Public Choice (2005) 123: 171–196 DOI: 10.1007/s11127-005-0266-0
C
Springer 2005
Social Capital and Endogenous Preferences ANDERS U. POULSEN1,∗ and GERT TINGGAARD SVENDSEN1 1 Department of Economics, The Aarhus School of Business, Silkeborgvej 2, DK-8000 Aarhus C, Denmark; ∗ Author for correspondence: E-mail:
[email protected]
Accepted 24 January 2004 Abstract. In this paper we analyze whether social capital can emerge endogenously from a process of preference evolution. We define social capital as preferences that promote voluntary cooperation in a one-shot Prisoner’s Dilemma game. We investigate how the endogenous preferences depend on the amount of information individuals have about each other’s preferences. When there is sufficiently much information, maximal social capital emerges. In general, the level of social capital varies positively with the amount of information. Our results may add to an understanding of the factors that determine a society’s ability to generate cooperative outcomes.
1. Introduction Consider a group of individuals who face a Prisoner’s Dilemma game. Assume there is no central authority (state) who can punish or reward behavior, and individuals cannot sign enforceable contracts with outside parties. Important examples are contributions to public goods and use of commonly owned resources. Nevertheless, individuals often manage to cooperate in these environments; see, for example, Ostrom (1990), Ahn, Ostrom, Schmidt, Schupp and Walker (2001), McChesney (2001), Schneider and Pommerehne (1981), Schneider et al. (1997), Schram (2000), van Winden (2002) and Zak and Knack (2001). How can we explain this cooperation? It is a well known and fundamental insight that if individuals are sufficiently patient, then long-term cooperation can be an equilibrium, even when individuals are completely selfish (see e.g. Osborne and Rubinstein, 1994). There is also an evolutionary literature showing how behavioral programs, such as ‘Tit-for-Tat’ (Axelrod, 1984), under certain conditions can be evolutionarily successful and give cooperative behavior. Other contributions point to sanctioning maintained by social norms, see e.g. Sethi and Somanathan (1996, 2003). Finally, a recent literature allows individuals to have social preferences, such as preferences for reciprocity or aversion to inequity. This can make cooperation rational even in a one-shot Prisoner’s Dilemma game. We refer the reader to Fehr and Schmidt (2002) and Fehr and Fischbacher (2002). All these cooperation-enhancing features have been summarized under the heading social capital. This concept may be defined as the ability to cooperate in a group, thus bringing about voluntary and informal collective good
172 provisions (Coleman, l988; Paldam and Svendsen, 2000). The social capital concept has attracted much attention, and the literature is large and growing; see, for example, Paldam (2000) and Svendsen (2003) for overviews. In this paper we consider the role played by social preferences in generating social capital. Indeed, we think of social capital as preferences that generate voluntary cooperation in a one-shot Prisoner’s Dilemma game. Rather than simply exogenously assuming such preferences, however, we ask: Can social capital, in the form of social preferences, emerge endogenously? This paper provides such an endogenization of preferences and demonstrates that there are evolutionarily stable populations where some individuals have social preferences and where some voluntary cooperation is generated. In this way we provide micro foundations for models that simply assume that individuals have social preferences. Preference evolution does not take place in a vacuum. Individuals interact with each other within institutions, such as markets, hierachies (state, work place) or local communities (families and residential neighborhoods). Institutions set the interaction structure within which individuals interact. Thus institutions determine the extent to which individuals with given preferences will cooperate, and hence determine social capital as we have defined it. Furthermore, whenever preferences in the long run are endogenous, as they are in our approach, the successful preferences depend on the institutional features. Thus, it is ultimately institutions that determine social capital. Throughout this paper our institution is a large population of individuals who are randomly matched in pairs and play a one-shot game Prisoner’s Dilemma game. Moreover, there are no punishment possibilities. We focus on such institutions since the endogenous emergence of social capital seem most relevant when the population is large and interaction is infrequent. One institutional aspect that we consider is the amount of information that individuals have about each other’s preferences. This amount of information can be interpreted as the degree of anonymity of interaction. The factors that determine the information are discussed in the following sections. It turns out that whenever there is some probability that individuals recognize each other’s preferences, some social capital emerges. Moreover, the more likely individuals are to be informed, the more social capital emerges. As soon as a certain information threshold is crossed, the preferences that emerge generate maximal social capital. These results may add to the literature that seeks an understanding of the factors that determine a society’s level of social capital. Note that we only consider a single aspect of social capital, namely the significance of preferences being endogenous. We also focus on a single institutional aspect only, the amount of information people have about other people’s preferences. The rest of the paper is structured as follows. The section, The Basic Model, sets up the model. In the Perfect Information section we assume that information about preferences is perfect. We derive the endogenous level of social
173 capital, discuss and interpret the results and study an extension of the model. In the section that follows, we determine social capital when there is imperfect information about other individuals’ preferences. Our results are discussed in Discussion section. The Empirical Relevance section discusses the empirical relevance of our results and gives an application to the Samaritan’s Dilemma (Buchanan, l975). Related contributions are described in the Related Work section. We conclude in the Conclusion section. All proofs are in the Appendix. 2. The Basic Model 2.1. The Prisoner’s Dilemma Throughout this paper we assume there is large (formally infinite) population of individuals. Time is continuous and each time instant individuals are randomly paired. Each individual then chooses an optimal action given his preference and given his information about the opponent. Any two matched individuals play the following simultaneous move oneshot Prisoner’s Dilemma game, shown in Table 1. Table 1. The money payoffs in the Prisoner’s Dilemma game C
D
C
1
b
D
a
0
where ‘C’ and ‘D’ stand for Cooperate and Defect, respectively. The money payoffs satisfy the conditions a > 1 > 0 > b. In the following, we think of social capital as preferences that, given the institutional set-up (in this paper, the amount of information) yield cooperation as a rational outcome in the one-shot Prisoner’s Dilemma game. Any preference distribution has some (possibly zero) social capital associated with it. We measure the level of social capital of a preference distribution by the frequency of mutually cooperative (C, C) outcomes in the population. We say that there is maximal social capital when this frequency equals one. 2.2. Preferences and Information 2.2.1. Preferences We consider the following preference types: • The Materialist (M) preference type prefers to defect, no matter what the opponent does.
