperate.

54

Schelling (1978). Schelling uses the term “multi-person prisoner’s dilemma” (MPD), but his definition is general, i.e. it covers the two-player case as well as the multi-player case. 55 Op. cit., p 218.

62

d

u(d,0)

Payoff

c

0

m

n -1

Number of others who co-operate

fig. 3.7

The assumption of uniformity is useful in the sense that it allows for a convenient graphical representation of the game. The interpretation of this assumption was briefly discussed in the previous chapter. It should be noted however, that it need not imply interpersonal comparability in a strong sense. Suppose that Si={ci, di} for all i∈N. Let ui(ci , j) be the payoff to player i for choosing her strategy ci , and ui(di , j) the payoff to i for choosing di, when there are exactly j others who choose their c-alternative. Then, payoffs are uniform iff, for every player h and every player i in N, there are some constants a and b, with b>0, such that for all j∈(0, n-1), ui(ci , j) = a + b uh(ch , j) and, ui(di , j) = a + b uh(dh , j) Given this assumption, we can have a common function u(s, j) which is taken to represent the payoffs for every player. Without this assumption, we would have to draw individual payoff-curves for each player. When the number of players is large, the graphical representation would be unreadable. For this reason we will often restrict attention to the narrower class of uniform, binary choice games. However, although convenient, uniformity should not be considered a strictly necessary condition for a problem to qualify as a Prisoner’s Dilemma

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Schelling also assumes (condition 3.) that payoffs are monotonically increasing with the number of cooperators. The same condition is invoked by Molander56. One might question whether this condition is necessary, however. Consider the following game (fig. 3.8). d

Payoff

c

m

0

n -1

Number of others who cooperate

fig. 3.8 This game does not satisfy Schelling’s third condition, since the payoff curve for Cooperation slopes downwards for some values of j. Still, however, Defection is strictly dominant for each player, universal Defection is the unique Nash-equilibrium, and universal Cooperation is strictly Pareto-superior to general Defection. Hence, rational agents are inevitably led to realize a strictly Pareto-inferior outcome. Michael Taylor57 suggests a wider and more general definition of the nplayer Prisoner's Dilemma than Schelling and Molander, a definition that does not require payoffs to be monotonically increasing. Taylor assumes that players face a binary choice between Cooperation and Defection, and also that payoff functions are uniform. However, the definition can easily be stated without the uniformity assumption.

56 57

Molander (1992). Taylor (1995).

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• Definition 3.2. n-player Prisoner’s Dilemma. Let G=〈N, (Si), (ui)〉 be a strategic game such that for each i∈N, Si = {ci, di}. Let ui(ci , j) be the payoff to player i for choosing her strategy ci , and ui(di , j) the payoff to i for choosing di, when there are exactly j others who choose their c-alternative. Then, G is an n-player Prisoner’s Dilemma iff, for all i∈N, and all j=0,1,..., n-1, (i) ui(di, j) > ui(ci, j) (Dominance of d) (ii) ui(ci, n-1) > ui(di, 0) (Pareto-inferior equilibrium) Taylor adds a third assumption, namely that ui(di, j) > ui(di, 0) for all j≠0. This assumption, Taylor claims, is “eminently reasonable”58. However, since (i) and (ii) are also claimed to be “necessary and sufficient conditions”, the additional assumption should probably not be considered a necessary part of the definition. However, there are some types of configurations that satisfy this definition, but which we might be uncertain whether they should be considered nplayer Prisoner's Dilemma games or not. For example, consider the following game (fig. 3.9). As before, we assume that there are n players facing the same binary choice between c and d, and we assume also that payoff functions are uniform. c

Payoff

d

Number of others who choose c

fig. 3.9

58

Taylor (1995) p 84.

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It is easily verified that this game satisfies Definition 3.2. Thus, it has one unique Nash-equilibrium, which obtains when each player adopts her strictly dominant strategy, d. This equilibrium is also strictly Pareto-inferior to the non-equilibrium outcome where each adopts her strictly dominated strategy, c. However, unlike the previous cases, universal Cooperation is not Pareto-optimal. Everyone is better off if some players Defect than if no one does. This type of case raises a further difficulty, namely that of deciding who should Cooperate and who should be allowed to Defect. If players are allowed to communicate prior to playing, they could, as Brian Barry says in a comment on Elster, “get together and work out some system – a rota, a lottery, or whatever – to deal in a systematic and coordinated way with the problem”. 59 Doing so, Barry contends, “ is far superior to either engaging in prodigies of individual computation in order to coordinate without communication (…) or following individual decision rules that make the prescribed action independent of any expectations about what others will do”. 60 The problem here is that there is no obvious answer to how such a system should be construed. We might, therefore, want to add a condition that allows us to avoid this difficulty. Not because these kinds of games are uninteresting. To the contrary, I believe that most interesting collective action problems in real life are such that, in addition to the problem of achieving cooperation, they also raise questions about how the burdens and benefits of cooperation should be distributed. This might be done by a lottery, as Barry suggests, or by a bargaining process, by submitting the case to an arbitrator, or simply by applying some generally accepted principle of fairness. Once this choice problem has been solved, it is often convenient to restate the problem of compliance with the agreement made in terms of a pure Prisoner's Dilemma. 61 For analytical reasons, however, we might want to 59

Barry (1985) p 157. The comment concerns Elster (1985). ibid. 61 David Gauthier’s Morals by Agreement develops an argument for such a case with only two players. What Gauthier does is to split the argument in two parts; one dealing 60

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separate the pure problem of achieving cooperation, from the problem of agreeing on the terms of cooperation. The monotonicity condition invoked by Schelling and Molander, of course, ensures that general Cooperation is Pareto-optimal, but there are weaker conditions that would do the job. For example, we might simply add the following condition to Def 3.2: For all i∈N, and for all j = 1,2…n-1, • j < n-1⇒ ui(ci, n-1) > ui(ci, j). For many interesting problems, monotonicity is a reasonable assumption, at least as an idealization. If the object of cooperation is the provision of a continuous collective good, then payoff curves are naturally monotonic. However, I believe that we should avoid restricting the definition more than necessary. Unless something else is said, the following conditions will be taken to characterize the general Prisoner's Dilemma in this work.. • Definition 3.3. n-player Prisoner’s Dilemma. Let G=〈N, (Si), (ui)〉 be a strategic game such that for each i∈N, Si = {ci, di}. Let ui(ci , j) be the payoff to player i for choosing her strategy ci , and ui(di , j) the payoff to i for choosing di, when there are exactly j others who choose their c-alternative. Then, G is an n-player Prisoner’s Dilemma iff, for all i∈N, and all j=1,2,..., n-1, (i) ui(di, j) > ui(ci, j) (Dominance of d) (ii) ui(ci, n-1) > ui(di, 0) (Pareto-inferior equilibrium) (iii) j < n-1 ⇒ ui(ci, n-1) > ui(ci, j) (Efficiency of cooperation)

3.5 Alternative games. The Prisoner’s Dilemma represents one very important type of collective action situation. The 2-player version of the Prisoner’s Dilemma is, without competition, the most frequently discussed formalization of a collective action problem, although the n-player generalization has been attracting an

with the bargaining problem and one dealing with the problem of compliance.

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increasing amount of attention. Not seldom, the general problem of collective action is simply identified with the n-player Prisoner’s Dilemma. However, there are many interesting problems, besides the Prisoner’s Dilemma, where individually rational behavior might tend to result in outcomes that are bad for all. Thomas Schelling claims that we should “identify as the generic problem”, not the inefficient equilibrium of prisoner’s dilemma, but all the situations in which equilibria achieved by unconcerted or undisciplined action are inefficient – the situations in which everybody could be better off , or some collective total could be made larger, by concerted or disciplined or organized or regulated or centralized decisions. 62

Defining collective action in this less restrictive way allows for a number of other types of game structures, besides the PD, sharing the property that individually rational behavior may sometimes yield suboptimal outcomes. Some of these other types of games are, as we shall see, of great interest for the study of cooperation and political authority. For example, there is the kind of situation, discussed in the previous section (cf. fig. 3.8.), where each player has a dominant strategy, where the dominant strategy equilibrium is strictly Pareto-inferior to some nonequilibrium strategy profiles, but where it is not obvious which of a player’s alternative strategies should be defined as the cooperative one. Note that, for this to be the case, it is not necessary that universal Cooperation is Pareto-superior to universal Defection. There might be cases where, as Barry says, Too many cooks might spoil the broth to such a degree that everybody agrees that it would be better than that to make and drink no broth at all. 63

An example of this kind of situation is given in fig.3.9. In this case, it would seem quite odd to speak of the strictly dominated pure strategy as Cooperation. Therefore, I use the more neutral labels s and s*, where s is 62 63

Schelling (1978) p 225. Barry (1985) p 156.

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the strictly dominant and s* the strictly dominated pure strategy. As before, we assume that payoffs are the same for all players. Although the dominant strategy equilibrium here is not Pareto-inferior to the dominated pure strategy profile, there is still a benefit to collect for everyone, if they manage to coordinate on a non-equilibrium (mixed) strategy profile. s*

Payoff

s

Number of others who choose s*

fig. 3.9 We could also consider the provision of a collective good, where the size of an individual’s contribution is not pre-determined. Cooperation is not always a matter of “all or nothing”; an individual can often choose whether to make a large or a small contribution. Another type of collective action problem that is not necessarily identical to a pure Prisoner's Dilemma is the one pictured in section 3.1; the tragedy of the commons. Whereas in the Prisoner’s Dilemma each agent always gains by sacrificing the collective interest, in the tragedy of the commons it is sufficient that there is always someone who gains by doing so. In the tragedy of the commons, the most affluent herdsmen might have an interest in avoiding overexploitation by stopping the growth of animal herds at the present level, but there is always someone, who has less, who has a greater interest in increasing her own share of the production than in the size of total production.

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In all these cases, there is an additional problem of coordination besides, and prior to, the compliance problem that is modeled by a standard nplayer Prisoner’s Dilemma. Players must coordinate on some collective choice device to determine which Pareto-optimal mixed strategy profile “cooperation” should target on. They must decide together how the burdens and benefits of their cooperative endeavor should be divided, e.g. who should be allowed to own how many cattle. This might be done, for example, by a bargaining process or by applying some generally accepted principle of fairness. Once this choice problem has been solved, however, it is often convenient to restate the problem of compliance with the agreement made in terms of a pure Prisoner's Dilemma. 64 3.5.1 The Assurance game In the Prisoner's Dilemma, each player has a strictly dominant strategy, i.e. she has an unconditional preference for one of her alternative strategies. However, there are many relevant collective action problems where this condition does not hold, where each player’s preferences are conditional on the behavior of other players. In some cases, players might prefer to cooperate provided that all, or nearly all, others cooperate as well, but prefer not to cooperate if all, or most, others Defect. In other types of cases, each player might prefer to Cooperate if all, or most, others Defect but prefer to Defect if all, or most, others Cooperate. Consider the payoff matrix of the following game (fig. 3.10).

64

David Gauthier’s Morals by Agreement develops an argument for such a case with only two players. What Gauthier does is to split the argument in two parts; one dealing with the bargaining problem and one dealing with the problem of compliance. Gauthier (1984).

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Player c

2 d 3

2

c Player 1

3

0 0

1

d 2

1 fig. 3.10

This is a type of situation that Amartya Sen has labeled “Assurance problem”. 65 Each player prefers the outcome where both cooperate, and each player also prefers to cooperate herself if she can be assured that the other will cooperate as well. Without such assurance, however, each player might prefer not to cooperate. There are two pure strategy Nash-equilibria in this game; mutual Cooperation and mutual Defection. One of these equilibria, mutual Defection, is strictly Pareto-inferior. There is also a mixed strategy (weak) Nash-equilibrium, i.e. the one where both players Cooperate with probability 0,5 and Defect with probability 0,5. The mixed equilibrium yields an expected payoff for each player of 1,5; hence, it is also Paretoinferior to mutual Cooperation. The generalization of the Assurance game to the n-player case is straightforward. Suppose that we have n players who face a binary choice between Cooperation and Defection. Then, the game is an n-player Assurance game, iff (i) all players prefer universal Cooperation to universal Defection, (ii) there is, for each player, i some positive integer ki, such that

65

Sen (1967).

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a) if the number of others who Cooperate is larger than ki, Cooperation is strictly preferred by i b) if the number of others who Cooperate is equal to ki, Cooperation is weakly preferred by i c) if the number of others who Cooperate is smaller than ki, Defection is strictly preferred by i, (iii) if a player Cooperates herself, she prefers that everyone else Cooperates as well. Condition (iii) here is analogous to condition (iii) in Definition 3.3 of the nplayer Prisoner’s Dilemma. It is included in order to avoid the difficulty, discussed above, of determining what outcome cooperation should aim for. Assuming that payoffs are uniform, an n-player Assurance game can be graphically represented as follows (fig. 3.11). Cooperate

Payoff

Defect

0

k

n -1

Number of others who Cooperate

fig. 3.11

The characteristic feature of an n-player Assurance game is that the payoff curves for Cooperation and Defection intersect, with Cooperation yielding a higher payoff than Defection to the right and Defection yielding a higher payoff to the left. Universal Cooperation and universal Defection are both strong Nash-equilibria. There is also a set of weak, mixed equilibria

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corresponding to the intersection point. The mixed equilibria are fragile, however, in the sense that, if anyone deviates from her equilibrium strategy, everyone has reason to follow her example. As soon as someone abandons her equilibrium strategy, the situation immediately tips into the direction of one of the two pure strategy equilibria. The n-player generalization of the Assurance game can be defined formally as follows. Definition 3.4. n-player Assurance game Let G=〈N, (Si), (ui)〉 be a strategic game such that for each i∈N, Si = {c, d}. Then, G is a n-player Assurance game iff, for all i∈N and all j=1,2,..., n-1, (i) ui(ci, n-1) > ui(di, 0) (Inefficiency of Defection) + (ii) For each i∈N, there is some ki∈Z such that j>ki ⇒ ui(ci, j) > ui(di, j) j=ki ⇒ ui(ci, j) ≥ ui(di, j) j ui(ci, j) (Intersecting payoffs) (iii) j < n-1⇒ ui(ci, n-1) > ui(ci, j). (Efficiency of c) This type of situation will play an important part in my analysis of the role of sanctions in collective action; one of the main arguments (see chapter 6) is that if sanctions are added to a situation that would otherwise be a nplayer Prisoner’s Dilemma, the resulting game will often have the structure of an n-player Assurance game. Is the Assurance game a genuine collective action problem? Jon Elster has suggested a strong and a weak definition of collective action. According to the strong definition the problem is identical to the n-player Prisoner’s Dilemma. According to the weak definition, a situation collective action problem iff (i) universal cooperation is strictly Pareto-superior to universal defection, (ii) that cooperation is “individually unstable”, i.e. each individual has an incentive to defect from universal cooperation, and

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(iii)

that cooperation is “individually inaccessible”, i.e. no one has an incentive to cooperate when everyone else defects.66

The Assurance game, as defined above, satisfies Easter’s conditions (i) and (iii) but not (ii). Universal Cooperation is better for all than universal Defection, Cooperation is “individually inaccessible”, but it is not “individually unstable”. Hence, according to Elster’s definition, the Assurance game is not a genuine collective action problem. Taylor presents a definition that appears to be less restrictive than Elster’s. According to Taylor, a collective action problem is a situation where, rational individual action can lead to a strictly Pareto-inferior outcome, that is, an outcome that is strictly less preferred by every individual than at least one other outcome.67

The word “can” here seems to suggest that Taylor considers Elster’s conditions (i) and (iii) to be sufficient. If cooperation is individually inaccessible, then it seems to be possible that a group might find itself locked into a sub-optimal, non-cooperative equilibrium. According to Taylor, however this is not the case. In Taylor’s view, if a game has multiple equilibria (as the Assurance game does) but one of them is strictly preferred to all others by everyone, then the Pareto-preferred one will be the outcome. On this view, rational action in an Assurance game does not lead to a Pareto-inferior outcome, so that this game is not a collective action problem. 68

I believe that matters are somewhat more complicated, however. Suppose that the payoffs in a uniform n-player Assurance game are those represented in fig 3.12.

66

Elster (1985) p 139. Taylor (1995) p 19, bold type added. 68 Ibid. 67

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Defect

Cooperate

Payoff

n-1

Number of others who Cooperate

fig. 3.12 Since the dominance principle does not provide an answer to the question of which strategy to choose, players might in this situation go for some alternative principle of rational choice. One such principle is the maximinprinciple, or the principle of maximizing one’s security level. In the above game, the maximin-principle clearly recommends Defection. Cooperation involves exposing oneself to the risk of ending up in the worst possible outcome. There are other alternative principles as well, such as the principle of minimizing regret, that will also recommend Defection in these cases. So, players who find themselves in such a situation, without ability to communicate, should at least consider it possible that others might choose to play it safe. Hence, in the absence of communication, it is plausible to assume that the Assurance game can lead to a strictly Pareto-inferior outcome. Communication, of course, changes the picture. If players can communicate effectively prior to making their choices, they might relatively easily coordinate on the Pareto-superior equilibrium. Once a reliable coordination agreement has been reached, it is in everyone’s interest to follow that agreement. In this respect, the Assurance game differs sharply from the Prisoner’s Dilemma, where communication offers

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no remedy, since players have no reason to comply with a pre-play agreement. 3.5.2 Chicken In the Assurance game, each player prefers to Cooperate on the condition that sufficiently many others Cooperate as well. Now consider the opposite situation, where each player instead prefers to Cooperate only if sufficiently many others do not Cooperate. Consider the pay off matrix of the following 2×2 game (fig. 3.13).

Player c

2 d 3

4

c Player 1

3

2 2

1

d 4

1 fig. 3.13

In this game, known as Chicken, each player prefers to Cooperate if the other player Defects, but prefers to Defect if the other player Cooperates. Both players prefer mutual Cooperation to mutual Defection. There are two pure strategy Nash-equilibria in this game, one where Player 1 Cooperates and Player 2 Defects, the other where Player 2 Cooperates and Player 1 Defects. There is also a mixed strategy equilibrium where both choose Cooperation or Defection with probability 0,5. Both of the two pure strategy equilibria are Pareto-optimal, but the mixed equilibrium is Paretoinferior to mutual Cooperation. In the mixed equilibrium each player’s expected payoff is 2,5 whereas with mutual Cooperation, each player gets 3.

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Maynard Smith and Price 69 used this game to model animal contests, but it probably serves as a reasonably good model of competition for scarce resources more generally. We might, for example, think of the players as competing firms involved in a price war, armies at war over a territory or children fighting over some toy. Each agent involved in such a conflict can choose between being a “Hawk”, i.e. being prepared to fight until the bitter end, or being a “Dove”, i.e. pretending to be prepared to fight but withdraw before there is risk of serious injury. When a Hawk meets a Dove, Hawk wins without risk of injury; Dove gets nothing but also escapes without injury. When two Doves meet they settle their conflict without serious fighting; perhaps someone simply gives in after having lost some time showing off. When two Hawks meet, they might both suffer injuries that go way beyond the value of the contested resource. There is a natural n-player generalization of this game as well. In Taylor’s formulation we stipulate that “each player prefers to Defect if ‘enough’ others Cooperate, and to Cooperate if ‘too many’ others Defect”70. Suppose that there are n players who face a binary choice between Cooperation and Defection. Then, the game is a n-player Hawk-Dove game iff (i) all players prefer universal Cooperation over universal Defection, and (ii) there is, for each player, i some positive integer ki, such that d) if the number of others who Cooperate is larger than ki, Defection is strictly preferred by i, e) if the number of others who Cooperate is equal to ki, Defection is weakly preferred by i, and f) if the number of others who Cooperate is smaller than ki, Cooperation is strictly preferred by i. (iii) if a player Cooperates herself, she prefers that everyone else Cooperates as well. What kind of situations satisfy these conditions? Consider, for example, provision of a continuous collective good with a strongly diminishing 69 70

Maynard Smith and Price (1973) , Maynard Smith (1982). Taylor (1995), p 42.

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marginal value. Suppose that the cost of making a contribution is constant, regardless of the number of contributions already made. To begin with, each additional contribution makes a great difference, such that the marginal value to an individual of making a contribution outweighs the cost of her contribution. As more and more people contribute, however, the marginal value goes down and eventually falls below the cost of making a contribution. Assuming that payoffs are uniform, the situation can be represented as follows (fig.3.14).

Cooperate

Payoff

Defect

Number of others who Cooperate

fig. 3.14 An interesting real-life example is vaccination against an infectious disease, when the vaccination as such involves some risk of negative sideeffects. When nearly no-one is vaccinated, the risk of being infected is fairly large. By being vaccinated, an individual reduces the risk of being infected to such a degree that it outweighs the risk of side-effects. When nearly everyone is vaccinated, however, the risk of being infected is very small, since the disease is almost eradicated. The risk of negative sideeffects may then outweigh the benefits of being vaccinated.71

71

This example was suggested by Wlodek Rabinowicz.

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Similar to the n-player Assurance game, the characteristic feature of this game is that the payoff curves for Cooperation and Defection intersect. However, in this case, contrary to the n-player Assurance game, Cooperation yields the higher payoff to the left, whereas Defection yields the higher payoff to the right. The only equilibria we find in this game are mixed ones, corresponding to the intersection point, where Cooperation and Defection are equally good for each player. Unlike the n-player Assurance game, however, the set of mixed equilibria is stable in the sense that, if someone deviates from her equilibrium strategy, others are not moved to follow her example but rather to change strategies in a way that compensates for the deviation and restores a mixed equilibrium. Does this game constitute a genuine collective action problem? Taylor claims that it does, since there is no clear answer as to what rationality prescribes. According to Taylor, Any outcome of a Chicken game, including the Pareto-inferior mutual Defection outcome, can be rationalized. Hence, on my account, rational action can lead to a Pareto-inferior outcome, so that on my account it is a collective action problem. 72

Still, Taylor holds, the prospects for Cooperation are a little more promising in Chicken games than they are in the Prisoner’s Dilemma. 73

There is another, interesting type of equilibrium that players might achieve in games like Chicken. Consider a two-player Chicken game with a payoff matrix like fig.3.13. In the mixed equilibrium, as was noted above, both players get 2,5. However, if both players are able to observe some signal prior to playing there is also a possibility of achieving a correlated equilibrium74, where the signal serves as a device to coordinate their expectations about each other’s 72

Taylor (1995) p 19. Op. cit. p 43. 74 Aumann (1974), Maynard Smith and Price (1973) and Maynard Smith (1982) show how asymmetries between players can be used as such signals to establish an ESS in animal contests. 73

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behavior. Such a signal might be the flip of a coin; if both players expect Player 1 to play Hawk if heads comes up, and Player 2 to play Hawk if tails, then both have reason to adapt their behavior to these expectations. It might be a traffic signal; if everyone expects all others to drive at red light and stop at green, then everyone has reason to drive at red and stop at green light. Or it might be some unintended random event in nature, or a some asymmetry of the players, e.g. who came first to the spot, who is larger etc. The important thing is that there is some observable sign that both contestors can react on. In the flip-of- a-coin example, provided that the coin is fair, the correlated equilibrium yields an expected payoff of 3 to each. In some cases, the correlated equilibrium, which allows the agents to take turns in playing Hawk according to a fixed pattern, might yield a higher expected payoff to both than mutual cooperation. To sum up. There are several different types of collective action problems. The Prisoner's Dilemma is interesting because it captures in a pure form an especially malign form of interaction problem, where a collectively devastating outcome seems almost inevitable. The Prisoner's Dilemma structure can sometimes be used as a simplification, to isolate a certain element of the logic of an interaction problem. This may be the case, for example, in situations where we want to isolate the issue of negotiating an agreement from the issue of compliance with the agreement that negotiations result in. However, as we will see in coming chapters, there are other game structures besides the Prisoner's Dilemma that are highly interesting in the analysis of social contract issues. In particular the Assurance game, but also Chicken, are games that should be given more attention in this context. One of the main conclusions of this essay will be that the Assurance game is a key to understanding the nature of a Hobbesian social contract argument.

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CHAPTER 4

THE STATE OF NATURE ______________________________________________________________________

If you occasionally wonder why we need a state at all, you should visit a place like Kosovo that has none. This has advantages, of course. For example, you don’t need to worry about speeding fines. But you can also get robbed or killed at night, and no-one will take any notice. Timothy Garton Ash75

4.1 The war of every man against every man. Thomas Hobbes’s account of the state of nature as a “war of every man against every man” is well known, not to say notorious. Not only would people in such a state have to live in constant fear of violence, and constantly be prepared to exert violence against others themselves, but because of the uncertainty that would prevail, there would be no room for constructive cooperative endeavors; “no place for industry; (…) no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; (…); no arts; no letters; no society” 76 Human life, as one can easily imagine, would be “solitary, poor, nasty, brutish, and short.”77 Other social contract theorists, like John Locke, have described the conditions of a state of nature in far less pessimistic ways; leaving much more room for altruistic motives, moral concerns and spontaneous cooperation. Whereas Hobbes claims that nothing can be unjust in the state of nature, since “the notions of right and wrong, justice and injustice have there no place" 78; Locke, on the other hand, maintains that people in the state of nature are bound by natural law, which “teaches all Mankind, who 75

“Kosovo – landet som icke är”, Dagens Nyheter (DN) 990731. The article was published in Swedish translation. Henrik Berggren at DN kindly supplied me with the English original. 76 Hobbes (1996) Ch 13, p 84. 77 ibid. 78 Op.cit., p 85)

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will but consult it, that being all equal and independent, no one ought to harm another in his Life, Health, Liberty or Possessions”. 79 Still, even the Lockean state of nature is supposed to be haunted by conflict. Irrationality and limited ability to understand and interpret the laws of nature correctly, will cause conflicts and violations of these laws. People will disagree on the interpretation of natural law, because of “selflove” they will be “partial to themselves and their friends” and because of “ill-nature, passion and revenge” they will be prone to punish offenders beyond what the laws of nature admit. When each is judge in his own case, Locke says, “nothing but confusion and disorder will follow.”80 Thus, there is need for government in order to overcome these inconveniences. In Hobbes’s case, the point of the account of the state of nature is not primarily to present a historically accurate theory of human life in a distant pre-social past. In fact, Hobbes admits that, probably, “it was never generally so, over all the world”, although he also believes that “there are many places, where they live so now.”81 Instead, the account of the state of nature serves as a point of comparison; its purpose is to substantiate the claim that the existence of government - even unlimited government with all its drawbacks for individual liberty - is advantageous to all. In order to evaluate the claim, that the existence of government is advantageous to all, we must know whether, in the absence of government, the prospects for mutually beneficial cooperation would actually be as gloomy as Hobbes’s account suggests. Thus, we need an accurate view of the nature of conflict in a Hobbesian state of nature. 4.2 The state of nature as a Prisoner’s Dilemma. Modern writers have often interpreted the basic interaction problem facing individuals in a Hobbesian state of nature as a Prisoner’s Dilemma. I will

79

Locke (1995) Ch 2, p 117. Op.cit. p 121. 81 Hobbes (1996) Ch 13, p 84. 80

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argue here that some, but not all, interaction problems that individuals might face in a state of nature have the structure of a Prisoner’s Dilemma. I believe, however, that there are many important aspects of such a state that are better understood in terms of some other type of game structure; Chicken games, coordination games and, in some cases, zero-sum games (games of pure conflict). Some of these other kinds of games, I think, have been given too little attention in the discussion of social contract issues. Moreover, I believe that a certain strategic diversity is essential for the Hobbesian solution to work. If the Prisoner’s Dilemma structure was pervasive in the state of nature, then it is difficult to see how a Hobbesian argument for the necessity and possibility of political authority could be sustained. As Brian Skyrms points out, the description of the structure of conflict in the state of nature must not be such that it precludes the realization of a social contract. A neo-Hobbesian theory of the social contract should explain why rational agents act in the way hypothesized in the state of nature without making it impossible for them to escape the state of nature. That is, both the “state of nature” and the civilized state under the social contract should be possible, natural, self-sustaining social states.82

The state of nature and the social contract, thus, must both be gametheoretic equilibria. According to Skyrms, the transition from the noncooperative, state-of-nature equilibrium, to the cooperative social-contract equilibrium must be explained without changing the payoffs of the game. Skyrms’s idea, which is only rather briefly sketched in Skyrms (1990), is to describe the basic problem in the state of nature in a way that allows people to move from a Pareto-inferior, but robust, non-cooperative equilibrium (the state of nature) to a Pareto-superior, perhaps less robust, cooperative equilibrium (the social contract) through a chain of belief revisions. Agents

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Skyrms (1990) p 140.

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are thought to play by habit, and continuously revise their beliefs about the other players’ habits over time, as they observe their behavior. 83 My approach is somewhat different. A central claim of this essay is that the problem of establishing sanctions is logically distinct from many other types of interaction problems that agents might face in the absence of such sanctions. Whereas the problem of achieving cooperation in the absence of such sanctions may frequently be a Prisoner's Dilemma, the problem of establishing and sustaining an effective sanction system, i.e. the problem of achieving a social contract, is not. This problem, I shall argue, has the structure of an Assurance game rather than that of a Prisoner’s Dilemma. In the Assurance game, as was seen in the previous chapter, general cooperation and general defection are both equilibria. Hence, it satisfies the demand that these should both be “possible, natural, self-sustaining social states” .The game structure that Skyrms brings in to illustrate his idea is, in fact, the exact intermediary between an Assurance game and a Prisoner's Dilemma. The difference between this game and the Assurance game is that, in the Assurance game, general cooperation is a strict Nash equilibrium. What I shall claim here, thus, is not that the problem of achieving a social contract is itself a Prisoner's Dilemma, but that, as long a group of individuals remains in the non-cooperative, state-of-nature equilibrium of the social contract game, some (but not necessarily all) of their other interactions will have the character of Prisoner's Dilemmas. As we shall see, moving from the non-cooperative to the cooperative equilibrium changes the payoff structure of these other interactions. The challenge, thus, is to explain how this move can itself be made, without invoking what Skyrms calls a deus ex machina. This challenge will be confronted in chapter 6. Here, we shall try to grasp the nature of the kind of problem that the social contract is supposed to be an answer to.

83

Op.cit. pp 140-1.

84

4.3 Resource competition. Not all Prisoner’s Dilemma interpretations of social interaction in a state of nature are equally convincing. In many cases, either the structure of preferences, or the set of alternatives, of the situation that they purport to describe is represented in a strained and implausible way. Consider for example the following account, suggested by Jean Hampton (1988). Suppose that there are two individuals, A and B, who live alongside each other in a state of nature. Each of them “has seized a number of goods, wants more, and hungrily eyes the goods seized by the other”84. Thus, each agent contemplates invading the other agent’s territory in order to lay hands on her goods. If A invades and B does not, then A will be able to seize B’s goods, and B might also loose her liberty. If B invades and A does not, then it is B who will be able to seize A’s goods. If no one invades the other, each player will be able to enjoy the goods she possessed originally. If both invade, the result is total war; each has an equal chance of winning or loosing all. On Hampton’s account, thus, each agent has two alternative strategies; Invade or Not invade. The payoff structure is the one represented in fig. 4.1, where 4 is best and 1 is worst85.

B Invade Not invade 2 1 Invade A

2 Not invade

4 4

1

3 3

Fig 4.1 84

Hampton (1988) p 62. Hampton uses the opposite notation, with 1 denoting the best and 4 the worst alternative. 85

85

It is easily verified that this payoff matrix represents a Prisoner's Dilemma. To invade is the strictly dominant strategy for each player, but mutual invasion is strictly Pareto-inferior to mutual non-invasion. However, I believe that the alternative strategies as well as the suggested preference rankings, in this story, are ill conceived. First, if A invades B’s territory, then the reasonable set of alternative responses for B should be to defend or not defend her goods, rather than to invade or not invade A’s territory. Thus, each player seems to have a choice between different defensive, as well as offensive, strategies. One might choose to not invade, but defend one’s original holdings if the other invades, or to invade and defend, or to invade but not defend or neither to invade nor defend. This problem can be avoided if we omit the assumption of distinct individual territories, and ignore for the moment the issue of who initially acquired what goods. We simply assume, then, that there is some bundle of goods that A and B compete for, regardless of who seized it first. Each agent can then choose to fight or not to fight. Reinterpreted in this way, Hampton’s account suggests that each agent, in such a resource competition, should always prefer to fight, irrespective of what the other agent does. However, this is hardly a plausible assumption. Assume, for simplicity, that both agents are roughly equal in fighting capacity, and hence have a roughly equal chance of being victorious in a battle. Then, if both agents choose to fight until one of them is unable to continue, each has a chance of winning and being able to seize the good, but also suffers a risk of being seriously injured or, ultimately, killed. Now, I believe that few goods in a state of nature would be worth taking such risks for. The situation, described in this way, is very similar to the type of animal contests that are modeled in Maynard Smith and Price (1973) and Maynard Smith (1982) (cf. section 3.5). They suggested that each player in such a contest could choose between playing “Hawk”, i.e. fight to the bitter end, or “Dove”, i.e. pretend to be a Hawk, but run away before any serious

86

fighting takes place. When Hawk meets Hawk they fight until someone is defeated and unable to continue. When Hawk meets Dove, Hawk wins without fight; Dove gets nothing but also escapes without injury. When Dove meet Dove, they settle their conflict without serious fighting; perhaps someone simply gives in after having lost some time showing off. “Invade” in Hampton’s model could then be taken to correspond to the Hawk-strategy in a Hawk-Dove game, whereas “not invade” corresponds to the Dove-strategy. However, Maynard Smith and Price suggests that each contester gains by playing Hawk if the counterpart plays Dove, and by playing Dove if the counterpart plays Hawk, i.e. their game has the preference structure of Chicken game (cf. fig. 3.13). Hampton, on the other hand, suggests that each player gains by playing Hawk, i.e. invade, regardless of what the other player does. In my opinion, Hampton’s version is the less plausible one. In extreme cases, when the contested good is of absolutely vital importance to both parties, it might be rational for both to take fight until the last drop of blood. In most normal cases, however, there are alternative, perhaps lesser or more distant, goods available. In such cases, at least one of the parties should be better off by swerving and going for the alternative. My conclusion, thus, is that many straightforward resource competitions in a state of nature would often have the structure of Chicken, rather than that of Prisoner's Dilemma. 4.4 The cold war. My own proposal for a Prisoner's Dilemma interpretation of the Hobbesian state of nature will take as its starting-point Hobbes’ contention that “the nature of war, consisteth not in actual fighting; but in the known disposition thereto”. 86 The war of every man against every man could be a cold war rather than a hot one, characterized by excessive war-preparations, high level of threat and occasional outbursts of actual violence.

86

Hobbes (1996) Ch 13, p 84.

87

Suppose that a group of individuals, being in a state of nature, frequently find themselves involved in resource competitions of the above kind. Suppose that both agents in such a contest are roughly equal in fighting ability. Then, if both choose to play Hawk, both will suffer a roughly equal, and substantial, risk of being seriously injured, or even killed. However, the fighting ability of an individual is not constant. It is plausible to assume that an individual can improve her fighting ability by physical training and practice in the use of arms, by production and care of arms, by alliance-formation, conquering of strategic positions etc. Likewise, an individual can sometimes worsen the fighting capacity of a potential opponent by wirepulling, pre-emptive strikes and conquering of resources that are strategically valuable to the other. By improving her fighting ability, or by worsening the position of a potential enemy, an individual may improve the odds of her being victorious in a battle. Hobbes speaks of such war preparations as “anticipation” and maintains that there is no way for any man to secure himself, so reasonable, as anticipation; that is, by force, or wiles, to master the persons of all men he can, so long, till he see no other power great enough to endanger him 87

Suppose that asymmetries in fighting ability are visible to the contestors. It is reasonable to believe that such asymmetries would then affect each player’s choice of strategy in a Hawk-Dove game. Intuitively, an agent should be more likely to play Hawk when she judges that she is the superior part than when she judges that she is the inferior one. 88 Consider the following Hawk-Dove game (fig. 4.2).

87 88

(Hobbes (1996) 13:4, p 83. See e.g. Maynard Smith (1982).

88

B Hawk 1

Dove 2

Hawk A

1

4 4

3

Dove 2

3 fig. 4.2

The matrix is to be interpreted in the following way. Two individuals, A and B, compete over some set of goods. There is a lesser good (value 2) and a greater good (value 4). If one of them plays Hawk and the other Dove, the one who plays Hawk will have the greater good, and the other one will have the lesser good. If the both play Dove, the issue is settled by some costless mechanism such that each has an equal chance of having the greater or the lesser good. If both play Hawk; again each has an equal chance of having the greater or the lesser good, but there is also an additional cost involved, in that both suffer a substantial risk of being seriously injured or killed. Suppose that this cost amounts to, on average, – 1 for the victorious part and, on average, –3 for the one who is defeated. Thus, playing Hawk when the other plays Dove yields 4, whereas playing Dove when the other plays Hawk yields 2. Playing Dove when the other plays Dove yields 0,5×2+0,5×4=3, and, finally, playing Hawk when the other plays Hawk yields 0,5 (4-1)+0,5 (2-3)=1. As was seen in chapter 3, none of the players has a dominant strategy in this type of game. If the other player plays Hawk, the best response is to play Dove. If, on the other hand, the other player plays Dove, the best response is to play Hawk. Thus, there are two pure equilibria (Hawk, Dove) and (Dove, Hawk), and one mixed equilibrium where each player plays Hawk or Dove with probability 0,5. In the pure equilibria, one player gets 4 and the other 2, whereas the mixed equilibrium yields 2,5 to both.

89

In this description of resource competitions, we have ignored the issue of who initially held what goods. However, initial holdings might well serve as a “signal” for a correlated equilibrium. Suppose that A plays Hawk whenever she came first to the contested goods, and Dove whenever the opponent came first. B ’s best response to that strategy is to do the same. Thus, mutual respect for initial holdings might constitute a correlated equilibrium, which, in this case (provided that each is equally likely to be the initial “owner”) yields an expected payoff to each of 3. 89 Now, suppose that one of the players, by preparing for war, can tilt the odds of winning a battle to her favor. How does this affect the payoff structure? Suppose that, when A has prepared but B has not, the probability of A being victorious if both play Hawk increases from 0,5 to 0,8. Provided that anticipation is costless, the new payoff matrix is the following (fig.4.3).

B Hawk -0,2

Dove 2

Hawk A

2,2

4 4

3

Dove 2

3 fig. 4.3

Hawk is now A’s dominant strategy. Thus, the new, modified game has one single equilibrium; i.e. A plays Hawk and B plays Dove. Note that this holds even if, prior to A’s preparing, A and B sustained a correlated equilibrium based on e.g. respect for initial holdings. In the new situation, A’s best strategy is to play Hawk regardless of who initially held the goods. By preparing for war, thus, A disables the correlated equilibrium and 89

However, the opposite, i.e. each plays Dove if first, Hawk if last, is also a possible correlated equilibrium.

90

ensures that she will get 4. If A has a choice between preparing and not preparing for war, she will, provided that she is rational, choose to prepare. However, in real life, war-preparations are not costless. To engage in warpreparations consumes time, energy, intellectual capacity and other valuable resources that could have been spent on more productive projects. For example, preparing for war means that an individual can use a smaller part of the resources that she is currently capable of laying hands on and keeping, for private consumption. Part of these resources will have to be spent on war preparations instead of consumption. However, the amount of resources that an agent is able to lay hands on and keep is not very tightly related to the amount of resources that she produces. In a Hobbesian state of nature there is, as Hobbes says, “no propriety, no dominion, no mine and thine distinct; but only that to be every man’s, that he can get; and for so long, as he can keep it”. 90 Thus whatever an individual could produce, for example by agricultural work or improvements of her place of living or hunting-grounds, just adds to a common stock of resources that is free to take by anyone who is strong enough. Hence, if everyone devotes the major part of their time to individual war preparations, it is reasonable to believe that the amount of resources available for satisfying basic desires will be substantially smaller for everyone than if no one does so. There will be no incentive to devote time and resources to productive work above what one can immediately consume or hold on to. These costs, in terms of decreased total production, are dispersed over the entire population There is a further kind of cost involved in anticipation. It is often advantageous to strike first, when a conflict is inevitable, or merely expected. Agents might therefore be tempted to direct pre-emptive strikes against potential opponents, in order to improve their chances of success in future competitions, or even to get rid of a potential competitor. However,

90

Hobbes (1996) Ch 13, p 86.

91

if it is rational to strike first, then each agent must expect her potential enemies to seek to strike first against her. Each agent, then, has an additional motive for pre-emptive attack.. She might choose to attack in order to protect herself from being the victim of pre-emptive attacks from them. Thus, the rationality of anticipation creates a security dilemma, such that whatever an agent does to protect herself, will also increase the threat perceived by all others, and hence motivate further anticipatory moves from their side. The combined effect of everyone’s rational self-protection is increased insecurity for everyone. Let us ignore, for the moment, the negative effects that a player’s preparation activities might have on the payoff matrix of the other player. Suppose that the cost of A’s anticipation in the above example is equal to decreasing A’s payoff with 0,2. The modified payoff matrix, after A’s anticipation is the following (fig. 4.4) B Hawk Dove -0,2 2 Hawk A 2 3,8 4 3 Dove 1,8 2,8 fig. 4.4 Still, A’s dominant strategy is Hawk and so the only Nash equilibrium is the one where A plays Hawk and B plays Dove. Thus, even if anticipation is costly, A can improve her own expected payoff by anticipating. 91 However, suppose that B can also improve her fighting ability, at the same cost as A. By preparing, B restores the equal odds of winning a battle, at the 91

Nothing essential is lost if we assume instead that war-preparations are costly only to the individual herself. The important factor is that the cost of preparing is smaller than what the agent gains by moving from the mixed, or correlated, equilibrium, to the equilibrium where she plays Hawk and the other plays Dove.

92

price of reducing her own payoff with 0,2. When both A and B have prepared, the modified payoff matrix is the following, which, again, is a Chicken game (fig. 4.5). B Hawk 0,8

Dove 1,8

Hawk A

0,8

3,8 3,8

2,8

Dove 1,8

2,8 fig. 4.5

The mixed equilibrium here yields 2,4 to both players, whereas a correlated equilibrium based on the flip of a fair coin yields 2,8 to both. Suppose that the expected payoffs in a game between two equally strong players is identified with either of these equilibrium payoffs. Then, by improving her fighting ability such that equal odds are restored, B increases her expected payoff from 2 to 2,4 or 2,8. Hence, if B is rational she will prepare for war. Now, suppose that two agents A and B are about to play a game like the one in fig. 4.2. Before playing, each has the opportunity of taking measures to improve her fighting ability, at the price of reducing the value of the contested goods. Each can choose between the two strategies Prepare, and Not prepare. By choosing to Prepare, a player will improve the probability of her winning with 0,3, but also reduce her own payoff with 0,2. The payoff matrix of this game of anticipation is the following (fig. 4.6), which is a Prisoner's Dilemma.

93

B Prepare Prepare A

Not prepare

2,8 2,8

Not prepare

2 3,8

3,8 2

3 3

fig. 4.6 Thus we see that, although pure resource competitions will often have the structure of Chicken rather Prisoner’s Dilemma games, pre-play preparations may turn the structure into that of a Prisoner's Dilemma. A similar argument could be formulated for large groups. Suppose that members of a group are frequently involved in resource competitions against each other, and suppose that asymmetries in fighting ability are visible to the players. Suppose, for simplicity, that in any given contest, a player is judged to be Superior if Hawk is her dominant strategy, and Inferior if Hawk is the dominant strategy of the counterpart. Further, players are judged to be Equal if none of the players has a dominant strategy, but each prefers to play Hawk when the other plays Dove, and Dove when the other plays Hawk. Assuming that “Superior to” is a strict semi-ordering92, we have a hierarchy, or pecking-order, of individuals in the group, based on their relative fighting ability, which affects each individual’s expected payoff. Provided that matchings are random and games symmetric, if i is superior

92

Let Q denote “Superior to”. That Q is a strict semi-ordering means that (1) if aQb, then it is not the case that bQa, (2) if aQbQc, then either aQd or dQc, and (3) if aQb and cQd, then either aQd or cQb. See Danielsson (1996).