174 • The Reciprocator (R) type prefers to cooperate if he believes the opponent will do the same, otherwise he prefers to defect. • The Altruist (A) preference type prefers to cooperate, no matter what the opponent does. In what follows we will, for convenience, often refer to a preference type as a “type”. The Materialist type always defects and the Altruist type always cooperates. The crucial difference between the Reciprocator and the Altruist type is that whereas the former is conditionally kind, the latter is unconditionally kind. 2.3. Information We assume initially that an individual, when matched with another individual, perfectly observes the opponent’s type before making a choice. In the Imperfect Information About Preferences section we relax this assumption. How should an assumption of perfect (or high) observability be interpreted? Since our context is one of a large population with random and short-lived contacts between individuals, it is not obvious that an assumption of perfect observability is appropriate. One possibility is that an individual’s propensity to cooperate can be deduced from the person’s facial expressions, body posture and from other signals that are (partially) outside the individual’s control; see Frank (1988). The experimental evidence on this is, however, mixed. We refer the reader to Brosig (2002), Frank, Gilovich and Regan (1993), and Ockenfels and Selten (2000). Another possibility is that our game is a reduced form of an underlying and more complicated game. Perfect observability of preferences occurs when the three prefererence types in this game behave in a way that reveals their types (a separating equilibrium). The challenge this interpretation raises is, of course, to study the underlying game and to show that such a separating equilibrium actually exists for plausible parameter values. See also the Discussion section. Finally, a related interpretation is that individuals form beliefs about other individuals’ preferences on the basis of observable signs such as income, skin color, area of residence and so on, and that these are sufficiently correlated with the individuals’ preferences. 2.4. Evolution Let π(i, j) denote the money payoff earned by an individual of preference type i when interacting with an opponent of preference type j, where i, j = A, R, M. These money payoffs depend on the features of the institution within which the individuals interact, and we derive them in the following sections. Let xi , where 0 ≤ xi ≤ 1, denote the proportion of individuals of preference
175 type i, where i = A, R, M. Formally, our goal is to endogenously determine these preference proportions. Let x = (x A , x R , x M ), where x A + x R + x M = 1, denote the preference distribution. Then π(k, x) = i x i π(k, i) is the expected payoff to a k type at the preference distribution x. Finally, π(x, x) = i x i π(i, x) is the average money payoff in the preference distribution x. The dynamic governing the time evolution of the proportion xi is x˙ i = xi [π (i, x) − π (x, x)], where i = A, R, M. This is the Replicator Dynamic (Taylor and Jonker, 1978). The growth rate of individuals of preference type i, where i = A, R, M, is positive if these individuals earn above-average money payoff. This makes preferences endogenous. Postulating such a dynamic on preferences is the “indirect” evolutionary approach, described in the introduction. The interpretation is that those preferences who give their users higher-than-average money payoffs tend to be adopted, or internalized, by more individuals over time. To survive in the economic system, a preference, or internalized norm, must give a sufficiently high level of material welfare. This assumption, that only money matters, does not, however, a priori bias the analysis toward the survival of materialistic preferences: Any sort of preference that leads individuals to a materially superior, or just reasonable, behavior will prosper. 3. Perfect Information In this section, we explore the emergence of social capital when information about preferences is perfect. 3.1. Derivation of Money Payoffs Suppose two Reciprocators meet. If each Reciprocator believes that the other will cooperate, then C is optimal. However, if each Reciprocator believes that the opponent will defect, then D is optimal. Thus there are two pure Nash equilibria, (C, C) and (D, D) (and a mixed one, which we ignore). Which equilibrium is selected? Let us assume that the (C, C) equilibrium Paretodominates the (D, D) equilibrium. Thus, on the basis of payoff-dominance, each Reciprocator plays C with probability one. However, the (D, D) equilibrium can risk dominate the (C, C) equilibrium. Thus, on the basis of the risk dominance criterion (Harsanyi and Selten, l988), each Reciprocator plays D with probability one.1 However, in what follows we assume that two Reciprocators select the payoff-dominant equilibrium with probability one. In the Imperfect Coordination section we allow for some coordination failure.