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to j, then i will, on average, be able to seize and keep more resources than j.93 If everyone else abstains from war-preparations, an individual agent might improve her position in the hierarchy by preparing, and hence successfully play Hawk more often. Doing so involves both a costs and benefits. The benefit is that the agent will be able to increase the relative share of resources that she is able to hold. The costs, as in the previous example, consist in decreased productivity, increased insecurity, and in the fact that the individual must spend some of her current holdings on preparing instead of consumption. Part of these costs must be borne by the individual herself, whereas other parts are borne by the entire group (i.e. decreased total production and increased insecurity). As long as the individual agent’s share of that cost is smaller than the benefit that she derives by improving her position, it is rational for each one to prepare for war. Thus, the mere possibility to improve one’s prospects in resource competitions by pre-play preparations might put people into a costly arms race, at the end of which everyone will be worse off than they were before. This account includes two of the “principal causes of quarrel” that Hobbes finds in a state of nature, namely “competition”, which makes agents fight “for gain”, and “diffidence” which makes agents fight “for safety”. Hobbes adds a third one, “glory”, which makes agents fight “for reputation”94. It might be tempting to try to describe the security dilemma as such, i.e. the second cause of quarrel, as a Prisoner's Dilemma. It would indeed be striking if individually rational self-protection alone could be shown to result in a Hobbesian “war of all against all”. However, I think it is necessary that there is a real source of conflict behind the security dilemma for the situation to constitute a genuine Prisoner's Dilemma. Or, more precisely, it is necessary that players believe that there is a real source of conflict.

93 94

Note that this is not a comparison of utility but of resources. Op.cit. 13: 6-7, p83.

95

If all agents were motivated exclusively by defensive concerns, and this was common knowledge among them, there would not be a Prisoner's Dilemma, but rather an Assurance game. If everyone could be assured that everyone else would abstain from war-preparations, then no-one would have reason to prepare herself. The Prisoner's Dilemma occurs only when there is an incentive to engage in war-preparations even if no one else does, or, rather, when all players believe that some, or all, of the others would have an incentive to engage in war-preparations even if no one else did. Gregory Kavka speaks of two types of agents that one might come across in a Hobbesian state of nature. “Moderates” anticipate only for purposes of self-defense. If they could be assured that they would not be subject to preemptive attacks from others, they would happily abstain from anticipation. But as long as the risk of being attacked is sufficiently large, moderates play it safe and anticipate. “Dominators” on the other hand “desire conquest, dominion, and power over others for its own sake”95, and hence prefer to anticipate regardless of what others do. In a population of “moderates” only, each whom knows that each of the others is “moderate”, each agent will have reason to anticipate if and only if she believes that the others will anticipate. But each knows also that each of the others, “being similarly situated”, will reason in precisely the same way. Thus, Kavka says, “[w]e have here a reflexive circle and an indeterminate outcome.”96 I would say that the preference structure, as described by Kavka, is an Assurance game. Each agent, provided that her preferences are “moderate”, will prefer to abstain from anticipation if and only if she can be assured that all, or most, others abstain from anticipation as well. If a sufficiently large number of others anticipate, each agent prefers to anticipate herself. The outcome is indeterminate in the sense that the game has two stable equilibria; that everyone anticipate and that no-one anticipate.

95 96

Kavka (1986) p 97. Op. cit. p 103.

96

Once a “substantial minority of dominators” is mixed into the population, however, the situation becomes radically different. For some number of expected attacks in the future, anticipation will appear as a more attractive strategy than lying low. Kavka suggests that the anticipation threshold, i.e. the level of expected attacks at which anticipation and lying low are equally attractive to an individual, may vary between different individuals. When the expected number of attacks by “dominators” grows beyond the anticipation threshold of a “moderate” agent, she will anticipate herself.97Thereby the number of expected attacks to the rest of the population increases slightly, perhaps beyond the anticipation threshold of another moderate. The expected violence will escalate as more and more moderates change from lying low to anticipation. “For most plausible distributions of anticipation thresholds,” Kavka argues, this escalation process will continue until everyone’s threshold has been passed.(…)At that point we have an active war of all against all, or nearly all against all.98

Kavka’s argument shows that it is not necessary to assume that everyone has an unconditional incentive to engage in war-preparations (anticipation). It is sufficient that some proportion of the players has such an incentive, or that sufficiently many believe that some proportion of the players has such an incentive, to trigger an escalating arms race, resulting in a Hobbesian war of all against all. This completes my interpretation of the nature of conflict in a Hobbesian state of nature. My conclusion is that there are important aspects of interaction in such a state that could plausibly be described in terms of the Prisoner's Dilemma.

97

In fact, it might be enough that some moderate agent mistakenly believes that the number of dominators is so large that her anticipation threshold is reached. 98 Op.cit. p 106.

97

4.5 A partial state of nature. There are less ambitious ways of perceiving the state of nature, however. In the Hobbesian story we are asked to consider a state where there is no political authority at all. The principal problem in such a situation is, of course, to provide basic personal security. Until this is achieved, few other projects might seem worthwhile. However, the state does more than to provide basic security against internal and external enemies. Government intervention is often seen as a remedy to a number of collective action problems apart from the basic problem of achieving peace and security. In Hume’s words bridges are built; harbours open’d; ramparts rais’d; canals form’d; fleets equip’d; and armies disciplin’d; every where, by the care of government, which, tho’ compos’d of men subject to all human infirmities, becomes, by one of the finest and most subtle inventions imaginable, a composition, that is, in some measure, exempted from all these infirmities.99

We might therefore want to consider a partial state of nature, a situation where political government exists and provides basic security of life and limb, but where certain domains of social interaction are, for some reason, excluded from government intervention. Thus, for example, fishermen at open sea might be held to be in a partial state of nature to the extent that internationally enforced fishing regulations are lacking. Similarly, if the government of a country decides not to interfere at all in certain types of interaction (e.g. a libertarian night-watch government, which decides not to interfere in the provision of other collective goods than defense against external enemies and protection of personal security and private property), then the population of that country may be in a partial state of nature as regarding these particular kinds of interaction, whereas they are not in a state of nature regarding other kinds of interaction. However, that certain types of interaction would, in the absence of government, constitute Prisoner's Dilemmas does not suffice to substantiate the Hobbesian claim that government is necessary to achieve cooperation. 99

Hume (1978), p 539.

98

We must also explore the possibilities for spontaneous cooperation to emerge in such situations. As we will see in the following chapter, the prospects for spontaneous cooperation to emerge in a state of nature might be a good deal better than Hobbes would want to acknowledge. 4.6 Egoism. To what extent does this account of conflict in the state of nature rest on the assumption that individuals are egoists? The issue of egoism was briefly touched upon in chapter 2. The game theoretical analysis as such rests on the assumption that players act so as to maximize satisfaction of their own interests, whatever these interests are. This need not imply that agents are egoists in the sense that they do not take an interest in the well-being of others. Players may have all kinds of other-regarding motives, but the game theorist assumes that all such motives are already taken into account in the payoff matrix of the game. That is, even an altruist is supposed to act so as to maximize her own payoff, since her altruistic motives are thought to be already reflected in the payoff matrix of the game. This position could be called “Tautological Egoism”100. However, the preceding account of conflict in the state of nature seems to require that agents are egoistic in a deeper sense. It seems to presuppose that their motives are egoistic rather than altruistic. That is, it seems to rest on the view that individuals are “Psychological Egoists”. 101 In principle, it is possible to derive a Prisoner's Dilemma game from nonegoistic motives. However, it seems unlikely that players who are motivated primarily by altruistic concerns would end up in the war of all against all that is supposed to characterize a Hobbesian state of nature.

100 101

Kavka (1986) p 35. Ibid.

99

Now, if the account of conflict in the state of nature would rest on the assumption that all agents are motivated only by concern for their own well-being, then an obvious objection would be that it is based on an implausible theory of human nature. After all, most of us have some concern for others, at least for friends and family members. However, Gregory Kavka shows that the Hobbesian account of conflict in the state of nature can be derived from a weaker assumption than pure Psychological Egoism, namely what Kavka labels “Predominant Egoism”102. Predominant Egoism is characterized as follows: 1. For most people in most situations, the “altruistic gain/personal loss” ration needed to reliably motivate self-sacrificing action is large.* 2. The number of people for whom altruism and other non-self-interested motives normally override self-interested motives is small. 3. The number of situations, for the average person, in which non-selfinterested motives override personal interest is small. 4. The scope of altruistic motives that are strong enough to normally override self-interest is, for most people, small, that is, confined to concern for family, close friends, close associates, or particular groups or public projects to which the individual is devoted.103

If the assumption of pure Psychological Pgoism is not a very plausible view of human nature, the assumption of Predominant Egoism is. While most people have some concern for others, and might be prepared to make some sacrifice in order to benefit others, few of us are prepared to sacrifice very much for the benefit of a complete stranger. In particular, few of us are prepared to sacrifice vital interests for others than our closest family Now, it seems clear that a Predominant Egoist in the state of nature would not abstain from anticipation. Doing so would mean to put her most vital interests – of life, liberty and means of subsistance - at risk, in order to benefit everyone. Thus, the objection that people are not purely egoistic does not overthrow the conclusion that a Hobbesian state of nature would bea state of war.

102 103

Op.cit. pp 64 ff. Op. cit. p 65. (* Kavka quotes Margolis, [1984])

100

CHAPTER 5

COOPERATION IN THE PRISONER'S DILEMMA. _____________________________________________________________________________________

The literature on the prospects for cooperation in Prisoner's Dilemma situations is voluminous. Most of this literature is devoted to the problem of cooperation in situations involving only two players. For this kind of interaction, results have been fairly positive; a large number of studies show that cooperative relations can be sustained in a two-player Prisoner's Dilemma, without interference by a coercive power, provided that the game is iterated an indefinite number of times between the same two players.104 Similar results have been obtained in two player interactions that occur repeatedly, not between the same two players, but in larger network of agents with overlapping relations.105 A different type of argument claims that cooperation can be sustained by rational players even in single-shot Prisoner's Dilemma games, on the assumption that rational agents can commit themselves to cooperation, and refuse to cooperate with agents who are not committed cooperators. 106 For the n-player version of the Prisoner's Dilemma, however, there is much less work done, and the results that have been obtained indicate that the prospects for successful cooperation are not as bright as in the two-player case. Michael Taylor has formulated an argument suggesting that rational agents might be able to sustain cooperation in n-player Prisoner's Dilemma games, on more or less the same grounds that they might sustain cooperation in 2 player games 107. Per Molander, however, applying an evolutionary model, argues that, in the absence of sanctions, free riding is

104

See e.g. Luce and Raiffa (1989), Shubik (1970), Axelrod (1984), Fudenberg and Maskin (1986), Taylor (1995) 105 Kandori (1992), Hardin (1993). 106 Gauthier (1986), McClennen (1985) (1990), Danielsson (1992). 107 Taylor (1995).

101

likely to prevail in Prisoner's Dilemma games with more than two players 108. In this chapter I will first discuss the two approaches to cooperation in the two-player Prisoner's Dilemma, and then consider the two different treatments of the n-player case.

5.1 The shadow of the future. According to the so called “Folk Theorems” for repeated games, stable cooperative relations can be sustained in a Prisoner’s Dilemma if the game is repeated an indefinite number of times between the same two players. Under these conditions, and provided that players are equipped with memory, each player can make future cooperation conditional on the previous behavior by the opponent. Such a strategy is based on a principle of reciprocity; it rewards current cooperation by cooperating in the future, and it punishes current defection by defecting in the future. If future interactions are sufficiently important, i.e. the discount factor is sufficiently large, mutual cooperation can be a stable equilibrium. There are many possible such strategies. One is the well-known Tit for tatstrategy, which was showed to be effective in a series of computer tournaments conducted by Robert Axelrod.109 Tit for tat begins by cooperating on the first move. On every subsequent move it does exactly what the opponent did on the previous move. If the opponent cooperates on the first move, then Tit for tat cooperates on the second move, if the opponent defects on the first move, Tit for tat defects on the second, and so forth. Thus, it immediately retaliates defection, but it also restores cooperation immediately when the opponent returns to cooperation.

108

Molander (1992). Axelrod (1984), Tit for tat was supplied to the tournament by the psychologist Anatol Rapoport. 109

102

Another reciprocal strategy is the less forgiving Tit for two tats, which defects twice for every defection by the opponent. A completely unforgiving strategy is the one that Axelrod labels Friedman110, sometimes also called Grim trigger111. Grim trigger begins by cooperating, but defects forever if the opponent defects even once. There are numerous intermediate possibilities. It is, of course, possible to play an unconditional strategy as well. All c cooperates on every move, regardless of what the other player did on previous moves, whereas All d defects on every move, regardless of what the opponent did. Suppose that two players, 1 and 2 play a series of Prisoner's Dilemma games, each of which have the payoff structure presented in fig. 3.2. The discount factor, wi, is a measure of the relative weight that each player i∈N attaches to payoffs in future moves of the game as compared to the payoffs of the current move. Since future payoffs should reasonably count for less than current payoffs, we assume that 0 1/3, continued cooperation is strictly better than All d. Similar reasoning applies when some of the players contemplate defecting only once. By defecting once, and then restoring cooperation, she will get 4+0+3w2+3w3+…=4+3w2/(1-w) Again, this is larger than 3/(1-w) only if w0]|n)=0. It is not obvious that (2) follows from (1). Further, there are strong reasons to believe that (2) will very rarely be satisfied. Consider what the condition “if there is an nth play” amounts to. If p1(n+1)=0, then, in most cases, it will also be, at least, extremely unlikely that there is an nth play. In the example above, the reason for assuming that p1(1000)=0 is that we are certain that people cannot live for 1000 years. But we are also certain that people cannot live for 999 years. Thus, if it were to turn out that people actually can live for 999 years, which must be the case if there is a 999th play, then our initial beliefs about possible lengths of life would have been refuted. It is reasonable to assume that if an agent were ever to experience her 999th birthday, she would then revise her beliefs, such that she would no longer hold that the probability of living for a 1000th year is zero.115 To avoid this argument, we must assume not only that there is some n such that p1(n+1)=0, but also that the occurrence of an nth round is compatible with the initial beliefs of both players. That is, we must assume that 115

This argument is developed in a (still unpublished ) paper by Rabinowicz and Jiborn “Reconsidering the Foole’s Rejoinder: Backward Induction in Indefinitely Repeated Prisoner’s Dilemmas”.

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(3) there is some n such that p1(n+1)=0, and p1(n)>0. If we assume that players change their probabilities in accordance with the conditionalization model, i.e. new probability assignments are formed from the old ones, by conditionalizing on new information, then (2) can be derived from (3) – but not from (1). Carroll is more precise than Kavka about this condition. Carroll’s argument is based on the notion of a p-function, p(t), where p(t) is interpreted as the probability that a certain game, gt, that takes place at time interval t, will be the very last game.116 Carroll is interested in games with terminating pfunctions, which is defined as follows p is a terminating p-function if and only if (i)

? p(t)=1, and

(ii)

there exists a natural number λ such that (a) p(λ)>0, and (b) for all natural numbers n>λ, p(n)=0.117

Given the above definition of a terminating p-function, Carroll proves the following theorem. In iterated Prisoner’s Dilemma games with terminating p-functions, there are only uncooperative equilibria. 118

However, the assumption that there is some specified time interval t such that a player’s (subjective) probability for an iterated game to extend over t is positive, whereas her probability for the game to extend over t plus one (arbitrarily small) unit (say a second) is zero, is less plausible than merely the assumption that there is some upper bound on the number of future 116

Note that Carroll’s definition of a p-function differs from the probability functions defined above. 117 Carroll, (1987), p 249. 118 Op. cit., p 252.

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interactions. Moreover, it is highly unlikely that that some such specified time interval should be common knowledge among the players. Thus, although formally valid, the argument does not extend to realistic cases. However, I believe that a more realistic version of Kavka’s argument can be formulated without relying on the assumption of a definite upper bound. We do not have to assume that there is some specified t such that probability for a game at t is positive, whereas the probability for one more game after t is zero. All that is needed is an assumption that the discount factor decreases continuously over time, and that, at some predictable point of time, the shadow of the future becomes too weak to motivate further cooperation. Suppose that two players are involved a series of Prisoner's Dilemma games. However, unlike in the example in section 5.1, we do not assume that the discount factor is constant, but it decreases continually over time. As long as both are young, the probability for one more game is high, but as they grow old, the probability for further games becomes smaller. The rate at which the discount factor decreases is predictable if players take statistical information on other human lives into account . Hence, both may know that, at some predictable point in the future, further cooperation will not pay, since the probability that they shall both live to play one more game is so small. From that point, backward induction enters and prescribes defection at every move. In this case, there is no reason to expect that players will revise their initial beliefs, since nothing is supposed to occur that is not compatible with these beliefs. Still, the argument rests on rather heavy assumptions of rationality and common knowledge. It is rarely the case that people are perfectly rational and absolutely clear about the probabilities for events in a distant future, and even less that these factors are common knowledge among them. If there is some measure of uncertainty, either about the exact structure of the game, or about the rationality or knowledge of the other player, rational

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players might be able to achieve cooperation even in a finitely iterated Prisoner's Dilemma. 119 In fact, an uncertainty at any level of the belief hierarchy can lead to predominantly cooperative behavior in the finitely repeated prisoner’s dilemma game, provided that the number of plays is large enough.120

Hence, Kavka’s argument is interesting because it highlights the importance of lack of common knowledge; it points out that the possibility of cooperation might, to some extent, depend on some measure of uncertainty, either about the precise structure of the game, or about the rationality and knowledge of the other player. 5.3 The evolution of cooperation. Section 5.1 ended with the observation that there are many alternative equilibria in an indefinitely iterated two-player Prisoner's Dilemma. However, the so-called “Folk Theorem” has often been interpreted as a strong support for the cooperative outcome. How can we motivate the assumption that people would actually coordinate on one particular, i.e. conditionally cooperative, equilibrium? Classical game theoretical analysis, with its focus on static equilibria, is not very well suited to answer this issue. But there are a number of evolutionary models that try to capture the dynamics by which a population might reach a cooperative equilibrium. The most well known of these is, without doubt, Axelrod’s The Evolution of Cooperation. Axelrod conducted a round-robin tournament between a number of strategies for playing the indefinitely iterated 2-player Prisoner's Dilemma. The participating strategies were submitted by a number of wellmerited game theorists, as well as computer experts and others. The winning strategy was Tit for tat. Even if Tit for tat was not the best strategy against every opponent, it turned out to be “robust” in a varied 119 120

Kavka (1983), Fudenberg and Maskin (1986), Skyrms (1998). Skyrms (1998), p 18.

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environment. Axelrod explains its success by it being (1) nice, i.e. it cooperates as long as the opponent does, without trying to exploit its opponent. In fact, all high-scoring strategies were nice in this sense. It is also (2) retaliatory, i.e. it immediately punishes any defection from the opponent, (3) forgiving, i.e. it reestablishes cooperation immediately when the opponent improves, and (4) simple to understand. Thus opponents are able to learn quickly that defection does not pay. Axelrod also tries to explain how Tit for tat can establish itself in a population, and, once established, be sustained. Suppose that the initial population state is entirely dominated by All d. Thus, everyone defects all the time. A single Tit for tat –strategist cannot do better than the population average, since it will find no one to cooperate with. However, since it only cooperates once, and then defects just as everyone else, Tit for tat does almost as well as the population average. A very small proportion of mutant Tit for tat in a population of All d can therefore survive, earning nearly the same payoff as everyone else. A somewhat larger proportion of mutants, however, might begin earning more, on average, than the population average. Since the mutants defect whenever they are paired with All d, they earn nearly the same as everyone else in most of their meetings. When two mutants are paired with each other, however, they cooperate and, hence, earn above average. The colony of mutants will start growing, and eventually take over the entire population. Thus, Tit for tat manages to invade and establish itself from an initial population state dominated by unconditional defection. Axelrod holds that Tit for tat is also collectively stable, a concept which closely resembles the concept of evolutionary stability. However, it can easily be shown that it is not an ESS. Suppose that everyone in a population has adopted Tit for tat. Non-nice mutants then do strictly worse than the population average, since the incumbent strategy punishes any defection immediately. Tit for tat is, thus, immune against invasion by non-nice strategies. However, since everyone cooperates in this state, there is no operational difference between Tit for tat and other nice strategies. Thus a small

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proportion of mutant All c can survive and reproduce. Random drift might cause the population to change into a state with a fairly large proportion of All c. At that stage, mutant All d can invade. Tit for tat thus is neutrally stable, i.e. it does at least as well as any other strategy when it dominates in the population, but it is not evolutionarily stable, since there are strategies that do equally well. However, the movement will be cyclical. Unconditional cooperators entering by random drift can upset the conditionally cooperative state. But once the unconditional defectors have entered, unconditional cooperators are exploited and do much worse than either All d or conditional cooperation. Thus, sooner or later, All c will be wiped out, and the dynamic process will move the population back to a state of general, conditional cooperation. Tit for tat has received very much attention in the literature since the publication of Axelrod’s study. Its simplicity is intuitively attractive and facilitates understanding of the underlying logic. However, it is not the only conditionally cooperative strategy that might succeed in establishing general cooperation, and it is not always the best one either. A number of studies have been made where different parameters of the problem are varied. For example, one might want to consider situations with some amount of “noise”, i.e. disturbances such that players might make mistakes either in their choice of action (e.g. Tit for tat-strategists sometimes defect by mistake) or in the interpretation of the other player’s action (e.g. Tit for tat-strategists sometimes punish cooperation because, mistakenly, they believe that the opponent defected.) Results from such complex environments indicate that more complex and sophisticated strategies than Tit for tat are sometimes more effective. 121 Although there are some question marks that remain to be straightened out, the overall picture from these studies is that mutual conditional cooperation 121

See e.g. Axelrod (1997), Lindgren (1995)

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is often a viable and likely outcome in indefinitely repeated 2-player Prisoner's Dilemma games. 5.4 Network cooperation. A natural extension of the previous kind of argument is to consider 2player games that are repeated between members of a community who change partners over time. That is, we do not assume that the same two players interact repeatedly, but the game is repeated between different individuals who belong to a network of overlapping relations. Michihiro Kandori shows that a group involved in this type of game can sustain cooperative relations by way of community enforcement, such that defection against one member of the community triggers retaliation by another member. 122 Provided that certain information processing is allowed within the group, general cooperation constitutes an equilibrium. Kandori supposes that each agent carries a “label”, conveying information about her previous behavior, and which her opponents can observe before playing. After each game, each player’s label is updated with information about her actions in that game. A label, thus, is a formalized version of a person’s reputation. In equilibrium, thus, each player has an incentive to cooperate in a current game (provided that her opponent is “innocent”), because current behavior will affect her reputation, and her reputation will affect her payoff in future games. 5.5 Constrained maximization. A completely different way of reasoning is suggested by e.g. David Gauthier and Edward McClennen. According to this view, mutual cooperation can be achieved among rational agents even in a single-shot Prisoner's Dilemma, provided that the agents are resolute, and know each other to be resolute. 122

Kandori (1992).