176 Table 2. The expected money payoffs in the evolutionary game
A R M
A
R
M
1 1 a
1 1 0
b 0 0
It follows that when two Reciprocators meet, each earns money payoff 1. The money payoffs arising from the other encounters are quite straightforward to derive.2 We thus obtain the following matrix. Entry (i, j) gives the money payoff earned by preference type i when meeting an opponent of preference type j, where i, j = A, R, M (Table 2). We see that an Altruist performs well against other Altruists and well against Reciprocators (they all cooperate), but is exploited by a Materialist. The Reciprocator performs well against others of his own kind and against Altruists, like the Altruist. However, unlike the Altruist, the Reciprocator is not exploited by a Materialist. Finally, the Materialist performs best when meeting an Altruist, but gets involved in mutual defection when meeting a Reciprocator. 3.2. Determining the Level of Social Capital We now determine the endogenous level of social capital. Consider a preference distribution where some individuals are Reciprocators and the remaining proportion are Altruists. In this preference distribution all individuals cooperate and hence each preference has the same material returns, a first condition for evolutionary stability. Second, as long as there are not too many Altruists, a Materialist cannot invade the preference distribution. This follows because the Reciprocators defect when meeting a Materialist, so when there are enough Reciprocators the Materialist’s money earnings are too low to invade the population. We have the following result, which characterizes the endogenous level of social capital: Proposition 1. Consider a preference distribution with maximal social capital, i.e., all individuals co-operate. That is, a proportion x A are Altruists, and the remaining proportion, 1 − x A , are Reciprocators: x = (x A , 1 − x A , 0). Then: (a) x is stable for the Replicator Dynamic when the proportion of Altruists satisfies xA
0, two Reciprocators can establish some cooperation. Denote again the money payoff that a Reciprocator earns when playing against another Reciprocator by π(R, R). We have π(R, R) = λ2 + λ(1 − λ)b + (1 − λ)λa + (1 − λ)2 · 0 = λ2 + λ(1 − λ)(a + b).
(1)
The matrix with the money payoffs is shown in Table 3. We make the usual assumption that the money payoffs satisfy a + b < 2. Then π(R, R) < 1 for all λ < 1.3 This implies that an Altruist invades the all-Reciprocator preference distribution. But an all-Altruist population is, as before, itself vulnerable to invasion by Materialists. Once there are sufficiently many Materialists, however, the Reciprocator performs best. Proposition 2. Fix λ ∈ (0, 1). There is a unique stable preference distribution, x ∗ , and all three preferences types are present: xi∗ > 0 for i = A, R, M. This equilibrium is a center, i.e., x ∗ is surrounded by closed orbits. Thus social capital fluctuates permanently. ∗ x ∗ = (x R∗ , x M , x ∗A ) −b(a − 1) (a − 1){1 − π(R, R)} −bπ (R, R) = , , , D D D
Table 3. The expected money payoffs in the evolutionary game
A R M
A
R
M
1 1 a
1 π(R, R) 0
b 0 0
(2)
180 where D = [π(R, R) − 1][1 − (a + b)) − ab and π (R, R) is given in Equation (1) above. Proof. In the Appendix. Whenever the initial preference distribution differs from x ∗ , the proportions of the three types fluctuate permanently. Since the implied cooperation rates fluctuate accordingly, we have fluctuating levels of social capital. It is consequently no longer true that stability of preferences implies maximal social capital. Proposition 2 is illustrated below (Figure 2) where a = 2, b = −1 and λ = 1/2. The essential logic behind and interpretation of Proposition 2 is the same as that of Proposition 1. There are fluctuations in preferences, and hence in social capital, over time. Now, however, the fluctuations do not depend on any evolutionary drift. In particular, whenever there are sufficiently many Reciprocators in society, the proportion of Altruists is sure to grow. Once sufficiently many Altruists have established themselves, it is the Materialists who perform best and hence the proportion of these types grows. The key to the result is that the Altruist induces a Reciprocator to cooperate with probability one. However, two Reciprocators cannot, due to the coordination failure, establish full cooperation. Thus the Reciprocator treats the Altruist better than he treats one of his own type. It underlines an advantage of the Altruist type: Since cooperation is always optimal for the Altruist, one never
Figure 2. Illustration of Proposition 2. a = 2, b = −1 and λ = 1/2. Equilibrium proportions: ∗ x ∗A = 1/4, x R∗ = 1/2 and x M = 1/4.
181 gets entangled in coordination problems with other individuals. Altruism is disastrous when one is surrounded by opportunists. But when most people are reciprocal, it is the Altruists who perform best. 4. Imperfect Information about Preferences In the previous sections we assumed that when two individuals met, each person perfectly observed the opponent’s preference type before deciding whether to cooperate or defect. In The Prisoner’s Dilemma section we listed some situations in which such an assumption is reasonable. In other situations, however, the assumption of perfect information can conflict with a large population and random matching. In this section we accordingly study the endogenous preferences and social capital when there is imperfect information about preferences. We model imperfect information about preferences in a very simple way. When two individuals meet, there are two possibilities: Each person observes the other’s preference type, or each person observes only the preference distribution, x = (x A , x R , x M ). In the latter case, an individual chooses the action that maximizes his expected subjective payoff, given x. The first happens with probability δ, where 0 ≤ δ ≤ 1, and the second happens with complementary probability, 1 − δ. We assume that it is common knowledge which scenario applies. Moreover, two Reciprocators, whenever they observe each other’s preferences, coordinate perfectly on the (C, C) equilibrium. 4.1. Optimal Behaviour When Uninformed An Altruist and a Materialist always plays C and D, respectively, regardless of their information. The Reciprocator type, when uninformed, chooses the action that maximizes his expected subjective payoff. To determine the optimal action, we choose specific values for the Reciprocator’s subjective payoff numbers. Let πiRj denote the Reciprocator’s subjective payoff from outcome (i, j), where i, j = C, D. We assume that the Reciprocator’s subjective payoff from the (C, C), (C, D) and (D, D) outcomes equal the material R R payoff. However, we set π DC = a R , with 0 < a R < πCC = 1. Suppose the preference distribution is such that all Reciprocators play C when uninformed. C is optimal for a Reciprocator when x A + x + bx M > a R (x A + x R ), or xM
x A + bx R + bx M , or xA
δ ∗ , where δ ∗ is given below. A preference distribution with only Reciprocators or Altruists, i.e., with maximal social capital, is stable
Figure 3. Reciprocators’ optimal behavior when uninformed about the opponent’s type. a = 2, b = −1, and a R = 1/2.