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The idea is, roughly, as follows. In a Prisoner's Dilemma both players prefer mutual cooperation to mutual defection. The problem is that, regardless of which action the other player takes, each of them has an incentive to defect. However, if each of them could commit herself to cooperation prior to playing, and then act on that commitment without yielding to the temptation to defect, they would both be better off. I will here consider Gauthier’s version of this argument. The solution to the Prisoner's Dilemma, according to this view, is to shift attention from the question whether it is rational to cooperate in a particular case, to the question whether, and under what conditions, it is rational to be disposed to cooperate. 123 Disposing oneself to cooperation without conditions is not a solution. If i is disposed to unconditional cooperation, then j’s best response is not to dispose herself to cooperation. Each agent might therefore consider the possibility of disposing oneself to cooperate on the condition that the opponent is similarly disposed. Gauthier defines a straightforward maximizer as “a person who seeks to maximize his utility given the strategies of those with whom he interacts”. A constrained maximizer, on the other hand, “has a conditional disposition to base her actions on a joint strategy, without considering whether some individual strategy would yield her a greater expected utility.”124 The constraints that a constrained maximizer observes have a certain, Kantian touch. She is “ready to cooperate in ways that, if followed by all, would yield outcomes that she would find beneficial and not unfair”. 125 In the Prisoner's Dilemma, a constrained maximizer is prepared to cooperate under certain conditions, whereas a straightforward maximizer always acts on her dominant strategy, i.e. defects. However, whether the constrained maximizer actually cooperates in a particular case, depends on what she believes that her opponent will do. If 123

Gauthier (1986) p 162. Op. cit., p 167. 125 Ibid. 124

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she believes her opponent to be a straightforward maximizer, the constrained maximizer refuses to cooperate. If, on the other hand, she believes that her opponent is a constrained maximizer, she will cooperate. Under what circumstances would constrained maximization (CM) be rationally preferred to straightforward maximization (SM)? Obviously, there must be some mechanism for detecting the strategy of one’s opponent. Gauthier suggests that agents should be considered to be translucent, such that their disposition can be detected by others with some probability, larger than 0,5 (opaqueness) but smaller than 1 (transparency). Gauthier considers a game with the following payoff structure (fig.5.1), where 0 u(d, 0). Groups of n players are randomly drawn from a large but finite population and play a series of games. At any move, the behavior of each player is determined by (1) the frequency of cooperation on the previous move and (2) the frequency of cooperation required by her current strategy. Some players will be prepared to cooperate when only a few others do, other players will be more intolerant and cooperate only if all or nearly all others do. Still others may not be prepared to cooperate at all. Each group continues to play until the game converges on a short term equilibrium where the frequency of cooperation satisfies the requirements of every

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player’s strategy, and no others would be prepared to cooperate given that frequency. When the group has reached its short-term equilibrium each member collects his steady state payoff. The process is then repeated with new random matching. The payoffs used in this model are not, thus, the accumulated payoffs from every move, but the steady state payoffs of the short-term equilibrium. The motivation for this assumption is that, if every group plays a sufficiently long series of games, payoff differences from the initial process of adaptation will be negligible. This assumption is actually favorable to cooperation, since it allows conditional cooperators to start by cooperating, without being punished. So there is a short-term dynamic process by which each player adapts her behavior to the level of cooperation in her current group. Each player retains her strategy throughout this short-term process. In the long run, however, there is another dynamic process by which the distribution of strategies in the population changes. In the light of the payoffs reached by various strategies, some players may want to switch to other strategies. This is the source of change at the evolutionary level of the game.139

The population state is thought to evolve in accordance with the replicator dynamics140. That is, the growth rate of the population share using a certain strategy sj is equal to the difference between the current average payoff to sj and the current population average. Consequently, the population state continues to change until it reaches an equilibrium where all strategies that are left earn the same average payoff. Given these preliminaries, Molander proves the following theorem. If m < n-1, 1. there is an asymptotically stable equilibrium consisting of a mix of Sm and Sn, which corresponds to an ESS; 139 140

Molander (1992), p 761. Taylor and Jonker (1978), see also chapter 2 in this work..

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2. there is a purely cooperative equilibrium set, not corresponding to ESSs; 3. there is no other ESS. If m = n - 1, the second condition listed holds, and there is no ESS.141

Thus, pure cooperation is an equilibrium, just as in the static analysis considered before. However, it is an unstable equilibrium. A population of intolerant conditional cooperators like sn-1 may be able to withstand invasion by sn since they are, as Molander says, “quick to retaliate“. But, since all cooperative strategies, in a state of pure cooperation, behave identically, there is no selection pressure against more tolerant cooperative strategies. Such tolerant cooperators may therefore move into the population by random drift, a process that will sooner or later make the cooperative population vulnerable to invasion by sn. The assumption that an equilibrium can be upset by drift implies that there is some mechanism of random change in the absence of selection pressure. Thus, there must be some mutation process that allows new strategies, or strategies that have once died out, to enter the population. Suppose that the current population state is dominated by sn-1. Suppose further that, in every new generation, there is some positive probability for s0-offspring to occur. Since nearly everyone cooperates nearly all the time, selection does not favor sn-1 over s0. The population share playing s0 will then fluctuate randomly over time. It may disappear entirely. Over a sufficiently long period of time it may take any value between 0 and 1. However, once the proportion of s0 becomes sufficiently large, a mutant sn will do strictly better than either of the cooperative strategies. Hence sn eventually invades, by exploiting the unconditional, or very tolerant, cooperators. This is the source of fragility of cooperation. Provocable cooperators can defend cooperation against invasion by unconditional defectors, but they cannot defend it against the naïve goodness of the unconditional cooperators.

141

Molander (1992), p 763.

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However, pure defection is not a stable equilibrium either. The reason is similar. A single conditional cooperator, in a population state of pure sn , earns the same steady state payoff as sn, since it refuses to cooperate unless there are others who cooperate as well. A small group of conditional cooperators may do strictly better than sn , since they refuse to cooperate when matched against unconditional defectors, but are sometimes matched against each other, and then succeed in establishing mutually beneficial cooperation. The only stable equilibrium is a mix of sn and sm, a mix which is “the most liberal one in the sense that it permits the maximal frequency of defection subject to the condition that the resulting degree of cooperation should be superior to universal non-cooperation”142This equilibrium is proved to be an ESS. Thus, in a way, one of Taylor’s results is confirmed; a small group of cooperators can maintain cooperation even if the majority defects. However, this yields little comfort, since each strategy in the mixed equilibrium earns an average payoff that is barely above the payoff for universal defection. In a sense it is even wrong to say that free riders exploit the cooperators, since all strategies in the equilibrium earn the same poor payoff. Everyone is worse off, even the free riders themselves. Further, the instability of the purely non-cooperative state seems to rest rather heavily on the assumption of steady state payoffs. With accumulated payoffs, the picture is even gloomier. Clas Pihlström has studied a similar game played in somewhat shorter series and with accumulated payoffs. The result shows that sn , in this setting, is an ESS.143 There are some restrictions in Molander’s approach that should be noted. First, and most seriously, the strategy set is not exhaustive. For example, there is no room in the model for strategies like sj,k , which cooperate as long as at least j others cooperate, otherwise defect, but then return to cooperation after l defections. Before the issue of cooperation in iterated n142 143

Molander (1992), p 763. Pihlström (1996).

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player Prisoner's Dilemma games is definitely settled, results must be tested against a wider variation of parameters, allowing for larger strategy sets, for noise of different kinds and so on. N-player games are still rather unknown territory, at least in comparison with the 2-player versions. Another restriction in Molander’s approach is that ”only the general frequency of cooperation is observable”. 144 The general applicability of this assumption is questionable, however145, and I will suggest, in the next chapter, that it is relaxed. Moreover, the condition seems to be inconsistent with a selection mechanism based on imitation. Molander does not explicitly say that strategies are transmitted by imitation. That players “want to switch” to other strategies, however, indicates that the evolutionary process is not driven by genetic inheritance, but by some other process whereby players choose strategies that they believe will be beneficial. It might, of course, be a pure process of “learning by doing”. However, in this kind of highly complex game, with a large number of possible strategies, I believe that Maynard Smith’s observation is more than appropriate: “One lifetime would not be long enough for such an inefficient learning process”146 If the process is driven by imitation, though, it must be possible to observe the behavior of at least some other players. How could successful agents be imitated, if all that can be observed is the general level of cooperation?

144

Molander (1992), p 768. Note that, for the restriction to be inapplicable, it is not necessary that each player has a complete record of each of the other players´ behavior. If it is possible, perhaps only with great effort, for someone to determine with some degree of certainty (>0,5) for any specified individual whether she did cooperate or not, the condition does not apply. 146 Op. cit. p 171. 145

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CHAPTER 6

SANCTIONS AND COOPERATION ______________________________________________________________________

No state could possibly compel people to obey all these rules at gun point; there would not be enough soldiers and policemen to hold the guns (a sort of Orwellian vision of society), they would have to sleep sooner or later, and then anarchy might break out. Anthony D’Amato 147

In the previous chapter we considered the prospects for cooperation to emerge in the absence of government, taking the Prisoner’s Dilemma to be representative of important aspects of social interaction under such conditions. It was seen that, in the case of indefinitely iterated two-player Prisoner's Dilemma, spontaneous cooperation is viable, provided that the value of future interactions is sufficiently large. In large-number interactions, however, the prospects for non-coerced cooperation were seen to be substantially worse. From Per Molander’s evolutionary analysis of an iterated n-player Prisoner’s Dilemma it was seen that, in the absence of coercion, a substantial proportion of noncooperative behavior is likely to prevail when the number of players is large. These results; Molander holds, seem to support a Hobbesian conclusion concerning the instability of cooperative regimes in the absence of coercion. 148

The issue to be considered here is whether, and how, the introduction of sanctions offers a solution to this dilemma. We shall consider a simple model of a limited sanction system, and see how the existence of sanctions affects the payoff structure of an n-player interaction problem.

147

Anthony D’Amato, “Is International Law Really ‘Law’?” Northwestern Law Review, vol. 79 (1984-85), pp 1293-1314, quoted from Hardin (1995) p 29. 148 Molander (1992) p 768.

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6.1 The power of coordination. According to a widely held view, law obedience, or compliance with a system of social norms, cannot be based on threat of sanctions alone, because no sanction system, however strong, would be capable of exercising threat against every individual all the time. In order to work, a sanction system must, on this view, possess some degree of “legitimacy” in the eyes of its subjects, i.e. a sufficiently large part of the subjects must be willing to comply voluntarily, for reasons other than fear of punishment. Thus, for example, H L A Hart contends that if a system of rules is to be imposed by force on any, there must be a sufficient number who accept it voluntarily. Without their voluntary cooperation, thus creating authority, the coercive power of law and government cannot be established.149

A central assumption behind this critique of the so called “gunman theory” of law150 is that the capacity of any sanction system, for gathering and processing information about the behavior of individual subjects, in order to detect violations, as well as for executing punishments against violators, is necessarily limited. No society could mobilize the resources necessary to be able to keep constant surveillance over each individual subject and intervene against every violation. No police force could master a situation where nearly everyone is unwilling to obey and hence must be coerced into obedience. Thus, according to this view, only when the major part of the population is willing to obey voluntarily, such that the resources for surveillance and punishment can be concentrated on a relatively small group of potential violators, will sanctions be effective. However, this critique seems to suggest that, in order to achieve large-scale compliance with a set of norms by fear of punishment alone - “at gunpoint” so to speak - it is necessary to actually execute constant surveillance and coercion against each individual subject. The model that will be developed 149 150

Hart (1994) p 201. This theory is ascribed to John Austin (1832), see also Hardin (1995).

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in this chapter aims to show that a less ambitious system of sanctions might suffice to make compliance individually rational. In order to deter an individual from violating a norm, it is not necessary to ensure that she will actually be punished if she violates. Often, it is sufficient that there is a certain (perceived) risk of being punished if she violates. Provided that nearly everyone complies, even a very limited capacity to enforce sanctions might suffice make the risk of being punished sufficiently large to make singular violations unprofitable. What a limited sanction system would not be able to cope with, however, is coordinated defection on a large scale. However, one of the main points that will be argued here is that, if the sanction system sustains a mutually beneficial cooperative equilibrium, it is in no one’s interest to try to achieve such coordinated defection. There is plenty of historical evidence, however, that even extremely oppressive regimes have been able to keep people in check for long periods of time. Methods of political control have been well developed by tyrants during centuries. The Videla regime in Argentine, the Nazi occupation of Czechoslovakia and Ceausescu’s regime in Romania is typical examples.151 There is a large number of regimes that could be added to that list; Pinochet’s regime in Chile, the European colonial rule in large parts of Africa and Asia, the Khmer Rouge government in Cambodia, the Indonesian oppression against the population of East Timor etc. In all these cases, people do have an interest in achieving coordinated defection. In order to stay in power, these regimes must therefore actively prevent people from achieving coordination. Successful coordination, Russell Hardin contends, “can create extraordinary power”. 152 A relatively small group of coordinated individuals might succeed in controlling a large group of uncoordinated individuals. Hardin therefore suggests that the “gunman theory” of law could also be labeled the “dual-coordination theory”. 151 152

Hardin (1995), p 28. Ibid.

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It depends on coordination at the level of government and on lack of coordination at the level of any potential popular opposition. The state need not compel everyone at gunpoint; it need merely make it in virtually everyone’s clear interest individually to comply with the law even though collectively it might be their interest to oppose the law.153

The power of coordination is beautifully illustrated by a prison interior from Sing-Sing, reported by Alexis de Toqueville from his journey to America, and retold by Anders Ehnmark in his essay on Toqueville; Slottet154. On his American journey, Tocqueville did not only take interest in the American democracy. On behalf of the French government, he also visited and reported from a number of American prisons, apparently in order to collect information for a reformation of the French penal system. In Sing-Sing, Tocqueville and his travelling companions met a scene that impressed them much. They saw hundreds of America’s most dangerous criminals, working side by side in a quarry, equipped with spades and pickaxes, with no chains or walls to keep them from escaping, and watched over by only a handful of guards. How was this possible? Why didn’t the prisoners revolt and escape? With joint efforts, they would have easily outnumbered the few guards, and most prisoners would then have been able to escape. The answer, according to Tocqueville, was silence. The prisoners were kept in isolation from each other by having to observe complete silence 24 hours a day. Any transgression of the prohibition to communicate was severely punished. Being deprived of the possibility to communicate, the prisoners were unable to coordinate their behavior. Thus, each individual prisoner who contemplated to revolt would find himself standing alone against the entire group of coordinated guards, and must therefore calculate on a very small chance of success, and a very large risk of being severely punished. The payoff structure of the situation that the Sing –Sing prisoners find themselves in could be represented as follows (fig. 5.1) 153 154

Op.cit. p 30. Ehnmark (1994)

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Suppose that there are n prisoners, each of whom has a choice between the two strategies Obey (o) and Revolt (r). Suppose further that there are x guards (with 0 u(o, n-1)

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Payoff

Revolt

Obey

k

n -1

Number of others who revolt

At some point, k, the two graphs intersect. Left of that point, continued obedience is individually rational, whereas, to the right, revolting is the rational choice. If all prisoners are rational and well informed, and if communication is effectively prevented, then general obedience can be stable for a long time, until, with a formulation from Hardin, “order is cracked by a tipping event or signal that coordinates an opposition”. 155 6.2 An Assurance game The idea of this section is to take some of Tocqueville’s Sing-Sing prison guards and place them in a population of "prisoners", captured in an ongoing n-player Prisoner's Dilemma, with the single task of detecting and punishing defections. The question is what effect the presence of these guards might have on the prospects for cooperation. A central, restrictive assumption in Molander’s analysis of the n-player Prisoner's Dilemma is that ”only the general frequency of cooperation is observable”. 156 As was argued in the previous chapter, the general applicability of this assumption is questionable. In any case, relaxing this 155 156

Hardin (1995) p 30. Molander (1992) p 768.

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condition is necessary for a sanction system to be able to operate. If it is not possible to discover defections, then, of course, it is not possible to impose sanctions against defections. What follows is a model of an n-person game, which is modified by a set of negative side-payments corresponding to a sanction system with limited capacity. The model does not assume that each player can easily monitor the behavior of each of the other players, and direct individual sanctions against each defector. It assumes only that it is possible in principle, perhaps only by considerable effort and with some degree of uncertainty, to obtain information about the behavior of any given individual. Since the capacity of the sanction system is assumed to be limited, the system will only be capable of detecting and punishing a small number of defections at a time. When defections are very frequent, a considerable number of defections will go unpunished, whereas, when defections are relatively infrequent, few defections will go unpunished. Thus, each individual who contemplates defection must take into account that she will face a certain risk of being caught and punished, where this risk monotonically increases with the number of others who cooperate. In order to model this more precisely, I suggest the following assumptions. Suppose that the sanction system is capable of carrying out a limited number, q, of simultaneous criminal investigations, where q is a positive integer smaller than n. When a defection occurs, an investigation is appointed to determine which player is guilty, provided that there are investigation resources available. If the number of defections, y, exceeds the available resources, there is a probability q/y for each defection that it will be the subject of an investigation. For each investigation, there is a certain probability, π, that it will succeed in determining which player is guilty of defection. Together, q and π define the investigation capacity of the sanction system. Further, let γ be the size of a punishment. A punishment is some harm that is inflicted on a player who is exposed as a defector. In the game matrix it

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is represented as a negative side payment. For simplicity, we assume that the size of a punishment is constant. Thus, a sanction system, Φ, is here defined by the triple (q, π, γ). Now, let ECd(j, Φ) the expected cost of sanctions that a defecting player must calculate on under the sanction system Φ when j others cooperate. For n-jq, the cost is ECd(j, Φ) = qπγ/(n-j).

Number of others who cooperate

fig. 6.2 Let G =〈N, S, u〉 be the uniform n-player Prisoner’s Dilemma157 represented in fig.3.7 and let GΦ =〈N, A, uΦ 〉 be a modification of G, such that N and S are identical in the original and the modified game, and uΦ (d, j) = u(d, j) - ECd(j, Φ). If πγ > u(d, n-1) - u(c, n-1), the modified game GΦ is an n-player Assurance game (see fig 6.3). The number k here is the smallest number 157

Since we assume that the game is uniform, individual qualifiers are omitted; each one’s payoff and strategy set is the same.

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such that if k or more others cooperate, cooperation is strictly preferred to defection, i.e. the first integer above the intersection point.

Payoff

c d

k

n -1

Number of others who cooperate

fig. 6.3 As was seen in chapter 3, general cooperation is a strict Nash-equilibrium of a n-player Assurance game. Hence, cooperation should be considered a viable outcome among rational players, even in a one shot game. Provided that one can be assured that nearly everyone else will cooperate, cooperation is individually rational. However, general defection is also a strict Nash-equilibrium. When few others cooperate, the expected cost of sanctions is small, hence defection is strictly preferred by everyone. The Assurance game, thus, illustrates the logic behind the dual coordination theory of power. It is possible that each agent cooperates by fear of punishment alone, as long as she expects everyone else to cooperate by fear of punishment. When everyone expects everyone else to cooperate, the expected cost of sanctions becomes sufficiently large to make cooperation individually rational. Thus, we do not have to assume that a substantial number of agents are prepared to cooperate voluntarily, i.e. disregarding the threat of punishment. Nor do we have to assume that sanctions are so far-reaching as to deter from defection regardless of the number of others who cooperate.

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If the values of q, π and γ can be chosen independently of other parameters of the game, then it is trivial that, for any n-player Prisoner’s Dilemma there are values of (q, π, γ) that guarantee that the modified game, GΦ , will be an Assurance game. However, we should not expect the values of q, π and λ to be independent of other parameters of the game. Sanctions are often costly, and these costs must normally be borne by the players themselves. James Buchanan argues that enforcement of a basic constitutional contract would be genuinely external only if there was a God who could be trusted to enforce man-made law. In the absence of such a punishing God, Buchanan writes; “man must rely on his own resources to pull himself from and stay out of the Hobbesian ’warre’”158 I can think of some cases (disregarding the possibility of a punishing God), where a sanction-enforcing agency could be considered as genuinely external. In the Dayton peace accord 1995, which ended the war in Bosnia, the conflicting parties mandated a multinational Implementation Force (IFOR) to enforce the terms of the peace agreement. The mission of IFOR was, thus, to enable the conflicting parties to achieve and comply with a (presumably) mutually beneficial peace-agreement, by punishing transgressions from any party. The countries that contributed to IFOR, financially or by sending troops, were not themselves parties to the conflict. The maintenance of the sanctioning agency, thus, was organized without relying on contributions from the conflicting parties. However, in most cases, a sanction system involves costs that must somehow be met by the players. Some players may, for example, have to devote their time and skills to the task of monitoring and punishing noncooperators instead of engaging in more constructive and welfare creating activities, thereby reducing the total payoff in the group.

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Buchanan J. M. (1986) p 130

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These costs could be distributed over the set of players in different ways. We may, for example, assume that costs are distributed equally over the set of cooperators. Alternatively, we may assume that costs are distributed equally over the entire set of players, defectors as well as cooperators. Which assumption is the more plausible one depends on the type of interaction problem, and the character of the costs involved. The latter might be a reasonable assumption when the object of cooperation is provision of a continuous collective good, and the main cost consists in decreased level of provision of the good. If some resources are spent on maintaining a sanction system instead of production there will be less of the collective good available for everyone. The former assumption, on the other hand, might be reasonable when maintaining sanctions requires some positive contribution by the players, e.g. financial support of the sanctionenforcing agency. The difference is not of decisive importance, however; all conclusions that will be derived below are equally valid whether we assume that costs are distributed equally over the set of players or over the set of cooperators.

Cost of sanctions

For a case where the cost of sanctions is distributed equally over the set of cooperators; suppose that the total cost of maintaining a sanction system Φ= (q, π , γ ) is δ Φ . The cost imposed on each cooperator is then δ Φ /(j+1).

Number of others who cooperate

fig. 6.4

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Since the maintenance of the sanctioning agency is here supposed to depend on contributions from the cooperators, we should assume that ECd(j, Φ) = 0 for j = 0. When no one cooperates, the sanction system cannot operate. There are some different possibilities here. Either the sanction system is conceived of as a step good; i.e. at some level of cooperation sanctions begin to operate, below that level there are no sanctions. Or the capacity of the sanction system is thought to increase continuously with the number of cooperators. Or, alternatively, the capacity increases continuously with the number of cooperators up to some level, and then levels out. Let G =〈N, S, u〉 be a uniform n-player Prisoner’s Dilemma as represented in fig. 3.7. Let GΦ =〈N, S, uΦ 〉 be a modification of G, such that N and S are identical in the original and the modified game, and, for all j∈N, uΦ (d, j) = u(d, j) - ECd(j, Φ) and uΦ (c, j) = u(c, j) -δ Φ /(j+1). If πγ - δ Φ /n > u(d, n-1) - u(c, n-1), the modified game GΦ will be an nplayer Assurance game (see fig 6.5).

c Payoff

d

k

Number of others who cooperate

fig.6.4

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n -1

When will this condition be satisfied? A reasonable assumption is that the cost, δ Φ ,of maintaining the sanction system Φ=(q, π, γ) is a monotonic function of its capacity. That is, δ(q, π, γ) = δ(q, π*, γ) iff π = π* δ(q, π, γ) = δ(q, π, γ*) iff γ = γ* δ(q, π, γ) = δ(q*, π, γ) iff q = q* The crucial factor is, I believe, the cost of effective surveillance. Once a defector has been exposed, it is normally possible for a coordinated group to inflict some harm to her at a moderate cost to themselves. In cases where surveillance is extremely difficult and costly, it might be technically impossible to design an effective sanction system at all, especially if the original temptation to defect is very large and n is small. Or, it might be possible only at a cost that is so large as to make everyone worse off in a state of near general cooperation, than if they would be defecting in a state of near general defection. However, if surveillance is possible at some determinate cost, there will exist values of π, γ and n such that πγ -δ(1, π, γ)/n > u(d, n-1) - u(c, n-1). When this condition is satisfied, if G =〈N, S, u〉 is a uniform n-player Prisoner’s Dilemma, it will be possible to construct a sanction system Φ such that the modification GΦ =〈N, S, uΦ 〉 of G is an Assurance game. The prospects for achieving effective sanctions, thus, seem to increase with the size of n. The model, thus, seems to indicate how a sanction system with limited capacity might be capable of enforcing cooperation in a large group. It is not necessary, as the critique of the “gunman theory” suggests, that, in order to make cooperation individually rational for everyone, a sanctioning system must have nearly unlimited capacity. It is not necessary that it is capable of intervening against every defection, regardless of the level of cooperation. It is sufficient that each agent, provided that nearly everyone

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else cooperates, suffers a sufficiently large risk of being punished, were she to defect. It is true, of course, that the system can operate only when a sufficient number cooperates, but it is not true that they must cooperate “voluntarily”, i.e. regardless of the risk of punishment. The “gunman theory”, thus, might be consistent with a fairly low level of actual force in a society. However, before these conclusions are brought home, the robustness of the cooperative outcome should be tested under an evolutionary dynamics. 6.3 Evolutionary analysis. Now consider an iterated game in an evolutionary setting, similar to that in Molander’s analysis of n-player Prisoner’s Dilemma. We make the same assumptions about strategy set and evolutionary framework as in Molander’s analysis, i.e. • S ={s0, s1, s2, ..., sn ) with sj denoting the strategy “cooperate iff at least j others cooperate, otherwise defect“, • groups are formed randomly from a large but finite population, • each player is programmed for a certain strategy sj∈S, • each group that has been drawn plays the game repeatedly until it reaches its short term equilibrium based on the strategy profile of its members, • payoffs are those of the steady state, • the strategy profile of the entire population evolves under pressure of a selection process in accordance with replicator dynamics. • apart from natural selection, the strategy profile may also change due to some random process (e.g. mutations). Thus we allow for drift. To abbreviate notation here, let cj = u(c, j) and let dj = u(d, j). Further, let ∆ be the set of possible probability distributions over S, and let x∈∆ be a population state such that x=(x0, x1,…,xn), where xj is the fraction of the population programmed for strategy sj. Let Aj(x) be the average payoff for strategy sj in population state x and let Ay(x) be the average payoff to a mixed strategy y in population state x.

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For n = 2 we have the following. S = {s0, s1, s2}. A0(x)=x0c1 + x1c1 + x2c0 A1(x)=x0c1 + x1c1 + x2d0 A2(x)=x0d1 + x1d0 + x2d0 Since c1 > d1 and c1 > d0, it follows that, x0 + x1 > 0 ⇒ A1(x) > A2(x). That is, for any x such that there is some positive proportion of cooperation, s1 does strictly better than s2. Thus, for any interior point of ∆ (i.e. any state where xj > 0 for all j), the selection dynamics will push the population towards a state of pure cooperation. Moreover, we have x0 + x1 = 0 ⇒ A1(x) = A2(x). Hence, since we allow for mutations, a state of pure defection can be invaded by s1. When a mutant s1 enters in a purely non-cooperative state, it earns the same payoff as the incumbent strategy sn. Once a small bridgehead of s1 has been established, s1 earns strictly more than sn. Hence, the selection dynamics will push the population to a state of pure cooperation. Moreover, since s1 does strictly better than s2 whenever there is a positive proportion of cooperation, once it is established s1 cannot be invaded by s2. However, since for x2 = 0 we have A1(x) = A0(x), s0 can drift into a population of s1. Eventually, this process of random drift might result in a population state y, where y1 = 0. Thus, s1 is not an ESS. As was said in chapter 2, a strategy set X ⊂ ∆ is evolutionarily stable if it is closed and nonempty and for every x∈X, there is some neighborhood W ⊂ ∆ such that, Ax(w) = Aw(w) for all w∈W, and Ax(w) > Aw(w) for w ∉ X.

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Let Xc={∀x: x∈∆, x2= 0}; i.e. Xc is the subset of ∆ consisting of only cooperative strategies. Since, for all for x∈Xc , A1(x) = A0(x), it follows that Ax(w) = Aw(w) for all x, w∈Xc. Let w be the mixed strategy corresponding to the population state that results when population state x is modified by the entrance of a small proportion, ε, of strategy s2. Thus, if x∈Xc we have w2 = ε and w0 + w1 =1-ε. Xc is an ES set iff for all x∈Xc, there exist some ε 2 ∈(0,1) such that, for all ε ∈(0, ε 2), Ax(w) > Aw(w). ⇔ Ax(w) > A2(w) Ax(w) = (w0 + w1)c1 + w2 (x0c0 + x1d0) A2(w) = w0d1 + w1d0 + w2d0 Denote by dmax the maximum payoff for d; i.e. dmax = max u(d, j). Thus, j∈(0, n) dmax = dj for all j∈(0, n). We then have, dmax = dj for all j∈(0, n) ⇒ w0d1 + w1d0 = (w0 + w1) dmax d0 > c0 ⇒ x0c0 + x1d0 = c0 Thus, (w0 + w1)c1 + w2 c0 > (w0 + w1) dmax + w2d0 ⇒ Ax(w) > A2(w) (w0 + w1)c1 + w2 c0 > (w0 + w1) dmax + w2d0 ⇔ (1 – w2)(c1 - dmax) > w2(d0 - c0) ⇔ (c1 - dmax) > w2((d0 - c0) + (c1 - dmax)) ⇔ w2 < (c1 - dmax)/ ((d0 - c0) + (c1 - dmax)) ⇔ ε < (c1 - dmax)/ ((d0 - c0) + (c1 - dmax)) Since both (c1 - dmax) and (d0 - c0) are positive numbers, it follows that ε ∈(0,1). Hence, for n=2, Xc is an ES set.

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Moreover, since A1(x) = A2(x) for all x, and A1(x) > A2(x) for all x such that x0 + x1 > 0, it is easily verified that no X∈∆ such that x2 > 0 for some x∈X can be evolutionarily stable. For n =3 we must distinguish between two types of cases. The first type of case (see fig. 6.5.) is defined by cn-1 > dj for all j∈(0, n).

d

Payoff

c

Number of others who cooperate

fig. 6.5

The second type of case (see fig. 6.6) is where cn-1 = dj for some j∈(0, n).

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Payoff

d

c

Number of others who cooperate

fig. 6.6

Case 1 For the first type of case, i.e. cn-1 > dj for all j∈(0, n), there is a purely cooperative equilibrium set that is evolutionarily stable, and no state x for which xn > 0 can be an ESS. Consider a case where dj > cj for all j dn-1. We know that An-1(x) = An(x) for all x, and An-1(x) > An(x) for every x such that xn < 1. Since sn-1 earns the same payoff as sn when xn = 1, and strictly more when there is some positive proportion of cooperation, a state of pure defection can be invaded by sn-1. Once a bridgehead of sn-1 has been established, the selection dynamics will push the population into a state of pure cooperation. Since An-1(x) > An(x) when xn < 1, sn is unable to invade as long as xn-1 > 0. However, since for xn = 0 we have Aj(x) = An-1(x), any cooperative strategy can drift into the population when sn has been extinguished. This process of random drift might result in any frequency distribution of cooperative strategies. Eventually, it might result in a population state x’, where x’ n-1 = 0. Thus, for pure cooperation to be evolutionarily stable, any mix of cooperative strategies must be able to resist invasion by sn.

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Let Xc={∀x: x∈∆, xn= 0}. Since Aj(x) = An-1(x) for all x∈Xc and all j < n, we know that, Ax(w) = Aw(w) for all x,y∈Xc. Thus, Xc is an ES set iff for all x∈Xc , there is some neighborhood W⊂ ∆ such that, Ax(w) > Aw(w) for w∈W and w∉Xc. Let w be the mixed strategy corresponding to the population state that results when population state x is modified by the entrance of a small proportion, ε, of strategy sn. Thus, when x∈Xc we have wn = ε and wc = 1-ε. Xc is an ES set iff for all x∈Xc, there exist some ε n ∈(0,1) such that, for all ε ∈(0, ε n), Ax(w) > Aw(w). ⇔ Ax(w) > An(w). Let xc=x0+x1+…xn-1. Let a = cn-1 – dmax, and b = dmax - cmin, where dmax = max u(d, j) and cmin =min u(c, j). Thus, dmax = dj for all j∈(0, n) j∈(0, n) j∈(0, n) and cmin = cj for all j∈(0, n). Since cn-1 > dj for all j∈(0, n) and d0 > c0, a and b are both positive numbers. For n = 3 we have the following. awc2 – b(1-wc2) > 0 ⇒ Ax(w) > A3(w). awc2 – b(1-wc2) > 0 ⇔ 2 wc (a+b) – b > 0 ⇔ wc > vb/(a + b) ⇔ 1-ε > vb/(a + b) ⇔ ε < 1- vb/(a + b).

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Since vb/(a + b) must be a positive number smaller than 1, it follows that there is some ε n ∈(0,1) such that, for all ε ∈(0, ε n), Ax(w) > A3(w). Hence Xc is an ES set for n=3. For the general case n = 3 we have, awcn-1 – b(1-wcn-1) > 0⇒ Ax(w) > Aw(w) awcn-1 – b(1-wcn-1) > 0 ⇔ n-1 ε < 1- vb/(a + b). Again, since n-1vb/(a + b) is a positive number smaller than 1, it follows that there is some ε n ∈(0,1) such that, for all ε ∈(0, ε n), Ax(w) > Aw(w). Hence Xc is an ES set for n =3. Moreover, since An-1(x) = An(x) for all x∈∆, and An-1(x) > An(x) for all x∈∆ such that xc > 0, no X∈∆ such that x2 > 0 for some x∈X can be an ES set. Thus, the following theorem is proved. THEOREM: Let G =〈N, S, u〉 be an iterated, uniform n-player Assurance game as defined above. 1. For n=2: There is a purely cooperative ES set, and no other ESS or ES set. 2. For n>2: If cn-1 > dj for all j∈(0, n), there is a purely cooperative ES set, and no other ESS or ES set. Case 2 For the second type of case, i.e. where cn-1 = dj for some j∈(0, n), the above proof does not apply since a = cn-1 – dmax is not a positive number. Thus we cannot conclude that there is a purely cooperative ES set. Moreover, since it is not necessarily true that An-1(x) = An(x) for all x∈∆, we cannot exclude the possibility of some ESS or ES set.

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Computer simulations that I have made with Clas Pihlström, setting cn-1 < dj for some j∈(0, n) and cm > d0 for some m < k, (where k is the smallest positive integer such that ck > dk ), indicate that there might be a mixed equilibrium state, x∈∆, such that xm>0 and xn > 0, i.e. a mixed equilibrium similar to the one found by Molander in the n-player Prisoner's Dilemma. An example, with n = 4, is shown in Appendix. Whether this equilibrium is also an ESS, I do not know yet; further analysis is required to determine the precise properties of this type of game. I am still rather uncertain about how to interpret this result. A cautious interpretation is to limit our conclusions about the viability of cooperation in the iterated n-player Assurance game to the subclass of games for which n=2 or cn-1 > dj for all j∈(0, n). This means that a slightly more efficient sanction system might be required in order to enforce cooperation. It is not sufficient that sanctions are effective when nearly everyone cooperates, i.e. that cn-1 - δ Φ /n > dn-1 - πγ. It must have some effect also at lower levels of cooperation, such that, for every j∈(0, n), cn-1 - δ Φ /n > dj - qπγ/(n-j). However, although more is required of the sanction system, nothing essential is changed regarding the main conclusions of this chapter. The same factors remain crucial for the possibility of establishing effective sanctions; i.e. the cost of surveillance and the size of n. Moreover, if it is possible for some n to set the values of π, γ such that the condition cn-1 - δ Φ /n > dn-1 - πγ is satisfied, then it is also possible for some n to set the values of q, π, γ such that the condition cn-1 - δ Φ /n > dj - qπγ/(n-j) is satisfied.

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Appendix to chapter 6 The example is of a 4 player game with the following parameters. S = {s0, s1, s2, s3, s4} c0 = 1, c1 = 3, c2 = 5, c3 = 7 d0 = 2, d1 = 20, d2 = 20, d3 = 4 It is assumed that the population state evolves in accordance with replicator dynamics. An evolutionary process is simulated from some different starting points within the mixed strategy space of the game. The initial population state will be called x and the resulting population state when the population reaches a stationary point will be called x’. For a large set of initial states, the following picture emerges. First, s4 prospers at the expense of tolerant cooperators like s0, s1 and s2. After a while, however, the most tolerant strategies are wiped out, then s4 declines, and finally a state of pure cooperation emerges, dominated by s3. This is the case in the following cases.

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Case 1. x0 = 0,5, x1 = 0,2, x2 = 0, x3 = 0,1, x4 = 0,2

x3 x4

x0

x1 Case 2 x0 = 0,45, x1 = 0,2, x2 = 0, x3 = 0,05, x4 = 0,3

x4

x3

x0

x1

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However, if x3 is very small, and x4 is large, then s3 is eventually wiped out, and something similar to the mixed equilibrium in the iterated n-player Prisoner's Dilemma appears. This is what happens in the following case. Case 3. x0 = 0, x1 = 0,2, x2 = 0, x3 = 0,1, x4 = 0,7

x4

x1

x3 If we analyze the payoffs, we find that in x’ s1, s3 and s4 earn the same average payoff (2,20016). If we modify x’ by adding a small proportion, ε 3 of s3 (e.g. ε 3 = 0,01), both s1 and s4 earn strictly better than s3. Thus the population state returns to a mix of s1 and s4 whereas s3 is again wiped out.

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CHAPTER 7

THE CREATION AND MAINTENANCE OF SANCTIONS ______________________________________________________________________

Why, for example, should the rational egoist pay his portion of the taxes that the state requires to maintain its police forces, etc., or why should the individual member of a community go to the trouble of punishing a free rider when he could be a free rider on the sanctioning efforts of others? Michael Taylor159

7.1 Introduction. The previous chapter introduced a model of a limited sanction system and showed how the existence of such a system affects the payoff structure of a collective action problem. If the original problem is an n-player Prisoner’s Dilemma, the game that emerges when payoffs are modified by a suitable set of sanctions (i.e. negative side-payments) is an n-player Assurance Game. In this game, unlike n-player Prisoner’s Dilemma, there exist a Pareto optimal Nash-equilibrium, which consists in universal Cooperation. Further, it was shown that the cooperative equilibrium is a robust and viable outcome of an iterated n-player Assurance game under evolutionary dynamics, and that the cooperative equilibrium that is reached by threat of sanctions may be better for everyone than the non-cooperative equilibrium that results if the threat of sanctions is lifted. This result holds even if we assume that sanctions involve costs and that those who cooperate must carry the cost of maintaining the sanction system. One might object, however, that the creation and maintenance of such a sanction system in a large group is in itself an example of a collective action problem160. Although the existence of a sanction system benefits everyone, it may be in no one’s interest to contribute to its maintenance. If 159

Taylor (1987) p 22. Buchanan (1975), Heath (1976), Taylor (1987), Molander (1992), Bicchieri (1997) Sober and Wilson (1998). 160

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sanctions are costly, and the benefit that an individual derives from their existence is (more or less) independent of her own contribution, there might be a strong incentive for each individual to be a free rider on the sanctioning efforts of others. Instead of solving the problem, introducing sanctions merely creates a second-order free rider problem. This chapter deals with this objection. I will argue that the payoff structure of the problem of forming an effective sanction system is not necessarily, or not even typically, that of a Prisoner's Dilemma. Rather, the creation and maintenance of sanctions typically presents an Assurance game, a game for which, as we have seen, the prospects for successful cooperation are much better. However, there are many different ways of organizing a sanction system. So far, we have left this issue to the side, considering the necessity and efficiency of sanctions in general. The Hobbesian argument, though, claims that a particular kind of sanctioning agency, i.e. a state, is necessary to secure conditions for successful cooperation. Here we shall consider some different ways in which a sanction system might be organized. We shall consider two different ways that a centralized sanctioning agency, i.e. a government, might be established, but also the possibility for a group to organize and maintain a decentralized system of sanctions. As we shall see, neither of these ways presents the kind of problems that the establishment of sanctions is often claimed to. In the beginning of the previous chapter, we briefly discussed the possibility of having sanctions imposed by a genuinely external agency, i.e. an agency that is organized and financed without participation from the players involved in the original collective action problem. As an example, the efforts by the international implementation force, IFOR, in Bosnia after the Dayton accord was mentioned. That a sanction-enforcing agency is external in this sense does not, of course, imply that it is immune against free rider considerations, only that these occur and must be handled at a different level. In the case of multinational peace-keeping and peace-enforcement operations; even if we

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assume that most states do have a stake in the preservation or restoring of international peace and stability, one could still question the rationality of contributing to such operations. Why should individual states be prepared to make sacrifices in order to promote a collective good such as international peace and stability? In most cases, however, the cost of maintaining sanctions must somehow be met by the players themselves. We shall thus consider how they might achieve this. 7.2 Commonwealth by acquisition Hobbes acknowledges two different ways that political authority might be instituted: by institution and by acquisition. A “commonwealth by institution” is established by a voluntary agreement between the prospective citizens, or subjects, of the commonwealth, in response to the threat that they perceive from each other. “A commonwealth by acquisition”, on the other hand, is one where the sovereign power is acquired by force; and it is acquired by force, when men singly, or many together by plurality of voices, for fear of death, or bonds, do authorize all the actions of that man, or assembly, that hath their lives and liberty in his power.161

Consider the n-player Prisoner's Dilemma of fig.3.7 and the n-player Assurance game of fig. 6.3. As we saw, the second game resulted from modifying the first one by a system of sanctions. It was shown that, provided that the costs of enforcing the sanctions were moderate, everyone would be better off in the cooperative equilibrium of the modified game than in the non-cooperative, or “shabby”, equilibrium of the original, unmodified game. In other words, the introduction of sanctions creates a surplus value. A smart political entrepreneur might realize that there is a possible profit to collect for someone who is able to enforce effective sanctions.

161

Hobbes (1996) Ch 20, p 132.

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One of the main points of chapter 6 was that coordination creates power. A relatively small group of coordinated agents can be capable of controlling a large group of uncoordinated individuals. Now in the state of nature, a small group of people, who realize the dynamics of the situation, might decide to unite forces in order to take control over a territory and enforce cooperation among its inhabitants. If they succeed, total payoffs in the territory, which they have conquered, will increase. The self-proclaimed rulers might confiscate a disproportionate part of the surplus value for their own consumption and still leave the inhabitants of the territory better off than they were in the state of nature. For this to be possible, the group of self-proclaimed political rulers must be capable of cooperating internally. They will have to establish an internal system of sanctions against disloyal behavior from group members, for example by expelling members who act against the group interest. Does this problem of internal loyalty within the group of self-proclaimed guards pose an n-player Prisoner's Dilemma? If it does, the proposed solution is invalid. However, I believe that it does not. First, the group is supposed to be relatively small, hence we might assume that they could relatively easily monitor the behavior of each other. If some member of the group does not do her part, for example refuses to participate in combat against opponents or to punish offenders of the rules that the group has decided to enforce, then this will, for the most part, be visible to her colleagues. Moreover, the good that coordinated behavior within the group creates for group members, i.e. being able to seize a disproportionate share of the surplus production, is not a collective, i.e. non-excludable, good. An disloyal member can easily be excluded form sharing the seized goods, or even expelled from the group, at little or no cost to other group members. Since, by refusing to do her part, the disloyal member increases the costs of sanction enforcement for other members of the group, these will have good reason to keep guard against, and punish disloyal behavior by group members.

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The problem of achieving cooperation in a large group in the provision of a collective good, can thus be solved by achieving cooperation in a much smaller group, by turning part of the benefit from large group cooperation into a non-collective reward, exclusively for loyal members of the small group. However, if taking control over a territory is very profitable, we might expect there to be more than one group of entrepreneurs competing for power. These competing groups should be expected to be at war with each other. The war of every man against every man will have changed to a war between competing warlords – a situation, which is hardly better for anyone than the initial state of nature. How is this situation to be solved? Robert Nozick has provided an answer to that question162. Nozick’s discussion concerns the development of so called “protective agencies” offering protection of the individual right of their “clients”. Nozick holds that in a situation where there are two protective agencies operating on the same territory, there are three possible developments. When conflicts occur between clients of the two agencies, the agencies fight. In these fights, either of these things will happen. (1) One of the agencies wins more often than the other. The winning agency will attract more support and hence grow stronger until it dominates entirely. “Since the clients of the losing agency are ill protected in conflicts with the winning agency, they leave their agency to do business with the winner.”163

(2) The two agencies have different geographical centers, and each agency wins more often close to its center. For the same reason as in (1) each inhabitant of the territory will either a) settle close to the center of the agency they prefer, or b) associate with the agency that is dominant in her local area.