183 when there are sufficiently few Altruists: x A < x¯ A (δ). There are no other stable preference distributions. (b) Suppose 0 < δ < δ ∗ . There is a unique asymptotically stable preference distribution, denoted x ∗ , where only Reciprocators and Materialists coexist (see Figure 3). The proportion of Reciprocators is is −b/(1−a R −b). There is some, but not maximal, social capital. (c) If δ = 0, individuals are either Reciprocators and Materialists in any stable preference distribution. There is zero social capital: All individuals defect in all encounters. a−1 a a−1 . x¯ A (δ) = 1 − aδ δ∗ =
(5) (6)
Proof. See the Appendix. Proposition 3 can be interpreted as generalizing Proposition 1. Part (a) shows that social capital emerges as long as individuals have sufficiently much information, δ > δ ∗ . We then have a set of stable preference distributions of the same nature as under perfect information. Reciprocators and Altruists coexist as long as there are sufficiently few Altruists. Indeed, when δ = 1, the set of stable distributions in part (a) is exactly equal to those identified in Proposition 1. However, there is again evolutionary drift. If the proportion of altruism exceeds the threshold x¯ A (δ) then any mutation to the M type takes the preference distribution into the strategy simplex, as described earlier. Considering the lower bound on information, δ ∗ , if the unilateral defection payoff a is small (close to one), the lower bound is close to zero. Social capital then emerges even when there is very little (but still some) information. In contrast, if a is large, the lower bound approaches unity. Social capital then emerges only when information is perfect. A similar logic governs the upper bound on altruism, x¯ A (δ): The larger is a the smaller the proportion of Altruism consistent with social capital. As δ become smaller, the upper bound becomes smaller, too. In the limit, as δ approaches (a −1)/a from above, x¯ A approaches zero, and stability is not compatible with any altruism at all. Part (b) of the Proposition describes the endogenous preferences when there is less, but still some, information available: 0 < δ < δ ∗ . In this case the set of Altruist–Reciprocator preference distributions in Part (a) is unstable, since a Materialist invades. Preference evolution instead leads to a unique preference distribution where there is a certain mix of Reciprocators and Materialists, but no longer any Altruists. In Figure 3, this preference distribution, denoted x ∗ , is at the right endpoint of the L line. Whenever two Reciprocators meet, they cooperate.4 Hence there is some social capital, but not maximal, social capital.
184 Finally, in the situation where there is no specific information about preferences, δ = 0, no social capital can emerge, as stated in Part (c). Here interaction is completely anonymous—a situation that seems as extreme as the one where individuals observe their opponent’s preference with probability one. Note that in the situation of total anonymity, some people can have social preferences in a stable reference distribution. However, since these people behave like the materialistic individuals (they defect), no social capital is created.
5. Discussion 5.1. The Determinants of Evolutionary Drift We have seen that for maximal social capital to emerge, the preference distribution must consist of Reciprocator and Altruist types. Propositions 1 and 3 show that such a preference distribution is stable when there are sufficiently few Altruists. If this condition is violated, Materialists will invade the population and there will be a decline in social capital, until it is endogenously restored. Whether the share of altruists will stay sufficiently below, depends on the evolutionary drift. There are two issues regarding this evolutionary drift: What forces determine the evolutionary drift, and when can it increase the proportion of altruism along the line segment Z (cf. Figure 1)? Second, in a preference distribution composed initially of Reciprocators only, how long time does it take for altruism to spread, such that the population moves leftward along line segment Z and reaches the critical threshold (x¯ A ) of altruism? Regarding the first question, one possibility is complexity costs. The Altruist’s preference is simpler than a Reciprocator preference in the sense that an Altruist treats everybody the same way, while a Reciprocator conditions his action on information about opponents. Thus from a computational point of view, the Altruist preference is cheaper than the Reciprocator preference. Assume that the evolutionary selection of preferences is lexicographic. This means that as long as the material returns of the Reciprocator type is higher than the Altruist type, complexity costs do not matter. However, if material returns are the same, it is computational cost that matters; see e.g. Rubinstein (1986). It follows that the A preference has a higher net expected payoff (net of complexity cost) at any preference distribution on the A–R edge. In Figure 1, the preference distribution therefore moves leftward along the A–R edge. Another factor that can influence evolutionary drift is the role played by important cultural actors, such as parents, teachers, politicians and rock stars. These send messages to children, and to people in general, about what behavior is desirable. If these messages are predominantly altruistic, then altruism spreads in any population on the line
185 segment Z . Thus, starting at the R corner, there is a movement toward the point z. There will, however, be a countervailing effect on the population. Whenever there are mutations to the M preference, and the population is in the interior of the Z line segment, then evolutionary selection brings it back to the line segment—and the proportion of Reciprocators will have increased somewhat. Thus if there are sufficiently many mutations to the M preference, the effect of evolutionary drift can be neutralized, and then social capital will remain at its maximal level. In sum, what matters is the relative size of the forces that promote altruism along the line segment Z , and the forces that inject M mutants into the population. We refer the reader to Binmore and Samuelson (1999) for a theoretical analysis of evolutionary drift. Turning to the second question posed earlier, we have no measure of time in our model. We may be talking about evolution in terms of months, decades or centuries. Nevertheless, it should be emphasized that since we are not here dealing with biological evolution, the time required for a significant change in the preference distribution can be short. This may well be the case for the application described in the Empirical Relevance section. 