162 163

Nozick (1991). See also Hampton (1988) p 168. Nozick (1991) p 16.

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(3) The two agencies win equally often and eventually agree with each other to solve their disputes by abiding by the decisions of a third, impartial judge. In either case, the result is that there is one totally dominant political regime in each distinct geographical area. The same reasoning applies to the political entrepreneurs that we discuss here. Either one of the competing groups will eventually win, and thereby establish itself as the sole sovereign of the territory, or they will somehow settle their dispute by dividing the territory or sharing power. In either case, there will finally be one dominant, self-proclaimed “government” in each distinct geographical area. To the inhabitants of the territory, this means that interactions that were formerly of Prisoner's Dilemma character have been transformed to Assurance games instead. Whether this is actually to the benefit of the inhabitants will, among other things, depend on the amount of the surplus production that their government seizes for its own luxury. Numerous examples, historical as well as contemporary, should warn us that the establishment of political authority, although it promotes social order and cooperation, is not always beneficial to the subjects. 7.3 Commonwealth by institution. A commonwealth by acquisition is established when someone, presumably with an eye to future profits, simply conquers power over a territory, and forces the inhabitants to finance its continuance. Hence, its establishment does not depend on the active participation of prospective subjects. Once it has established itself, subjects participate in financing its maintenance because they will otherwise be punished. Commonwealth by institution, on the other hand, does require that the prospective subjects actively participate in its establishment. Could such participation be achieved in a state of nature? Consider a group of people situated in a state of nature. Lacking government, they are captured in a war of every man against every man,

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which has the character of an n-player Prisoner's Dilemma. Now, let us assume that these people understand their predicament and realize that they would all be better off if they managed to terminate their war. Suppose, at least, that some wise people realize this, and have called to an “assembly of institution”. During this assembly, the group will jointly try to decide on an arrangement that will allow them to end war and maintain peaceful cooperation. The first proposal is optimistic: “Listen”, the first speaker says, “ this state of war is destructive to all of us. We would all be better off if everyone would surrender her arms, and quit threatening and attacking her neighbors. Therefore, let us agree to surrender our arms here, to be destroyed by some trustworthy person among us, and then we can all walk home in peace.” The second speaker disagrees: “This proposal is not only optimistic”, she says, ”it is utopian. If everyone else surrender their arms, someone who does not will be able to play Hawk in all subsequent competitions. If I follow the proposal and lay down my arms, but others do not, then I just expose myself to their exploitation. Why should I do that? The proposal is not a solution, it merely brings out the Prisoner's Dilemma structure of our situation.” A third speaker enters the scene and tries to mediate. .“The previous speakers are both right”, she says. “We should all surrender our arms and engage in peaceful cooperation. However, since each is tempted to defect from such an arrangement, we need sanctions that make compliance with the agreement individually rational. Let us therefore also elect a sovereign and assign her the power to enforce our agreement.” But the second speaker is not convinced: “Sanctions are expensive”, she says. “Who shall pay for it? The third speaker tries to solve a collective action problem, but the solution, which she suggests, requires that the problem has already been solved. Why should people contribute voluntarily to the maintenance of sanctions? If sanctions are effective, everyone will benefit, non-contributors as well as

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contributors. If sanctions are not effective, contributions will be wasted. Hence, regardless of whether others contribute or not, each of us has reason not to contribute herself. The situation is still a Prisoner's Dilemma.” Now a fourth speaker enters: “Listen”, she says. “We need to establish peace, and in order to do this we need someone to enforce peace for us. However, the objection raised by the previous speaker must be taken seriously. If we do not, all of us, contribute to maintaining the sovereign’s ability to enforce sanctions, it is not going to work. However, believing that everyone will contribute voluntarily to the maintenance of sanctions is as utopian as the initial proposal that everyone should voluntarily lay down their arms. If people are supposed to contribute to maintaining the sanctioning agency, such contributions cannot be voluntary. Thus, we need to impose a system of sanctions against those who refuse to contribute to the sanction system.” What the fourth speaker suggests here is the introduction of a meta-norm: “punish those who fail to assist in punishing offenders of the original norm”. It has been suggested that such meta-norms may offer a solution to the incentive problem involved in norm enforcement. 164 Cristina Bicchieri (1997) claims, however, that meta-norms do not solve the punishing dilemma Instead, it “only shifts the problem one level up: Upholding the metanorm itself requires the existence of a higher-level sanctioning system.”165 If norm enforcement is costly, then so is, presumably, enforcement of a meta-norm. If people could not be expected to contribute voluntarily to the enforcement of the first order norm, then why should they be expected to do so when it comes to the second order norm? These questions will be treated below. First however, we shall consider the possibility of establishing a decentralized sanction system, i.e. a system where the enforcement of sanctions is not left to a central agency but is

164 165

Maynard Smith ( 1982), Axelrod (1986). Bicchieri (1997) p 23.

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carried out by the agents themselves. As we shall see, both these suggested solutions face the charge of Bicchieri’s infinite regress argument. 7.4 Overlapping relations The evolutionary analysis of the n-player Prisoner’s Dilemma (chapter 6) rests on the assumption that groups are formed randomly from a very large population. This set-up suggests that interactions take place in a social vacuum, with no other relevant relations between the agents than that they currently happen to find themselves trapped in an n-player Prisoner’s Dilemma together. In real life, interactions often take place within a web of ongoing and overlapping relations. Rather than interacting with people, who are randomly drawn from the entire population of Homo Sapiens, we tend to interact repeatedly – and on many different issues – with a more limited set of people; neighbors, colleagues, business partners etc. Farmers who release their cattle to graze on common land in Garrett Hardin's “Tragedy of the Commons“ are not, in general, randomly drawn from a very large population, but are likely to be linked together in numerous other ways; as neighbors, through trade and by family relations. As Russell Hardin puts it: “If we are members of a large group facing an ongoing Prisoner’s Dilemma, we are likely to have dyadic and small-number relationships on other issues.“ 166 If this is the case, there is a possibility that non-cooperative behavior at one level of interaction may trigger sanctions against the non-cooperator at another level. Suppose that a group is involved in an ongoing n-player Prisoner’s Dilemma. Since the situation is an n-player Prisoner’s Dilemma, we assume that it is not possible to exclude non-cooperators from enjoying the benefits of eventual cooperative efforts in the group. However, an agent contemplating defection must take into account the fact that she will have to face members of the group in other types of interactions in the future, and that she may be excluded from enjoying the benefits of cooperation in these interactions. 166

Hardin (1982) p 196.

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Villagers who persist in enlarging their herds above a sustainable level may, perhaps, not be excluded from releasing their herds on the commons, but they may suffer other kinds of sanctions from their neighbors. They may not be invited to feasts and celebrations, people may refuse to trade with them or refuse their sons and daughters to marry their children, they may be excluded from the various systems of mutual assistance that exist in many rural communities etc. If social relations are important enough, the risk of being subject to such social sanctions might restrain the temptation to defect in the large group. Some kinds of sanctions - e.g. exclusion from being invited to feasts - can probably be executed without costs, but other kinds involve costs for the punisher as well as the punished. Suppose that two individuals, i and j, who are both parts of a larger group involved in an iterated n-player Prisoner’s Dilemma, also have a dyadic relation with each other. Suppose, further, that their dyadic relation is such that j could choose to reduce i’s payoff, but only at the price of reducing her own payoff as well. They might, for example, have ongoing and mutually beneficial exchange relations. Now, if i were to defect in the large group, j has a possibility to retaliate i’s defection by withdrawing from cooperation in their dyadic relation. However, why should j do that? If, by punishing i, j would increase the general level of co-operation in the large group, j would thereby produce a benefit to the entire group. By punishing i, however, j also incurs a cost. While the benefit that is derived from increased cooperation in the large group is dispersed over the entire group, j must carry the whole cost of the sanction herself. So, even if the total benefit from increased large-group co-operation is larger than j’s loss, j’s share of that benefit may still be smaller than her loss. It is a classical collective goods problem, although what we might call a “second order” collective goods problem167. While it is in everyone’s interest that people be willing to execute punishments, it is in no one’s individual interest to do so herself. If there 167

Sober and Wilson (1998).

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was a temptation to be a free rider in the original, first-order, public goods problem, there seems to be an equally strong temptation to be a free rider in the second-order problem of providing mutually beneficial sanctions. One way of escaping this negative conclusion would be to argue that the execution of sanctions need not involve any substantial costs at all, at least not costs that agents take on intentionally. 168 Russell Hardin argues that this might be the case if there are several alternative partners available for dyadic co-operation. Suppose that a set of agents N={1, 2, …, n} are involved in an ongoing multi-player Prisoner’s Dilemma, and that they have dyadic exchange relations with each other as well. Suppose further that player i∈N chooses to defect in the multi-player Prisoner’s Dilemma. Now, although i cannot be excluded from enjoying the benefits from co-operative efforts in the large group interaction, there are some players in N\{i} who can withdraw from co-operation in their dyadic relations with i. If the set of players who withdraw from dyadic cooperation make up a substantial part of i’s potential exchange partners, i will suffer a large loss, since there will not be a sufficient number of alternative partners left to compensate for the lost exchange relations. Those players who withdraw, on the other hand, will only loose one of many potential partners; they will be able to substitute the lost transactions at a negligible cost, by trading with someone else instead.169 A somewhat different consideration draws on an argument from chapter 4. Suppose that previous behavior is taken as evidence on a player’s strategy, and that most people are conditional cooperators who cooperate only when they believe that their counterpart is a reliable “constrained maximizer”. Agents might then refuse to cooperate with someone who has a record of previous defection, not in order to punish her, but because previous defection is taken as evidence that she is not a reliable “constrained maximizer”. Also, agents might be motivated to act as if they were “constrained maximizers”, because it is important to have a reputation for being reliable. It is having a reputation for being a reliable “constrained 168 169

Hardin (1985), Sober and Wilson (1998). Hardin (1985).

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maximizer” that increases an agent’s expected payoff, not her actually being one. However, in a world where present behavior affects future reputation, the effect is that people should act like constrained maximizers. For example, the reason that banks and credit institutions usually demand a credit report on applicants before granting credits is not that they intend to punish people who have previously failed to repay their debts to other creditors, but that they want to avoid loosing money themselves. The effect, however, is that people who have previously failed to pay their debts are punished by not having credits granted in the future. Similarly, defections in an n-player Prisoner’s Dilemma might be taken as evidence that an agent is not a reliable “constrained maximizer”. In the example above, where i defected in a n-player Prisoner’s Dilemma, j∈N\{i} might withdraw from cooperation in their dyadic relation, not in order to punish her but in order to avoid being exploited by a person who has demonstrated that she is not a reliable partner. If this is the case, then j might not perceive her punishing i as being costly to her at all. The effect is the same. Cooperation is motivated by the importance of future interactions; defection in an n-player Prisoner’s Dilemma now will induce others to withdraw from future dyadic cooperation. Such an explanation, of course, might seem less plausible in cases where i and j already have a long record of successful dyadic cooperation. Why would j take i’s failure to cooperate in the large group as evidence that i is also unreliable in dyadic interaction when j has plenty of evidence to the contrary? On the one hand it might seem unreasonable that j should count i’s behavior in multi-player situations as stronger evidence about i’s dispositions in two-player situations than what j has already been able to observe about i’s behavior in two-player situations. On the other hand, it is possible that there is an asymmetry between positive and negative evidence that might affect j’s judgements. Whereas trust is established slowly and with difficulty, it can be immediately destroyed by negative evidence. 170 170

Persson (1999), see also Slovic (1993).

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However, although we acknowledge that exercising sanctions needs not always involve intentionally borne costs, there are undoubtedly cases where it does. The scope of arguments based on the possibility to exercise sanctions at a negligible cost is limited. We need therefore to consider whether a system of decentralized sanctions could be maintained when sanctions are costly. 7.5 Meta-norms and the infinite regress Sober and Wilson (1998) tell a tale of two imaginary cultures, the “squibs” and the “squabs”. Squabs are complete egoists who live according to the norm “Solve your own problems.” Squibs, on the other hand, are conditional “altruists”; they follow the norm “Be altruistic to fellow squibs, punish those who don’t, and punish those who fail to punish.”171 The social norm of the squibs includes a meta-norm - “punish those who fail to punish.” As was noted above, meta-norms have been suggested as a solution to the problem of norm-enforcement, but this solution is also contested since the same problem seems to reappear as a jack-in-the-box at a new level. Consider again the example above of where N={1, 2, …, n} are involved in a repeated multi-player Prisoner’s Dilemma, and they have dyadic exchange relations with each other as well. Suppose that i∈N chooses to defect in the multi-player interaction. Now j∈N\{i} has an opportunity to punish i, but doing so involves a cost to j. Why should j be prepared to carry this cost in order to produce a collective benefit? The meta-norm solution is to suggest that those who fail to punish are themselves punished. If j fails to punish i, then there must be some player h∈N\{i, j}who can punish j. But if punishing j involves a cost to h, then why should h be prepared to do that? It seems that we need a meta-meta-norm, to the effect that those who fail to punish those who fail to punish should also be punished. If h fails to punish j here, then the meta-meta-norm requires some player k∈N\{i, j, h} to 171

Sober and Wilson (1998) p 151.

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punish h. But why should k be prepared to do that? Presumably, because there is a meta-meta-meta-norm that requires…And so on, ad infinitum. The counter-argument against the meta-norm solution is, thus, that invoking meta-norms simply leads to an infinite regress. I shall try two different strategies in order to meet this counter-argument. Let us first consider how the infinite regress occurs. To begin with, there is a first-order norm, ν 1, which says “Cooperate!” or “Be altruistic to fellow squibs!” or something similar. Second, there is some well-defined set T(ν 1) of acts that count as transgressions of ν 1. Further, there is a second-order norm, ν 2, which require agents to punish all proven transgressions of ν 1. Now if ν 2 is a norm, there must be some set T(ν 2) of acts that count as transgressions of ν 2, and the meta-norm solution therefore suggests that there is a third-order norm ν 3 which requires agents to punish all proved transgressions of ν 2. If ν 3 is a norm, however, there must be some set of acts that count as transgressions of ν 3, so there must also be a fourth-order norm ν 4 which requires agents to punish all proved transgressions of ν 3. And so on. Thus, what we get is the following, infinite, sequence of norms and meta-norms, ν 1, ν 2,… (i) ν1 = … (ii) ν i+1 = punish every proved τ∈T(ν i), where T(ν i )is the set of all acts τ that count as transgressions of ν i. 7.6 The package-solution My suggestion is that this objection can be met by offering first-order, second-order and higher-order norms simultaneously, not one by one, but as a composite package. First, we define a set, N, of norms that a society tries to uphold. Second, we define T as the set of all acts,τ, that count as transgressions of any norm ν∈N. Third, we define a meta-norm, ν ’, which says “punish every proved τ∈T”. Finally, we state that ν ’∈N. Thus, we have: (i) N = { ν 1, ν 2,…} (ii) N ⇒ T = {τ1, τ2,…}

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(iii) (iv)

ν ’= “assist in punishing any proved τ∈T”. ν ‘∈ N

Now, what happens if a set N of norms that satisfy (i) – (iv) is offered as a package? How do individual payoffs look when agents are offered a choice between following every ν∈N or not following any ν∈N? Let c be the strategy of compliance, and let d be the strategy of noncompliance. Suppose that everyone else chooses d. If the first-order norm was a norm of contributing to the provision of a public good, we may assume that the public good will not be provided at all, or that it will be provided only to a very limited extent. There will be plenty of punishing to do for the norm-follower, however, so the individual cost of choosing c will be very high. Thus, u(d,0) > u(c, 0). Suppose instead that nearly everyone else chooses c. The public good will be provided. Thus, we assume that u(c,n-1) >u(d,0). Further, since choosing c implies that one is prepared not only to comply with the first order norm, but also with the meta-norm of punishing non-cooperators, free riders suffer a substantial risk of being punished. Still further, since the meta-norm ν ’ refers to itself as well as to all other norms in N, one cannot benefit by being a free rider on the punishing efforts of others. Those who do not punish suffer a substantial risk of being punished themselves. Thus, u(c,n-1) > u(d,n-1). As we can see, the meta-norms game that emerges when norms and metanorms are presented as a composite package does not lead to an endless regress of higher-level n-player Prisoner’s Dilemma. The n-player Prisoner’s Dilemma is characterized by u(c,n-1) >u(d,0) u(d,0) > u(c, 0) u(d,n-1) > u(c,n-1).

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Here we have instead u(c,n-1) >u(d,0) u(d,0) > u(c, 0) u(c,n-1) > u(d,n-1). This is the structure of preferences that is typical of a n-player Assurance game. The problem of establishing effective sanctions, thus, is the problem of achieving a sufficient level of coordination around a suitable package of norms and meta-norms, for norm-compliance and norm-enforcement to become individually rational strategies. Two potential problems must be considered here. First, we should note that this solution rests on a self-referring norm, ν ’. The meta-norm, ν ’,states that one should punish every transgression of every norm ν∈N, including transgressions of the meta-norm,ν ’,itself. Is such self-reference malignant? Not necessarily. Although it is well known that self-referring propositions can sometimes generate paradoxical results, this is not always the case. Self-referring conditions and principles are found in many areas, without paradoxical or otherwise troublesome effects. On the front paper of the edition of Leviathan that I have used, the following condition is found: This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.172

I believe that law normally contains a number of self-referring principles. A constitution, for example, should contain principles for how the constitution can be changed. Since these principles are part of the constitution, they are also principles for how they can be changed.173 Selfreferring norms, thus are common and do not generally cause problems. 172

My emphasis. This example of self-reference was suggested by Wlodek Rabinowicz. This example was suggested by Lena Halldenius. A few years ago, an organisation that I worked with decided to change its statutes. The new statutes contained a rule stating that statutes could only be changed at the ordinary annual meeting. Some people noted - with more than a slight dash of irony - that the new statutes were passed at an 173

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Let us now return to the assembly of people who were about to terminate their war of every man against every man and enter into a commonwealth. The fourth speaker had just suggested that they needed both a system of sanctions against non-cooperative behavior in general, and a system of meta-sanctions against refusal to contribute in the maintenance of sanctions. The above analysis of meta-norms and the infinite regress applies to their situation as well as a group of people who seek to escape the curse of their n-player Prisoner’s Dilemma by way of a decentralized sanction system. If the system of norms and meta-norms is appropriately designed, such that first-order norms and higher-order norms are offered as a package rather than one by one, the second-order free rider trap can be avoided and an effective sanction system successfully be established. Thus, the objection that maintaining sanctions is itself a Prisoner’s Dilemma can be met and rejected; the meta-sanction game need not degenerate into an endless series of n-player Prisoner's Dilemma games. 7.7 Is government necessary? Everything that has been said so far applies to decentralized sanctions, i.e. sanctions that are organized and enforced by the players themselves, as well as sanctions imposed by a central authority. If the web of overlapping relations that tie people together is strong enough, a group of individuals may develop a suitable system of norms and meta-norms that enable them to effectively police each other. This can be done without assigning the tasks of surveillance and execution of sanctions to a centralized agency. However, whereas such a web of overlapping relations is likely to exist in many kinds of small or medium-size groups, such as a neighborhood, a community, a working team, a company or a tribe, it becomes less likely when the interacting group grows very large. In many cases, we are forced to interact with large groups of more or less anonymous people. Perhaps we

extra-ordinary annual meeting.

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have ongoing dyadic relations only with a small fraction of the large group. Could we then maintain cooperation by way of decentralized sanctions? It is plausible to believe that, at some level of group size, the web of relations becomes too weak and too fragmented to be capable of supporting a self-policing system of norms and meta-norms. Also, for large groups, the information problem might become difficult to handle. The meta-norm solution to the incentive problem involved in maintaining sanctions requires that defections become publicly known. Consider again a case where N = {1, 2, ..., n} are involved in an iterated multi-player Prisoner's Dilemma, and where there is also a network of overlapping, dyadic relations between the players. That is, we do not have to assume that every player i∈N has an ongoing relationship with every other player j∈N\{i}, only that each player has ongoing dyadic relations with some subset of N, and that these subsets overlap. Now suppose that some player i∈N defects, and that this is observed by some other player j∈N\{i} who has an ongoing, dyadic relationship with i. According to the meta-norm, j ought to punish i, although doing so is costly also to j. However, j has reason to punish i only if she expects some other player, h∈N\{i, j} to punish her in case she fails to punish i. But h should be expected to punish j for failing to punish i only if (1) h knows that, according to their social norm, j ought to punish i, i.e. h knows that i defected, and (2) h expects some player k∈N\{i, j, h} to punish her if she fails to punish j. This, in turn, implies that k must know that h ought to punish j, which implies that k must know that i defected and that j failed to punish i, etc. In small groups, where everyone knows and interacts with everyone else, such publicity might be fairly easy to achieve. In large-scale interactions, however, where players are more or less anonymous to each other, it is more likely that there are serious gaps in this chain of knowledge. It is probably wise not to underestimate the efficiency of decentralized sanctions in small group interactions. Empirical studies, for example Elinor Ostrom’s studies of water reservoir regulations in California, show that

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decentralized solutions to collective action problems may often be surprisingly efficient.174 Some of the factors that have been shown to affect the level of cooperation in empirical studies are, • Group size. The level of cooperation decreases with increasing group size. • Communication. If players can communicate before choosing, the probability for successful cooperation increases. • Visibility. If players believe that their choices are visible to others, the level of cooperation increases. If players believe that their choices are anonymous, cooperation decreases.175 Thus, that sanctions are needed to ensure successful cooperation in n-player Prisoner's Dilemma games does not, by itself, imply that a central authority is necessary. To take this further step, which is crucial to the Hobbesian enterprise, the Hobbesian must argue that decentralized sanctions are, in some important cases, insufficient to ensure an acceptable level of cooperation. Group size is apparently a crucial factor here; the need for a centralized sanctioning agency arises with the need to organize cooperation in very large groups, where the visibility, communication and overlapping relations of the members are insufficient to support a system of decentralized community enforcement. However, if we accept the description of the state of nature as a cold war, where every agent must take measures to improve her relative fighting ability, then this involves a dynamic process that is likely to lead to continuously growing group size. An effective way to improve one’s relative power is to form alliances with others. Hobbes acknowledges the possibility of forming alliances in the state of nature in many passages of Leviathan. Thus, for example, Hobbes holds that 174 175

Ostrom (1996). Dawes (1980), Ostrom (1996), Eek (1999).

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if one plant, sow, build, or possess a convenient seat, others may probably be expected to come prepared with forces united, to dispossess, and deprive him, not only of the fruit of his labour, but also of his life, or liberty.176

In his much discussed reply to “the foole”, who holds that compliance with agreements is never rational, even when the other party has already performed her part, Hobbes’s argument suggests that people can join together in protective confederacies, and that it is rational for an agent to comply with an agreement to help her confederates, provided that they have already performed their part, because she may otherwise be excluded from the confederacy. in a condition of war, wherein every man to every man, is an enemy, there is no man can hope by his own strength, or wit, to defend himself from destruction, without the help of confederates; where every one expects the same defence by the confederation, that any one else does: and therefore he which declares he think it reason to deceive those that help him, can in reason expect no other means of safety, than what can be had from his single power.177

It has been held that Hobbes’s views on this subject are inconsistent with his description of the state of nature as a state of war. If it is rational to comply with an agreement after the other party has already complied, then, provided that both agents are rational and know each other to be rational, it must also be rational to comply first. 178 However, in the context of forming small size protective confederacies, Hobbes views seem to be in line with the analysis made here. In a small group, each member’s behavior can easily be monitored. Moreover, the good that the protective confederacy provides is not a non-excludable good. Hobbes’s argument for compliance with agreements in this particular case is - correctly - based on the fact that those who fail to comply can be excluded from the protection that the confederacy provides. Thus, the small size protective confederacy possesses both the information and the means to sustain cooperation by community enforcement. 176

Hobbes (1996) ch 13, p 83. (My italics.) Op. cit., ch 15, p 97. 178 See Hampton (1988) 177

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Unfortunately, the possibility of forming small size protective confederacies does not terminate the state of war. Instead of a war between single agents we have a war between small bands of people. Hobbes was clear about this, and he also saw the potential for an arms race involved in the situation. Nor is it the joining together of a small number of men, that gives them this security; because in small numbers, small additions on the one side or the other, make the advantage of strength so great, as is sufficient to carry the victory; and therefore gives encouragement to an invasion. The multitude sufficient to confide in for our security, is not determined by any certain number, but by comparison with the enemy we fear; and is then sufficient, when the odds of the enemy is not so visible and conspicuous moment, to determine the event of war, as to move him to attempt.179

Thus, in response to the threat posed by other alliances, each protective alliance has a motive to grow larger and stronger. At some level, alliances become too large to be able to sustain internal order and coordinate everyone’s behavior effectively without some central authority. Ultimately, then, the Hobbesian argument boils down to this. A confederacy that is sufficiently large to guarantee an acceptable degree of security to its members cannot keep together without some common power. It will be divided, and its strength will be reduced by internal conflicts. In the face of an enemy it will be weak due to lack of coordination, and in the absence of a common enemy, it will break down as its members “make war upon each other, for their particular interests”. 180 It is certainly a rather cynic and non-idealistic view of the foundations of the state that the Hobbesian argument presents; the need for a state arises to make it possible to mobilize armies.

179 180

Hobbes (1996), ch 17, p 112. Ibid.

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CHAPTER 8

THE IDEA OF A SOCIAL CONTRACT _____________________________________________________________________________

We are all players in the game of life, with divergent aims and inspirations that make conflict inevitable. In a healthy society, a balance between these differing aims and aspirations is achieved so that the benefits of cooperation are not entirely lost in internecine strife. Game theorists call such a balance an equilibrium. Sustaining such equilibria requires the existence of commonly understood conventions about how behavior is to be coordinated. It is such a system of coordinating conventions that I shall identify with a social contract. Ken Binmore181

In previous chapters it has been argued that the establishment of a system of sanctions, offers a solution to certain malignant collective action problems, and hence makes cooperation possible in situations where it would otherwise not be. Such a sanction system can be based on either decentralized community enforcement, or enforcement by a central sanctioning agency, i.e. political government. In the previous chapter it was argued that decentralized sanctions could be effective in small groups, where players are united by a web of ongoing relations at different levels, and where choices are visible so that defections become publicly known. When groups are very large, choices are made anonymously, and the web of relations is weak or fragmented, community enforcement is less likely to succeed in supporting an acceptable level of cooperation. Further, it was argued that the establishment and maintenance of sanctions, whether enforced by a central authority or based on decentralized community enforcement, is itself viable, given the motivational assumptions behind the statement of the original collective action problem. Once established, general obedience and support to such a sanction system

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Binmore (1994) p 6.

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constitutes a stable equilibrium. When such an equilibrium is realized in a population, it can be said to constitute a social contract. However, a critic might object to this use of social contract terminology. In the argument presented, the act of contracting plays no explicit role. Contracting normally involves signing papers or uttering certain words, thereby taking on special obligations. According to the argument presented here, however, a sanction system can be established through a dynamic process, without such explicit signs of agreement or even without a conscious plan among the prospective citizens of the political society. Further, the critic might object, the point of a social contract argument, as this is normally perceived, is to provide justification for political authority, not to explain it. But stability is not legitimacy. The argument that has been presented here, thus, lacks a normative element that is normally associated with the concept of a social contract. 182 In this chapter I will present and discuss an interpretation of the notion of a social contract, which attempts to deal with these objections. According to this interpretation, the notion of a social contract is related to - but not identical to - Lewis’s (1986) notion of a convention, in the sense that it is seen as a special kind of coordinated equilibrium. Starting from Lewis’s account of the relation between convention and social contract, I will discuss in what sense these two concepts coincide or differ, and also to what extent a social contract, viewed as a coordinated equilibrium, could be said to answer questions about legitimacy.

8.1 The idea of a social contract. The idea that political legitimacy can be derived from voluntary agreement has deep historical roots. Thus for example, in Plato’s dialogue Crito, Socrates contends that he is under an obligation not to escape from the death penalty, to which he has been sentenced by an Athenian court of law,

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This objection was raised by Lena Halldenius.

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because he has previously accepted to live under, and benefit from, the laws of Athens 183. According to Socrates’ argument, each citizen, by remaining within the jurisdiction of the laws of her state, tacitly agrees to the legitimacy of these laws. That Socrates, in a particular case, finds the verdict of the court of law to be wrong, does not change the fact that, by not protesting or leaving the city before, he has implicitly acknowledged the validity of the court’s decisions and accepted an obligation to obey. Political legitimacy and obligation, thus, are thought to arise from a tacit contract between each citizen and the laws of the state. Modern social contract arguments often trace their origins back to the classical theories of Thomas Hobbes, John Locke or Jean Jacques Rousseau. According to these theories, political authority is instituted through an original contract, entered into by individuals in a pre-political state of nature in order to establish peace and security and facilitate productive cooperation among them. On Hobbes’s account A commonwealth is said to be instituted, when a multitude of men do agree, and covenant, every one, with every one, that to whatsoever man, or assembly of men, shall be given by the major part, the right to present the person of them all (that is to say, to be their representative;) every one, as well he that voted for it, as he that voted against it, shall authorize all the actions and judgments, of that man, or assembly of men, in the same manner, as if they were his own, to the end, to live peaceably amongst themselves, and be protected against other men. 2. From this institution of a commonwealth are derived all the rights, and faculties of him, or them, on whom the sovereign power is conferred by the consent of the people assembled. 184

John Locke presents a similar view. Men being, (…), by nature, all free, equal and independent, no one can be put out of this estate, and subjected to the political power of another, 183 184

Plato (1997) Hobbes (1996) Ch 18, p 115.

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without his own consent. The only way whereby anyone divests himself of his natural liberty, and puts on the bonds of civil society is by agreeing with other men to join and unite into a community, for their comfortable, safe, and peaceable living one amongst another, in a secure enjoyment of their properties, and a greater security against any that are not of it. (…) When any number of men have so consented to make one community or government, they are thereby presently incorporated, and make one body politic, wherein the majority have a right to act and conclude the rest. 185

In Hobbes’s case, it is not entirely clear whether the act of original institution should be taken literally, as something that has actually taken place, or whether it should rather be interpreted as a metaphor. Although Hobbes speaks of “The act of instituting a commonwealth” and “the consent of people assembled” in a way that seems to suggest that it should be taken literally, there are also passages that could be taken to support a metaphorical reading. Thus, for example, Hobbes writes of the original covenant that it is, as if every man should say to every man, I authorize and give up my right of governing myself, to this man, or to this assembly, on this condition, that thou give up thy right to him, and authorize all his actions in like manner.186

Moreover, as we saw in chapter 4, Hobbes admits that there probably never was a time when the world as a whole was characterized by the conditions of a state of nature. The state of nature, in Hobbes theory, is not primarily a description of a pre-historical period. As Russell Hardin says, we could interpret the state of nature in Hobbes’s theory as being “not an origin but a possible destination of society.”187 Its function is rather to serve as a point of comparison, in order to make us realize the necessity of government. As was noted in the previous chapter, Hobbes also acknowledges that political authority could be established through acquisition by force. According to Hobbes, there is no difference between the legitimacy of a sovereign whose authority is derived from original institution and that of a

185

Locke (1995) sect. 95, p 163. Hobbes (1996) Ch 17, p 114, bold type added. 187 Hardin (1991) p 166. 186

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sovereign who has acquired her power by force. The only difference between the two forms of political authority is this, that men who choose their sovereign, do it for fear of one another, and not him whom they institute: but in this case, they subject themselves, to him they are afraid of. In both cases they do it for fear 188

This is a rather striking claim for a theory that purports to derive political legitimacy from voluntary agreement. If agreement at gunpoint is a valid source of political legitimacy, then how could any exercise of force ever fail to be legitimate? Again, it not entirely clear whether Hobbes’s argument here is that people actually have given their consent by submitting to the conqueror, and that it is such actual, although tacit, consent that creates political legitimacy. I shall suggest that there is an alternative reading of the social contract argument, which renders Hobbes’s views on this matter more natural (and less offensive). It is often taken for granted that, whatever justifying force a social contract argument might have, is based on “the moral principle requiring that we do as we have agreed to do ”189. On the interpretation of the Hobbesian argument that I will defend here, the effective substance is mutual advantage, rather than a moral duty to carry out what we have promised. In fact, I shall hold, the alleged moral obligations that might be thought to arise with the act of promising do not play an essential part in the argument. Hence, the conditions, under which the original agreement is thought to be made, need not be very important. In Locke’s case, matters are quite different. Locke explicitly holds that it is actual consent as such that creates political legitimacy and political obligation. According to Locke, there is nothing that can make a man member of a political society “but his actually entering into it by positive engagement, and express promise and compact”. 190 188

Ibid. Kavka (1986) p 392 190 Locke [1995], sect. 122, p 178. 189

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To Locke, therefore, it is essential that he can render it credible that existing states have actually been founded by acts of positive agreement. As Jean Hampton says, If it turns out that throughout history states have been created in ways that have not involved subjects’ consent and people have taken themselves to be subject to political authority for reasons other than that they have consented to such authority, then the consent that the contract argument requires in order to explain and legitimate that authority has simply not occurred – and the contract argument would have failed to locate the source of political authority.191

8.2 Objections against social contract arguments. There is a set of standard objections that are frequently raised against social contract arguments. First, if the justifying force of a social contract is thought to derive from the mutual exchange of promises, then the most obvious objection is that no such mutual exchange of promises ever occurred. Political authority has not, in general, been founded by people voluntarily deciding together to confer their rights and powers to a common authority in the way that social contract theorists like Locke suggest, but more often by force and conquest. 192 Thus, even if the argument as such was valid, it would fail to provide justification for actually existing states. Moreover, if such an act of agreement had occurred in some distant past, why should such an agreement, made between our distant ancestors, have any normative impact on us now? Consider Locke’s claim that there is nothing that can put a man “into subjection to any earthly power, but only his own consent”193. If this is taken seriously it should be considered necessary that each new generation actually consents to the political authority that they are required to submit to. But this is rarely the case,

191

Hampton (1997) p 65. Lena Halldenius argues for a different interpretation of Locke’s social contract argument. According to Halldenius, Locke’s argument can be reconstructed in terms of hypothetical consent. See Halldenius, “Aspects of ‘political freedom’” (unpublished manuscript, Department of Philosophy, Lund University). 192 See e.g. Hume (1986) p 34, Hampton (1988) p 266. 193 Locke (1995) sect 119, p 176.

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citizenship is not something that we freely choose, but something that we are born into. One possible defense against these objections is to hold, with Socrates, that merely remaining within the borders of a state, and submitting to the authority of its laws without protesting, amounts to tacit agreement. Although, historically, states have been founded by force, people have, since then, submitted to, benefited from and hence implicitly accepted their authority. Arguments from tacit agreement avoid the most obvious objections based on historical inaccuracy, but face a number of other problems 194. Thus, for example, it is unclear why merely remaining within the territory of a state, should count as tacit consent. If the legitimacy of the state is contested, then taking its authority over a certain territory for granted, and requiring dissenters to leave that territory, seems to beg the question. Further, claiming that a person tacitly agrees to the authority of the state, unless she leaves its territory, implies that there is a realistic exit option available to her. This may often be questioned. Thus, for example, Hume asks, [c]an we seriously say, that a poor peasant or artisan has a free choice to leave his country, when he knows no foreign language or manners, and lives, from day to day, by the small wages which he acquires? We may as well assert that a man, by remaining in a vessel, freely consents to the dominion of the master; though he was carried on board while asleep, and must leap into the ocean and perish, the moment he leaves her. 195

And what if a person explicitly disagrees, but obeys the laws out of fear of being punished? Can such a person really be said to have tacitly agreed to the exercise of force against her, in spite of her explicit disagreement? Locke develops a tacit agreement argument, which is somewhat more restrictive than the one ascribed to Socrates. According to Locke, simply 194 195

See e.g. Kavka (1986) pp 391-398. Hume (1986) p 36.

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remaining within the state’s territory and submitting to its laws does not amount to tacit consent. It is necessary that there is also some “positive engagement” such as possession of land or actively enjoying something that is within the state’s dominion, such as “travelling freely on the highway”196. We shall leave it an open question whether there are versions of the argument from tacit consent that effectively disarm the objections that are raised against it. Going deeper into this issue here would only take us too far off on a sidetrack. As we shall see, there is another way of understanding the social contract, which better fits with the Hobbesian line of reasoning, and which also avoids the objection of historical inaccuracy. Instead of interpreting the social contract as a historical fact, it could be treated as a hypothetical device, intended to demonstrate what individuals ought rationally to agree to, given certain counterfactual conditions. This is the kind of argument that John Rawls presents in A Theory of Justice.197 On Rawls’s account, idealized agents, placed in a fictitious Original Position, are to agree on the principles of justice that are thereafter to govern their society. The point of the argument is to spell out the implications of the moral ideals of fairness and equality. The fictive Original Position, which is very far from a Hobbesian or Lockean state of nature, is specially designed so as to “incorporate commonly shared presumptions” about justice. Modern commentators have sometimes suggested a similar, metaphorical, reading of Hobbes’s social contract argument. Thus, for example, Gregory Kavka holds that Hobbes is essentially a hypothetical social contract theorist. For him, the social contract is not an actual historical event, but a theoretical construct designed to facilitate our understanding of the grounds of political obedience.198

196

Locke (1995) sect 119, p 176. Rawls (1972) 198 Kavka (1986) p 22. 197

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I shall follow Kavka in this. Or rather, I shall hold that a metaphorical reading of the Hobbesian social contract argument is possible and meaningful, leaving the issue of what Hobbes’s own real intentions was to the side. The principal aim of this essay is not to defend a particular reading of Hobbes’s text as the historically correct one, but to investigate the plausibility of a certain type of argument for the legitimacy of the state. Unlike Rawls’s argument, however, the point of the Hobbesian hypothetical contract is not to demonstrate the implications of certain preconceived moral presumptions, but rather to spell out the implications of individual rationality in a complex strategic situation. The validity of arguments from hypothetical consent is quite controversial, however. For example, Ronald Dworkin has formulated a counterargument that is often taken to be devastating. A hypothetical contract, Dworkin says, is not simply a pale form of an actual contract; it is no contract at all.199

Jean Hampton expresses the same thought, in paraphrasing Samuel Goldwyn. Hypothetical consent, she says, “is not worth the paper it’s not written on”200. Dworkin contends that, since hypothetical contracts cannot be morally binding, the idea of the social contract adds no independent argument for the principles of justice that Rawls puts forward. Hypothetical agreement is simply a device that is used ”to make a point that might have been made without that device”. 201 However, the objection that hypothetical agreement is no agreement at all, has its bite from the assumption that the eventual justifying force of a social contract argument is derived from a moral duty to fulfil promises. If this assumption is accepted, then the fact that no promises were ever given is,

199

Dworkin (1989) “The Original Position” in Norman Daniels (ed.) Reading Rawls, p

18. 200

Hampton (1997) p 66. In a note, Hampton claims that the paraphrase “is often attributed to Robert Nozick.” 201 Dworkin (1989) p 18.

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of course, devastating. One cannot, reasonably, be morally bound by promises that have not been given. If we reject that view, however, it is less obvious that its hypothetical character must necessarily render a social contract argument invalid. What we must do then, however, is to focus on some other aspect of the argument, which might account for the justifying force that we ascribe to it. The following section will discuss the relation between the notion of a social contract and that of a convention, as defined by Lewis (1986), and propose an interpretation of the social contract, which makes it related to, but not identical to, a convention. Since the social contract argument, so interpreted, does not appeal to a moral duty to fulfil promises, it is immune against the standard objections that have been considered here. The strength of the argument must, hence, be judged from other points of view. 8.3 Contract by convention. David Lewis (1986) characterizes the general concept of convention by quoting Hume. “Convention”, Lewis says, “turns out to be” a general sense of common interest; which sense all the members of the society express to one another, and which induces them to regulate their conduct by certain rules. I observe that it will be to my interest [e.g.] to leave another in the possession of his goods, provided he will act in the same manner in regard to me. When this common sense of interest is mutually expressed and is known to both, it produces a suitable resolution and behavior. And this may properly enough be called a convention or agreement betwixt us, though without the interposition of a promise; since the actions of each of us have a reference to those of the other, and are performed upon the supposition that something is to be performed on the other part.202

Lewis’s own discussion starts from the theory of games of pure coordination. For a very simple example of a pure coordination game, consider a situation where two cars meet on a highway. Each driver can 202

Lewis (1986) p 4, referring to Hume: A Treatise of Human Nature III.ii.2. In Hume (1995), which is the edition that I use, there are some marginal differences, e.g. concerning spelling.

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choose to drive in the left or the right lane of the road. Each driver’s sole interest, we assume, is to avoid collision; i.e. no one has a special preference for either right or left. The situation can be illustrated as in fig 7.1. Left

Right 1

0

Left 1

0 0

1

Right 0

1 fig. 7.1

This game has two pure strategy equilibria, (Right, Right) and (Left, Left), and one mixed strategy equilibrium where each choose Right with probability 0,5 and Left with probability 0,5. The mixed strategy equilibrium is obviously worse for both than any of the pure strategy equilibria. There are several ways, however, that agents who face this kind of situation can improve their chances of coordinating on one of the pure strategy equilibria. First, if there is a possibility of pre-play communication, they may simply agree that either both drive to the right or both drive to the left. Once such an agreement is made, it is self-policing in the sense that no one has an interest in deviating from it. Second, even without communication, players might sometimes be led to coordinate their behavior because some particular choice of strategy, for some reason, appears to be naturally salient. 203 Third, if there is no preplay communication, but similar situations have occurred in the past, each of the players might form expectations about the behavior of other on basis of the antecedents. If the situation is repeated a sufficient number of times, we should expect that sooner or later a certain regularity will be established, such that all drivers go either to the left or to the right. When 203

See Schelling (1960)

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such regularity has been established, such that nearly everyone conforms to it, and it is common knowledge that nearly everyone conforms to it, it constitutes what Lewis labels a convention. Conventions may, of course, emerge in a group of just two players. A married couple, or a couple of best friends, may develop a number of conventions among themselves, such as private linguistic codes and behavioral rules. Most (politically) interesting conventions, however, involve larger numbers of agents. Consider the following representation of an n-player coordination game (fig. 7.2), which might be thought of as an n-player generalization of the game of driving to the right or to the left on the highway. Left

Payoff

Right

n -1

Number of others who choose R

Fig 7.2 Again, the game has two pure strategy equilibria, all choose Left or all choose Right, as well as a mixed equilibrium corresponding to the intersection point. The horizontal line can be taken to represent the payoff for the mixed strategy of going to the right with probability 0,5 and to the left with probability 0,5. However, the intersection equilibrium is unstable in the sense that if one single player were to deviate from her equilibrium strategy, then everyone else would have reason to follow her example. If, for example, by chance, one extra player chose to drive to the right, then right driving yields a strictly higher payoff than either left-driving or the

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mixed strategy. Hence right driving is each players best response to the new situation. Thus, if the game was repeated a number of times, and each player based her expectations about the behavior of other players on the previous play, we should expect that the situation would, relatively soon, stabilize on one of the two pure equilibria. Again, once such regularity has been established, it constitutes what Lewis calls a convention. The same effect could, of course, be reached immediately if players were given the opportunity to communicate prior to playing. As Lewis notes, “a convention is a regularity in behavior which holds as if in consequence of an agreement so to behave.”204 But actual pre-play communication, i.e. actual agreement, is not necessary for the coordinated equilibrium to occur. Lewis defines the concept of convention in the following way A regularity R in the behavior of members of a population P when they are agents in a recurrent situation S is a convention if and only if it is true that, and it is common knowledge in P that, in almost any instance of S among members of P, (1) almost everyone conforms to R; (2) almost everyone expects almost everyone else to conform to R; (3) almost everyone has approximately the same preferences regarding all possible combinations of action; (4) almost everyone prefers that any one more conform to R, on condition that almost everyone conform to R; (5) almost everyone would prefer that any one more conform to R’, on condition that almost everyone conform to R’, where R’ is some possible regularity in the behavior of members of P in S, such that almost no one in any instance of S among members of P could conform both to R’ and to R.205

It follows from this definition that if nearly everyone would conform to R’ instead of R, then R’ would be a convention. This is an aspect of convention that Lewis emphasizes: “there is no such thing as the only

204 205

Lewis (1986) p 88. Lewis (1986), p 78.