5.2. Altruism Our result that altruism can survive when there is enough information seems at odds with evidence from experiments. In public goods experiments, for example, there are typically few unconditional cooperators. Instead there is a mix between Reciprocators and Materialists (see, for example, Keser and van Winden (2000), Clark and Sefton (2001), and Fischbacher et al., (2001)). However, we believe one should be careful in directly comparing our results with such experimental findings. First, most public goods experiments have a larger action space than the very simple two-action model that we study here, and altruism in the experiments ought to be defined differently than the stark unconditional cooperation used by us. Second, most experiments take relatively short time, and hence do probably not allow for the evolutionary adjustment process that is crucial in our analysis. Third, an experimental finding that most subjects would defect when believing, or informed, that the opponent defects does not, we believe, negate our finding that altruism can survive when no opponents defect. Indeed, the crucial feature of the evolutionary drift identified and discussed in the previous sections is precisely that there is no (or very little defection). Finally, no experiment has, to our knowledge, incorporated the advantages that altruism has relative to reciprocity, such as co-ordinating on cooperation and low computational cost (see the Imperfect coordination and The determinants of evolutionary drift sections). Notwithstanding these arguments, however,
186 the data from public goods experiments are not inconsistent with our results. Indeed, when there is sufficiently little information, the data are qualitatively captured by our results in the section Imperfect Information About Preferences. 6. Empirical Relevance What is the empirical relevance of the fluctuations in social capital that we identified in the previous sections? One interpretation and application is the Samaritan’s Dilemma (Buchanan, 1975). This game describes a situation where an individual, the potential Samaritan, can help someone else (say a poor person). The Samaritan wishes to help the poor, but realizes that the more help he extends, the lower the incentive of the poor person to take care of himself (find a job). Consider the following societal Samaritan’s Dilemma. There is a distribution of A, R, and M preference types. A person is a tax payer (or employed) or a social welfare recipient (unemployed). Recipients receive transfers from the tax payers. We assume, for simplicity, that individuals can switch between the two roles. To make the following simple we assume that an Altruist and Reciprocator always chooses to be a tax payer, and that a Materialist always chooses to be a recipient.5 If recipients earn more than tax payers there is a inflow into the group of recipients, and conversely. Assume also that there are two possible policies for the welfare state. One, the reciprocal policy, conditions payments to recipients on certain actions the recipients take (say job search). The other, altruist, policy does not condition payments on any such actions. We assume that the extent to which the welfare state implements one or the other policy is proportional to the relative proportions of Reciprocators and Altruists: The more Altruists, the more altruistic the welfare state is (we can think of the tax payers as voting about the policy). Suppose all individuals are tax payers, and that all of them are reciprocal. Then no tax payer can gain from becoming recipient, since the mutant will receive little net assistance. However, a reciprocal tax payer can mutate to the Altruist preference and neither gain nor suffer. That is, there is again drift among tax payers, which can cause tax payers (or their politicians) to adopt an Altruist preference. This accordingly makes the state more altruistic. If sufficiently many tax payers have become Altruists, an individual who mutates to the M type gains from becoming a recipient and is followed by others. As sufficiently many have become recipients, however, revenues, and hence transfers, fall so much that recipients switch back to becoming tax payers again (In Figure 1, the P curve is crossed in a rightward movement). Eventually the system returns to a situation where all, or sufficiently many, tax payers are reciprocal. The empirical data, such as the post WW
187 II growth in the Western welfare states and the general rise in (internal and external) transfers and numbers of recipients, is broadly consistent with this dynamic. We do not, of course, claim that our preference dynamic is the only relevant factor. However, to the extent that a Prisoner’s Dilemma model identifies crucial features of real world decision processes, our characterization can be helpful in thinking about changes in preferences and behavior. 7. Related Work The literature on social capital and its determinants is very large and growing. In this paper we only explored one potential determinant, namely the extent to which social preferences generating cooperation can emerge over time. For some good general discussions of the social concept, we refer the reader to the references given in the Introduction, and to Bowles and Gintis (2002) and Glaeser, Laibson and Sacerdote (2002). There are several papers who use evolutionary methods to explore the emergence of social capital, broadly conceived as cooperation in social dilemma games and trusting behavior in “games of trust.” See, for example, G¨uth and Kliemt (1994), G¨uth (1995), G¨uth, Kliemt and Peleg (2000), Guttman (2003), Sethi and Somanathan (2003), and the references in these papers. Our model is probably closest to Guttman (2000). There individuals play a one-shot Prisoner’s Dilemma game, and there are two preference types: A Reciprocator and an Opportunist type; the latter corresponds to our Materialist. Guttman shows that when information about preferences is perfect, only a preference for reciprocity survives. Since there are no Altruists in his model, the allReciprocator preference distribution is asymptotically stable, and there is no evolutionary drift. He also analyzes the case where an individual does not know other individuals’ preference, but can, at a cost, monitor the opponent. Monitoring means that the opponent can protect himself against exploitation by the opponent. When the cost of monitoring is sufficiently low, he shows that the stable preference distribution contains both Reciprocators and Opportunists.6 There are three differences between Guttman’s and our model. First, we allow, in addition to a reciprocal and a materialistic preference, for an altruist preference. This inclusion is nontrivial since when there is sufficiently much information about preferences, our results differ from those in Guttman (2000). In particular, altruism survives and there can be fluctuations in aggregate cooperation frequencies. The second difference lies in the institutional set-up. In Guttman’s model an individual who co-operated and whose opponent defected can, at a cost, retract his choice and defect, too. As long as the cost of retracting is not too high, reciprocity survives even when individuals cannot learn their opponent’s preference. One interpretation is that individuals
188 can buy insurance against exploitation or that they can punish the opponent. This is not possible in our model. In Guttman’s model the institutional set-up is thus more elaborate than in our model.7 Third, Guttman assumes that two Reciprocator individuals always coordinate on their preferred, cooperative equilibrium. We, on the other hand, also study the case where there is some (arbitrarily small) probability that they fail to do this, and both defect. This modification turns out to make a difference for what preferences are stable (cf. Imperfect coordination section). Our finding that the survival ability of social preferences is weakened when information about preferences is imperfect, resembles the findings by other contributions. See, for example, Engelmann (2001), G¨uth (1995) and Ok and Vega-Redondo (2001). Engelmann (2001) studies interaction between altruistic, reciprocal and materialistic preferences like us, but uses a different model of information transmission. In his model, when two person of the same preference type meet, each perfectly recognizes the other’s preferences. However, if two persons with different preferences meet, neither learns anything specific about the opponent’s preference. In G¨uth (1995), Nature provides information to an individual with a probability that is independent of the probability that the opponent is informed, whereas in our model a pair of individuals are either both informed or uninformed. Adopting the approach in G¨uth (1995) would, however, give essentially the same results in our model. The essential assumption of either approach is that there is some positive probability that both players recognize each other’s preference. In Ok and Vega-Redondo (2001), individuals know only the aggregate distribution of preferences, and they show that only selfish behavior can be evoltuionarily stable. These models of imperfect information can be interpreted as reduced forms of more complicated games with different opportunities for signalling and learning. The information transmission process studied in our paper could be a situation where a separating equilibrium is played (cf. the discussion in the Preferences and information section), but where there is an exogenous risk of communication breakdown, in Engelmann (2001), on the other hand, the interpretation is that individuals with say, a reciprocal preference, take certain actions that allow them to recognize themselves. However, people with other preferences do not observe these actions, or do not know that these actions are relevant and related to preferences at all. Thus non-Reciprocators cannot recognize Reciprocators. They can, however, recognize their own type, on the basis of the actions that are specific to their own type. We refer the reader to the discussion in Engelmann (2001). There are clearly many other ways to model information transmission about preferences and about individuals’ types in general. We believe it is an important task for future research on social preferences and cooperation to study exactly how information is transmitted and processed.
189 8. Conclusion A society’s stock of social capital can be thought of as the characteristics of individuals and the institutions within which individuals interact that promote voluntary cooperation in social dilemma situations. In this paper we think of social capital as preferences that induce individuals to voluntarily cooperate in a one-shot Prisoner’s Dilemma game. In order to analyze whether such social capital can emerge endogenously, we endogenize preferences. Those preferences that give higher than average material returns are adopted by more individuals over time. We show that as soon as there is sufficiently much information about other individuals’ preferences, (some) social capital emerges. The exact configuration of preferences, and the resulting level of social capital, that emerges depends on the exact amount of information. When there is sufficiently much information, we can get fluctuations in society’s cooperation rates over time. A preference for reciprocity tends to be replaced by altruism and the latter is replaced by materialism, after which the pattern can repeat itself. The level of social capital can thus fluctuate. One application of our results is to societal Samaritan’s Dilemma. We give empirical correlates of these endogenous fluctuations. When there is less information available, the process of preference evolution settles down to a configuration of reciprocal and materialistic preferences. There is less social capital than under perfect information. Our findings can be interpreted as providing micro foundations for the assumption or claim that non-selfish preferences can survive in an evolutionary context and that, in general there can be preference heterogeneity. 9. Appendix Proof of Proposition 1: (a) let x denote any preference distribution where a proportion x A are A types and the remaining proportion 1−x A , are R types then everybody cooperates, so average payoff π (x, x) = 1. At x an M-individual earns expected payoff π(M, x) = x A a. when π (M, x) ≤ π(x, x), or x A ≤ 1/a, (x, x) is a symmetric Nash equilibrium. Consider any x satisfying x A ≤ 1/a. To show that x is stable for the Replicator Dynamic, it stuffices to show (see e.g. Weibull, 1995) that x is a Neutrally Stable Strategy (NSS). This means that x satisfies the following two conditions: (i) π (x , x) ≤ π(x, x) for all x and (ii). For any x = x satisfying π(x , x) = π (x, x) we have π(x, x ) ≥ π(x , x ). We already know that condition (i) holds whenever x A < 1/a. To check condition (ii), we note that π (x , x) = π (x, x) only for those x that assigns positive probability to strategy A or strategy R. But for any such x we have π(x, x ) = π (x, x) = 1. Thus condition (ii) is also satisfied, so we may conclude that any x with x A < 1/a is an NSS and hence stable for the Replicator Dynamic. (b) Any interior preference distribution
190 is unstable, since the Reciprocator earns a strictly higher expected payoff than the Altruist type. Thus it remains to consider the edges. Any preference distribution on the A–M edge or on the M–R edge is unstable since expected payoffs differ. Similarly, at any preference distribution on the A–R edge with x A > x¯ A is unstable, since a Materialist can invade. Deriving the Phase Portrait in Figure 1: Here we give the computations for the phase portrait in Figure 1, were a = 2 and b = −1. We compute π(A, x) − π(x, x) = −x R2 + (2 − x A )x R + x A − 1, π(R, x) − π(x, x) = π(M, x) − π(x, x) =
−x R2 −x R2
(7)
+ (1 − x A )x R ,
(8)
− x A x R + x A.