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possible convention.”206 Thus, every convention is arbitrary in the sense that there is at least one alternative possible convention. Peter Vanderschraaf suggests that the concept of convention can be defined more generally as a “correlated equilibrium (Aumann 1974, 1987) satisfying a public intentions criterion: Every agent wants his intended action to be common knowledge.”207 The class of conventions defined in this way contains the class of conventions that is comprised by Lewis’s definition, but also includes situations that are not comprised by Lewis’s definition. I find Vanderschraaf’s generalization very reasonable. However, since I intend to start from Lewis’s discussion of the relation between convention and social contract, I will here stay with Lewis’s classical definition. I believe that this restriction is not of essential importance to the conclusions, but it simplifies the discussion. Now, what is the relation between convention and social contract? Lewis holds that the two concepts “overlap heavily” but are not equivalent.208 Lewis as defines a social contract any regularity R in the behavior of members of a population P when they are agents in a situation S, such that it is true, and common knowledge in P, that: Any member of P who is involved in S acts in conformity to R. Each member of P prefers the state of general conformity to R (by members of P in S) to a certain contextually definite state of general nonconformity to R, called the state of nature relative to social contract R.209

Consider a situation where we live in general conformity with the regularity R of obeying a sovereign. This state is called Status Quo (SQ). Now, Lewis asks us to consider the preference relations each of us has between SQ and two possible alternative states of affairs, namely the State

206

Op.cit., p 70. Vanderschraaf (1995) p 65. 208 Op.cit., p 88. 209 op.cit. p 89. 207

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of Nature (SN), i.e. no one conforms to R, and Lone Disobedience (LD), i.e. everyone else conforms but I do not. If R is a convention, Lewis holds, then according to the above definition of convention, the preference rankings between the alternative states must be such that, for nearly everyone, SQ ? LD. Thus the only linear preference rankings that occur are the following (1) SN ? SQ ? LD, (2) SQ ? LD ? SN, and (3) SQ ? SN ? LD.

If R, on the other hand, is a social contract, then, according to Lewis’s definition, the preference rankings must be such that SQ ? SN. Consequently, the only possible linear preference rankings that occur are (1’) LD ? SQ ? SN, (2’) SQ ? LD ? SN, and (3’) SQ ? SN ? LD.

A is easily seen (2) and (2’) respectively (3) and (3’) are identical, so there is an area where the two concepts might overlap. If R is both a convention and a social contract, then the only preference rankings that occur are (2)/(2’) and (3)/(3’). However, although the two concepts overlap, they are not identical. R might be a convention but not a social contract, and vice versa. In the former case, there are some people who have the preference ranking of (1). Lewis also holds that “If R is a social contract but not a convention, some or all of us have the preference ranking [LD? SQ? SN]”210. Strictly speaking, however, this is not entirely true. Lewis’s definition of convention includes a number of restrictions over and above the preference rankings of SQ, SN and LD. If R is a social contract, and the only preference rankings that occur are those of (2’) and (3’) it might still fail to be a convention because there is no alternative, potential convention R’ that 210

op.cit. p 93. Lewis has a slightly different way of writing preference orderings, which I have here adjusted to fit with the usage of this essay.

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satisfies condition (5) of Lewis’s definition of convention. We shall return to the implications of this restriction below. However, let us first see how these considerations apply to the kind of games that have been considered in the previous chapters of this essay. Consider the following game (fig. 8.3), which is an n-player Assurance game. Suppose that Status Quo is that nearly everyone conforms to R. We could, for example, think of R here as compliance with a package of norms such as the one discussed in the previous chapter, i.e. (v) N = { ν, ν’ } (vi) ν = “contribute to the provision of collective good G” (vii) ν ’= “assist in punishing proved transgressions of norms in N”. The State of Nature, of course, is that no one conforms to R. The preference ordering in this case is SQ ? LD ? SN. Hence R, in this case, might be both a convention and a social contract. Not Conform

Conform

SQ

Payoff

LD

SN

Number of others who Conform

Fig 8.3

Consider instead the following game, which could be taken to represent the conditions of the Sing-Sing prisoners of chapter 6. R here means obedience to the guards’ commands. As before SQ is that nearly everyone conforms to R, SN that nearly no one does. The preference structure in this case is SN

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? SQ ? LD. R might, hence, be a convention but it cannot be a social contract. Conform

Not Conform

Payoff

SN

SQ

LD

Number of others who Conform

Fig 8.4 Finally, consider the following game (fig. 8.5), which is an n-player Prisoner's Dilemma. Here R might be taken to mean compliance with a set of norms that are not backed up by effective sanctions. Again, we suppose that SQ is general compliance with R. In this situation, the preference structure is LD ? SQ ? SN, which implies that R is a social contract but not a convention.

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Not Conform

Conform

LD

Payoff

SQ

SN

Number of others who Conform

Fig. 8.5 Unlike the social contract of fig. 8.3, however, this social contract is not self-policing. We must assume that agents conform to R against their rational self-interest. Hence, we must assume that there is some other kind of motivation than self-interest that accounts for the general compliance. Or, we could of course deny that such a social contract is at all viable. Let us, tentatively, define a contract by convention as a behavioral regularity R that satisfies the conditions for being both a convention and a social contract. By interpreting the social contract as a certain kind of convention, we restrict attention to a particular class of possible agreements, namely those that are self-policing, i.e. those that could be sustained by a group of rational egoists. Agreements, whether actual or hypothetical, to conform to the regularity R in situations like fig. 7.5 are rejected on the grounds that they are utopian. Moreover, invoking such an agreement in order to justify political authority, requires that we attach moral importance to the act of contracting as such, a position that, as was seen in the previous section, is vulnerable to a number of well known objections.

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By considering contracts by convention, we also restrict attention to a particular class of conventions, namely those that are mutually advantageous, which could result from a voluntary choice by a group of rational egoists in a state of nature. Conventions, such as the convention to obey the prison guards of Sing-Sing, are rejected on the grounds that they are not mutually advantageous. Such a concept of contract by convention, thus, invokes self-interest in two different ways. First, it insists that the establishment social order through a social contract must be in everyone’s interest, and second, it insists that social order agreed upon must constitute a stable equilibrium, such that compliance with the terms of the contract, once it has been established, is in accordance with each one’s rational self-interest. Since the social contract is supposed to be self-policing in this way, there is no need to fall back on an alleged moral principle “requiring that we do as we have agreed to do”. Hence, the standard objections against social contract arguments for being historically inaccurate, or for suggesting that make-believe contracts are morally binding, have no bite against this view. A social contract, according to this view, is not merely a stable equilibrium. It is a stable equilibrium, based on mutual expectations, the existence of which everyone benefits from, and hence one which rational individuals might voluntarily agree to establish if it did not exist, and which they have a common interest in preserving, once it does exist. Thus, regardless of how such a convention actually came about, it is meaningful to hold that it is as if it had been established by voluntary consent. Thus, claiming of a currently operating convention in our society that it constitutes a social contract, is to assert that we have a common interest in preserving the structures on which it rests. Holding, for example, that the convention of general obedience to law is a social contract, is to assert that we have a common interest in preserving the structures, i.e. the institutions of justice, which make law obedience individually rational. In contrast to this, if a currently operating convention in our society does not constitute a social contract, it is still the case that we have prudent

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reasons to conform to it, but we do not have a common interest in preserving the structures that make conformity individually rational. On the contrary, we might sometimes, like the prisoners of Sing-Sing, or the subjects of the Ceausescu regime in Romania, have a common interest in having these structures removed or destroyed (although it might not be in anyone’s individual interest to act so as to destroy it). However, interpreting the social contract as a kind of convention is also somewhat problematic. It means that we restrict attention, not only to possible agreements that constitute stable equilibria, and hence are selfpolicing, but also to possible agreements that are arbitrary. Now, arbitrariness might be a reasonable condition for a regularity to be a convention, but it is certainly an odd restriction to put on a social contract. Consider the state of general non-compliance in the Assurance game of fig. 8.3. It might seem to follow from the above discussion that this state, i.e. the state of nature, constitutes a potential convention. However, this is not the case. Although, in a state of near general non-compliance, nearly everyone prefers not to comply herself, it is not true that nearly everyone prefers that “any one more” does not comply. Taking R to be the regularity of complying and R’ to be the possible regularity of not complying, thus, we see that R’ does not satisfy condition (5) in Lewis’s definition, and hence fails to be a potential, alternative convention. Therefore, if the social contract of fig. 8.3 is a convention, there must be some conceivable alternative set of norms, or political regime, that could be established instead of R. In most realistic cases there are, of course, some conceivable alternative to the current regime that amounts to a potential convention (and possibly also a potential, alternative social contract.) Still, it seems odd that the applicability of an argument, which purports to be relevant for the issue of justification, should depend on a criterion of arbitrariness. Intuitively, one might think instead that if a certain regime were the only possible alternative to the state of nature, that would strengthen the justifying argument for that regime. Thus, the concept of contract by convention, as defined above, is too restrictive for our purpose here. One possible response is to relax condition

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(5) in the definition of convention. Instead of requiring that nearly everyone would prefer that any one more conform to some alternative regularity R’, on condition that almost everyone conform to R’, we require only that nearly everyone prefers to not to conform to R given that that almost no-one conforms to R. We thus define a contract by convention* as a regularity R in the behavior of members of a population P when they are agents in a recurrent situation S such that it is true that, and it is common knowledge in P that, in almost any instance of S among members of P, (1) almost everyone conforms to R; (2) almost everyone expects almost everyone else to conform to R; (3) almost everyone prefers that any one more conform to R, on condition that almost everyone conform to R; (4) almost everyone would prefer to not to conform to R in a state of nearly general non-conformity with R. (5) each member of P prefers the state of general conformity to R (by members of P in S) to a certain contextually definite state of general nonconformity to R, called the state of nature relative to social contract R.

8.4 Commonwealth by acquisition. Consider again Hobbes’s contention that a commonwealth could be established by acquisition as well as by institution, and that a sovereign who has acquired her power by force is as legitimate as one who has been given power by voluntary agreement between the prospective subjects. How does this view fit with the idea of political legitimacy as being derived from a social contract? If the force of social contract argument is thought to derive from a moral obligation to obey one’s promises, commonwealth by acquisition is hard to reconcile with the contractarian view. Generally, we do not think that

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agreement under threat, e.g. when the pilot of an airliner agrees to follow the orders of an armed skyjacker, justifies the use of threat. If, on the other hand, the social contract is interpreted as a kind of convention, it is the implications of political power that is important, not the process by which it was established. The point of the social contract argument, on this view, is not to derive political legitimacy from a certain historical process, but in Lewis’s words, to provide “a fictitious dramatization of our present reasons for conforming.”211 Suppose that a group of agents have seized power over a territory by force, in the way that was described in the previous chapter. They enforce cooperation and confiscate part of the surplus value that general cooperation yields. Provided that they leave enough of the surplus value for everyone to be better off than they were before, general obedience to the self-proclaimed rulers satisfies the conditions for being a social contract as defined in the previous section. Although the regime didn’t actually obtain authority through a voluntary agreement between their prospective subjects, one might claim, it is as if they had obtained it by voluntary agreement. If the inhabitants of the territory had been given a free choice between the new regime and the state of nature, then they should rationally have chosen the new regime. Moreover, if the inhabitants were given a free choice of whether the group of rulers should resign from power, leaving the population of the territory in a renewed state of nature, they ought rationally to vote against such a proposal. If the subjects have an opportunity to coordinate their efforts in order to get rid of their masters, it seems that they should reject it. Regardless of how the regime actually came to power, once it is established, the inhabitants not only have prudential reasons to obey its commands, but they also have a common interest in preserving the structures that make obedience individually rational. 211

Lewis (1986), p 88.

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However, there is a problem here that will be further considered in the next section. The claim that the inhabitants of the territory should reject an opportunity to get rid of their self-proclaimed masters rests on the assumption that they can only choose between the current regime and the state of nature. If they can by a coordinated effort get rid of their masters and at the same time institute a more advantageous social contract by convention, then they should not reject it. A group of coordinated agents might, of course, take control over a territory, enforce cooperation among the inhabitants and confiscate the entire surplus value for their own consumption, or even leave a considerable part of the inhabitants worse off than they were before. However, although obedience to such a regime might be a convention in Lewis’s sense, it does not qualify as a social contract. If the inhabitants are worse off than in the state of nature, then, although obedience is individually rational, they do not have a common interest in preserving the structures that make continued obedience individually rational. Thus, the hypothetical social contract argument that has been sketched here defines sharp limits as to how political power can legitimately be used. A thoroughly oppressive regime, e.g. a slave owning elite, an exploiting colonial power or a regime, which, like the Rwandan Hutu-regime of 1995 or the German nazi-regime of the 1930s and 40s, indulges in a policy of genocide against parts of the population, cannot claim that its authority is based on a social contract. These limits are, in fact, implicit even in the theory of Hobbes, although he seems quite reluctant to admit it. Hobbes’s purpose is to provide arguments for absolute sovereignty. He has no interest in discussing the eventual limits to the sovereign’s power, since the power of an absolute sovereign is, per definition, unlimited. Still, there are passages in Leviathan that indicate that there are limits. Thus, for example, Hobbes holds that The obligation of subjects to the sovereign, is understood to last as long, and no longer, than the power lasteth, by which he is able to protect them. For

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the right men have by nature to protect themselves, when no one else can protect them, can by no covenant be relinquished.212

If the end of government, and the motivation for subjects to consent, is taken to be the protection and security of the subjects’, then that should also put limits to the uses of political power that can be considered justifiable. A government might fail to protect its subjects for different reasons. It might be too weak to provide effective protection, in which case Hobbes, presumably, would admit that the subjects take refuge to a stronger power if they can. Or an oppressive government might itself become the main threat to parts of the subjects. Hobbes even admits that a group of people, the lives of whom are threatened by the sovereign, might justly join together and take up arms to protect themselves against the sovereign’s power. 213 Modern commentators, as well as Hobbes’ contemporaries, have noted that, contrary to Hobbes’ explicit defense of absolute power, Leviathan leaves much room for, and actually provides arguments, for rebellion. Jean Hampton, quoting the 17th century critic of Hobbes’ theory, Bishop Bramhall, contends that Leviathan could actually be read as a “Rebel’s Catechism”. 214 If Hobbes is reluctant to admit the limits of political power that follows from deriving legitimacy from self-interest, David Hume is much clearer over the implications of such a derivation. Although Hume is commonly known as a harsh critic of social-contract arguments, and explicitly rejects the idea of a social contract as the foundation of government, Hume’s account of the origins of, and motives for, government comes very close to the account of contract by convention discussed in the previous section. What Hume does reject is the idea that government is founded by or justified through any kind of promise, either express or tacit, between the citizens. On Hume’s account the origins of government are to be found 212

Hobbes (1996) ch 21 p 147. Op.cit. pp 145-6. 214 Hampton (1988) pp 197-207, (1997) pp 51-2. 213

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directly in individual self-interest. The reason for inventing government is that it is mutually advantageous; the reason that individuals obey their government is that it is in their individual interest to do so. But then, according to Hume, As interest, therefore, is the immediate sanction of government the one can have no longer being than the other; and whenever the civil magistrate carries his oppression so far as to render his authority perfectly intolerable, we are no longer bound to submit to it. 215

8.5 Some further objections and limitations. We have seen, thus, that interpreting the social contract as a kind of convention avoids a number of the standard objections against social contract views. The social contract, interpreted in this way, is not just a stable equilibrium. It is a stable and mutually advantageous equilibrium, an equilibrium that people, under certain conditions, could have agreed to coordinate on if they had been able to communicate in advance. It was also shown that the concept is sharp enough to demarcate a clear limit against certain, intuitively unjustifiable uses of political power. The critic might insist, however, that the concept lacks a normative element that is normally associated with political legitimacy and social contract arguments. According to the argument that has been sketched here, if the regularity R is both a social contract and a convention, then nearly everyone has self-interested reasons to conform with R, e.g. obey the laws, provided that nearly everyone else does. However, that a political authority or system of laws is legitimate is normally taken to imply a moral obligation to obey, an obligation that goes beyond the demands of rational self-interest. The argument that has been presented here, though, does not imply such moral obligations. We normally assume, the critic might hold, that if the laws of our society are legitimate, we are under an obligation to obey, even in situations where disobedience would be individually beneficial, e.g. where the risk of being 215

Hume (1978) p 551.

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detected and punished is negligible. The argument that has been presented her does not substantiate that assumption. The argument is entirely based on rational self-interest and cannot support the assumption of moral obligations that go beyond, or against, rational self-interest. This objection is entirely correct, I believe. One effect of accepting contract by convention as a criterion for political legitimacy is to break up the conceptual link between legitimacy and obligation. However, this does not mean that the concept of legitimacy is emptied of all meaningful content. It is perfectly meaningful, I think, to separate the question of what justifies the state’s coercion against its subjects from the question of what is the basis of our (alleged) moral obligation to obey the commands of a legitimate authority. It is possible to argue that the state is justified in exercising coercion against us, but that we have no moral obligations to obey over and above what rational self-interest dictates. It is also possible to argue that we are morally obliged to obey the commands of our government, but that the government is not justified in enforcing sanctions against us if we do not obey. These are two separate issues. However, the social contract argument provided here does not imply that there are no moral obligations above self-interest. Consider, for example, the coordination game of driving to the right or to the left. Once a convention is established, e.g. nearly everyone drives to the right, it is in nearly everyone’s interest to conform to that convention. But apart from these self-interested reasons one might hold that there is also a moral obligation to conform with the convention that is operating in one’s society, since nonconformity will often be harmful to others. By driving on the left side of the road (in Sweden) I risk the lives of my fellow road-users as well as my own. Nothing in the argument that has been presented here excludes the existence of such moral obligations. However, the existence of such moral obligations remains an open question, which must be addressed with other arguments.

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There is another, even more serious, limitation to the kind of social contract argument that have been presented her. Suppose that left driving is, for some reason, slightly preferred to right driving by nearly everyone. However, the convention currently operating, i.e. SQ, is that nearly everyone drives to the right. Instead of fig. 7.2 we would have the following situation (fig. 7.6). Left

Right

Mixed strategy

Payoff

SQ SN

LD Number of others who choose Right

Fig. 7.6 Is the right-driving convention in this case a social contract, although everyone would prefer the alternative, left driving, convention? According to Lewis’s definition it is. Hence, a social contract, although, by definition, Pareto-superior to the state of nature, need not be Pareto-optimal. That this is a rather serious objection is seen by considering the following case. Suppose that there are two alternative claimants, A and B, to political authority over a certain territory. Everyone prefers A to B, but everyone also prefers B to the state of nature. Now, suppose that B is in power and actively and effectively prevents people from coordinating to put A into power. Perhaps individuals who support A are punished. According to the above definition, general obedience to B amounts to a social contract, provided that B’s rule is better for all than a state of anarchy.

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However, this is counterintuitive. It is reasonable to demand that, in holding that a certain regime is mutually advantageous, we should compare with all relevant alternatives, not just one, particularly ugly, alternative. If there are alternative conventions that everyone prefer to the actual one, then it seems odd to hold that it is “as if” the actual convention would have been voluntarily agreed to. If agents had been able to agree in advance, they would, presumably, have agreed on the Pareto-superior convention, not the Pareto-inferior one. For an even worse example, consider the following case. Suppose that people in a certain territory live under a political authority that they have freely chosen and are generally satisfied with. Now, a conqueror enters the scene, grabs power by force, enforces obedience to her commands, and leaves the population of the territory substantially worse off than under their previous regime, but slightly better off than they would be under pure anarchy. In what sense could such a sovereign claim that it is “as if” power had been given her by voluntary agreement? This objection can be dealt with by making the definition of contract by convention more restrictive. According to the definition that was quoted from Lewis above, a social contract is, any regularity R in the behavior of members of a population P when they are agents in a situation S, such that it is true, and common knowledge in P, that: Any member of P who is involved in S acts in conformity to R. Each member of P prefers the state of general conformity to R (by members of P in S) to a certain contextually definite state of general nonconformity to R, called the state of nature relative to social contract R.

We now add the following condition. There is no alternative regularity R’ such that, (1) R’ satisfies the conditions for an alternative convention, and (2) general conformity to R’ is preferred by everyone to general conformity to R. This restriction allows us to avoid the above difficulties. The conqueror that discharges a popular political leader, and establishes a rule that is

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worse for everyone, but still better than pure anarchy, cannot claim her authority to be based on a social contract. As it stands, however, this is a very problematic restriction: In most cases, however good a given regime is, there might exist a political solution that is slightly Pareto-better than the one that exists. So perhaps (2) above should be modified: there should exist no potential convention R’ such that general conformity to R’ is much preferred by everyone to general conformity to R. But how much is much? The social contract argument, as sketched here, provides no answer. As a criterion of political legitimacy, the idea of social contract argument turns out to be extremely thin; it offers almost no guidance at all in a choice between alternative political regimes. At best, it provides us with a conditional argument against anarchy. This objection can be illustrated further by the following example. Suppose that there are two potential regimes that compete for power over a certain territory. Some of the inhabitants prefer regime A to regime B, whereas others prefer regime B to regime A. However, everyone prefers that one of these candidates is in power to the anarchy of a state of nature. Suppose that B is actually in power. Does general obedience to B’s commands amount to a social contract? Presumably it does, just as general obedience to A’s commands would amount to a social contract, if A were in power. Obedience to B’s commands is individually rational, everyone prefers B’s rule to the state of nature, and (provided that A and B are the only alternatives) general obedience to B’s commands is Pareto-optimal. The situation cannot be changed without someone being worse off. The same is true for the case where A is in power. However, the differences between A’s and B’s rule might be huge. Perhaps A’s rule would be characterized by equality, whereas B’s rule is strongly aristocratic, with a small group living in luxury, and the majority living in poverty.

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Thus, even with the more restrictive definition of a social contract, there might be several alternative, non-equivalent, social contracts. The Hobbesian social contract argument, as it has been presented here, is ultimately incapable of selecting between these alternatives. Thus, although it might say something about the reasons for accepting political authority of some kind, it says very little, if anything, about how that authority should be organized.

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