(9)
The roots of the first equation are 1 − x A and 1, and we have π(A, x) > π(x, x) iff x R ∈ (1 − x A , 1); this, however, is impossible since x R ≤ 1 − x A . Thus we have x˙ A < 0 at any preference distribution where x A > 0. Next, we have π(R, x) > π(x, x) iff x A + x R < 1, which holds whenever x M > 0. Thus x˙ R > 0 at any interior preference distribution. We finally consider the equation for type M. We have π(M, x) > π(x, x) for x R ∈ [0, [x¯ R ), and π(M,√x) < π(x, x) for x R ∈ (x¯ R , 1 − x A − x M ), where x R = −(1/2)x A + (1/2) x A (x A + 4), and x¯ < 1. The preference distributions satisfying x R = X¯ R corresponds to the P curve in Figure 1; to the left of the P curve we have x˙ M > 0, and to the right we have x˙ M < 0. Proof of Proposition 2: For convenience, set π(R, R) ≡ π. At any x = (x A , x R , x M ) the expected money payoffs to the A, R and M types are (cf. Table 3): π(A, x) = x R + bx M + x A , π(R, x) = π x R + x A , and π(M, x) = ax A . Solving the system π(A, x) = π(R, x) and π(R, x) = π(M, x), using x A = 1 − x R − x M , gives the solution x ∗ in Equation (2). It is straightforward to verify that 0 < xi∗ < 1 for all i = A, R, M when 0 < λ < 1. There are four equilibria for the Replicator dynamic: The three vertices x = (1, 0, 0), (0, 1, 0) and (0, 0, 1), and x ∗ . We examine the stability of each in turn. The two vertices (1, 0, 0) and (0, 1, 0) are not Nash equilibria, hence they are unstable. The equilibrium (0, 0, 1) is, in fact, a Nash equilibrium. Still, it is unstable. To see this, suppose that some small perturbation takes (0, 0, 1) to x = (0, , 1 − ) where > 0 is a small number. At any such x the R-type earns a strictly higher expected monetary payoff than M. Thus at x the proportion of R types increases at the expense of M types. This is a contradiction of stability. We next consider the interior equilibrium x ∗ . On subtracting 1 (1) [b]from the first (second) [third] column from the matrix in Table 3 in section Imperfect coordination we obtain the equivalent matrix:
191
A R M
A 0 0 a−1
R 0 π −1 −1
M 0 −b −b
Or, in abbreviated form,
A R M
A 0 α δ
R 0 β ε
M 0 γ θ
Denote this matrix by A, with typical element ai j , where i, j = A, R, M, and let x = (x A , x R , x M ) be a column vector. Hofbauer (1981) shows that the three-dimensional Replicator dynamic, x˙ = x[Ax − x T Ax], is equivalent to the two-dimensional Lotka-Volterra dynamic, x˙ = x[a M R + a M M x + a M A y] y˙ = y[a A R + a AM x + a A A y], where x = x R /x A and y = x M /x A . If (x, y) is an interior equilibrium of the Lotka-Volterra dynamic, then ∗ ) = (1/(1 + x + y), x/(1 + x + y), y/(1 + x + y)), x ∗ = (x ∗A , x R∗ , x M
is an equilibrium for the Replicator Dynamic. Conversely, if x ∗ is an interior equilibrium for the Replicator Dynamic, then ∗ /x ∗A ). (x, y) = (x R∗ /x ∗A , x M
(10)
is an equilibrium for the Lotka-Volterra dynamic. Moreover, results about the stability of equilibria for the Lotka-Volterra dynamic carry over to the Replicator dynamic and conversely, via the two transformations given above. See Hofbauer and Sigmund (1998, Section 7.5). Our strategy is to use the transformation (10) on x ∗ and to study the stability of the corresponding equilibrium (x, y) for the Lotka-Volterra dynamic; the equilibrium x ∗ then has the same stability properties under the Replicator Dynamic.
192 Using the transformation in Equation (10), the equilibrium for the LotkaVolterra dynamic is x R∗ 4b(a − 1) ∗ = xA 4bπ ∗ 4(1 − a)(1 − π ) xM . y= ∗ = xA 4bπ x=
(11) (12)
In characterizing the stability of (x, y), we use the characterization of equilibria given in Bomze (1983). Bomze (1983) shows that if βx + θ y = 0, then (x, y) is a center. We compute 4b[−(1 − π )(a − 1) + (a − 1)(1 − π )] = 0. 4bπ
βx + θ y =
(13)
Thus we may conclude that (x, y) is a center for the Lotka-Volterra dynamic. This implies that x ∗ is a center for the Replicator Dynamic. Proof of Proposition 3: In Region C the expected money payoffs are π(A, x) = x A + x R + x M b
(14)
π(R, x) = x A + x R + x M (1 − δ)b
(15)
π(M, x) = x A a + x R (1 − δ)a.
(16)
In Region D they are π(A, x) = x A + x R [δ + (1 − δ)b] + x M b
(17)
π(R, x) = x A [δ + (1 − δ)a] + x R δ
(18)
π(M, x) = x A a.
(19)
We note that for δ = 1, we obtain the expected payoffs from the Perfect information section. Part (a): Suppose δ > (a − 1)/a. We show that a preference distribution x = (x A , 1 − x A , 0) that satisfies x A < x¯ A (δ) (given in (6) above) is a Neutrally Stable Strategy, and hence stable for the dynamic in Region C. We have π(x, x) = 1 for any x with carrier in {A, R}. Moreover, π(M, x) = ax A + a(1 − δ)(1 − x A ), and π (x, x) < π(M, x) is equivalent to x A < 1−a(1−δ) = x¯ A (δ). When this holds, (x, x) is a Nash equilibrium. Moreaδ over, any alternative best reply to x has carrier in {A, R}. For any such alternative best reply, x , we have π(x, x ) = π (x , x ) = 1. Thus x is a NSS. To show that no other preference distribution is stable, consider any interior preference distribution x in Region D. Here type R earns a strictly higher
193 expected payoff than typer A, so x is unstable. Furthermore, any x on the A– M edge or on the R–M edge is unstable. Similarly, in Region C any interior preference distribution is unstable since π (R, x) > π (A, x). Any x on the A–M edge or, on,the R–M edge is also unstable. The same is true for those distribution on the A–R edge with x A > x¯ A (δ), since an M type can invade. Part (b): Consider first a preference distribution x in Region C. From Equations (14–16) we see that R strictly dominates A for all δ < 1. Moreover, type M strictly dominates R when (1−δ)a > 1, i.e., δ < (a−1)/a = δ ∗ . Then M strictly dominates the two other preference types, so the Replicator dynamic raises the proportion of M types at any preference distribution in Region C. Thus, whenever the initial distribution is in Region C, we eventually reach a point on the L line (cf. Figure 3). Suppose then the preference distribution is in Region D. We see from Equations (17) and (18) that R strictly dominates A when δ < 1. Then any interior distribution in Region D is unstable, and any stable distribution is on the R–M edge. But δ > 0 implies that any distribution on the R–M edge in Region D is unstable, too. This follows since R earns strictly higher expected payoff than M on the part of the R–M edge above the L line. Thus we can conclude that whenever the initial preference distribution is in Region D, it eventually reaches the distribution x ∗ = (0, x R∗ , 1 − x R∗ ), with x R∗ = (−b)/(1 − a R − b) (cf. (3)), on the R–M edge (this is where the L line meets the R–M edge). Part (c): Suppose δ = 0. In Region C, M strictly dominates the two other preference types, so any preference distribution is taken to the L line (cf. Figure 3). In Region D, the M and R types earn exactly the same expected payoff, since they behave identically. Moreover, each type strictly dominates the A type. The set of stable distributions is the entire R–M edge above the L line. Here all individuals defect in all encounters. Acknowledgements An earlier version of this paper was presented at the European Public Choice Society’s 2003 Meeting. We gratefully acknowledge financial support from the Danish Social Science Research Coucil (SSF) and thank the Editor and an anonymous referee for their very constructive comments that significantly improved the results and their exposition. We also thank Niels B. Christensen, Martin Paldam and the late Mancur Olson. Of course, any remaining errors are our own. Notes 1. Let πiRj denote the Reciprocator’s subjective payoff from outcome (i, j), i, j = C, D. R R Suppose the Reciprocator’s ranking is πCC = 1 > π DC = a R > π DR D = 0 > πCRD = b. Then the (D, D) equilibrium risk dominates the other equilibrium when a R > 1 + b.
194 2. A-A: Here both persons cooperate. Thus each gets a money payoff of unity. A-R: The Atype plays C; the R-type does the same, since the R-type observes that the opponent will play C and hence his best reply is to do the same. A-M: Here the A type gets πC D = b and the M type gets π DC = a. R-M: Since the M-type always plays D, the R-type does the same. M-M: Here both individuals defect. 3. To see this, note that we can write π(R, R) as λ2 + 2λ(1 − λ)[(1/2)(a + b)] + (1 − λ)2 · 0. The weights λ2 , 2λ(1 − λ) and (1 − λ)2 sum to one, so the conclusion follows. 4. Since the preference distribution is on the L line, an uninformed Reciprocator is indifferent between C and D. We assumed that all Reciprocators played C when uninformed. If instead Reciprocators played D when uninformed, Reciprocators would cooperate only when recognizing each other. 5. Clearly, this may or may not hold in a more realistic model where individuals freely choose their roles given their preferences. However, we believe our simplification is defensible, since our purpose in this section is to illustrate the possibility of drift and fluctuations. 6. Other models assume that information acquisition is costly. We refer the reader to G`‘uth and Kliemt (1994) and Giith et al. (2000). 7. See also Guttman (2003) for an analysis of preference evolution when there is (finitely) repeated interaction between individuals.
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