[6] Paul D., Clio and Economics of QWERTY,. Amer.Econ.Rev.Vol.75, pp332-337, 1985. [7] Rosenschein J S. & Zlotkin G., Rules of Encounter, Designing.
The Complexity of Collective Decision Saori Iwanaga & Akira Namatame Dept. of Computer Science, National Defense Academy, Yokosuka, 239-8686, JAPAN, E-mail: {g38042, nama}@nda.ac.jp Tel: +81-468-41-3810 (ext. 2432), Fax: +81-468-44-5911 Abstract This paper discusses the complexity of collective decisions, which consists of simple individual decision. The models we present are particularly valuable on helping to understand situations where collective outcomes do not seem intuitively consistent with the underlying individual motivation preferences. Usually we tend to observe mainly outcomes and assume that the motivations generating them were consistent with rather than opposed or unrelated to them. The large-scale collective behavior emerged through the purposive interactions among self-interested agents. We provide the aggregate models for describing collective behavior in the long run, and show the complex and interesting collective behavior which emerges through the interactions of agents with micromotives. Keywords: contingent behavior, collective decision, complexity, nonlinear dynamics 1. Introduction An agent’s behavior is called purposive behavior, if it is based on the notion of having preference, pursuing a goal, or maximizing his interest. However, in many situations, the goal or purpose of an agent relates directly to other agents. Therefore, their behaviors are also constrained by an environment consisting of other agents who are also pursuing their goals or interests. We call this type of behavior as contingent behavior with micromotives that also depends on what others are doing. We consider the collective decision problem among heterogeneous agents in Fig.1. As shown in this figure, what agents do affects what other agents do, and agents’ behaviors or choices depend on the behavior or the choices of other agents, and which usually do not permit simple summation or extrapolation to the aggregate [8]. To make the connection we usually have to look at the interaction between agents and their environment, that is, between agents and other agents or between agents and the collective. We formulate each local interaction as a game problem, and provide the aggregate model for describing collective behavior. The evolutionary consideration is taken to model the dynamics of the collective behavior. The growing literature on evolutionary models, however, treats agents as automata, merely responding to changing environments without deliberating about decisions or motivations of agents’ [7]. Within the scope of our model, we include models which agents make deliberate decisions by applying rational procedures that guide their reasoning about what to do and also how to decide. A simple relation between collective results and individual motives, that if most agents of a group make the same behavioral decision, we can infer from this that most ended up sharing the same norm or belief about the situation, whether or not they did so at the beginning. However, it will be clear even in the simplest versions of these models that the collective behavior can seem
paradoxical, that is, intuitively inconsistent with the intentions of the individuals who generate them. Conventional models treat the aggregation of individual preferences, and they do not consider how individuals happen to have the preferences they do. There is still a great deal to be done, and those outcomes cannot be determined by any simple counting of motivations. This will be particularly clear in cases where a very small change in the distribution of preferences generates a large difference in the outcome. Analysis focusing only in determination of motivations could not explain such a phenomenon. The collective decision process is guided by the self-interest seeking of many heterogeneous agents. The mechanism has a strong similarity to the nature of the self-organizing and growing process [5]. The growth starts from the set of the unstructured decisions based on different micromotives of agents. However, they are let to self-organize into the whole collective decision by establishing the harmony as a whole. We also argue here that knowing preferences, motives, and beliefs of agents as the components in can only provide a partial condition for the explanation of outcomes of the collective behavior. We also need models of how these individual motivations interact and aggregate. Collective Decision
Agent with micromotive
Fig.1: Concept of collective decision of heterogeneous agents with micromotives 2. Rational Decisions of Heterogeneous Agents A models of collective behavior is developed for situations where actions have two alternatives and the costs and/or benefits of
each depend on how many other agents choose which alternative. The key concept is that of threshold; the number or proportion of others that must make one decision before a given agent does so. The models of this paper treat binary decisions problem where each agent has two distinct and mutually exclusive behavioral alternatives. In most cases the decision can be thought of as having a positive and negative side, deciding to do a thing or not to. The individuals in these models are assumed rational, that is, given their goals and preferences, and their perception of their situations, they act so as to maximize their utility. As shown in Fig.2, at any given moment, a small fraction of the population is exogenously given opportunities to observe the exact distribution in the group, and to take the best response against it. We assume that there are many numbers of random matches within each time period, so that each agent’s average payoff in that period is equal to the expected payoff. There are two features distinguish our approach from traditional approach [3]. First, each agent is not assumed to be rational. Second, each agent is not assumed to be knowledgeable enough to correctly anticipate the other agents’ choices. It cares only about the distribution of behavioral pattern of agents in an organization.
payoff for each strategy of Agent Ai (1 ≤ i ≤ n) is given as shown in Table.1. The expected utilities he votes for ( S 1 ) and he votes against ( S 2 ) are then obtained as follows: U i ( S1 ) = p(t )U i1 + (1 − p (t ))U i2
U i ( S2 ) = p(t )U i3 + (1 − p(t ))U i4
(2.1)
Since each agent makes his decision in order to maximizing his utility, he votes for ( S 1 ) if (2.2) p(t )U i1 + (1 − p(t ))U i2 ≥ p(t )U i3 + (1 − p(t ))U i4 otherwise he votes against ( S 2 ). By rearranging (2.2) we have following relation; p(t ) ≥ (U i4 − U i2 ) /(U i1 + U i4 − U i2 − U i3 )
(2.3)
Agent Ai ’s rational strategy is then given as the function of the common
information
p(t )
and
his
personal
attributes U i1 , U i2 , U i3 and U i4 , which is described: (I) p(t ) ≥ (U i4 − U i2 ) /(U i1 + U i4 − U i2 − U i3 ) : votes for ( S 1 ) (II) p(t ) < (U i4 − U i2 ) /(U i1 + U i4 − U i2 − U i3 ) : votes against ( S 2 ) (2.4)
Agent A
An organization of agents
The parameters in the payoff matrix in Table1 can be also given as follows: U i1 = α i + β i , U i2 = α i
Fig.2: The modeling of a decision problem of an agent in an organization One’s decision to vote for a particular candidate, for instance, may depend heavily on how many others have already decided to do so, partly because of social influence, partly because one does not want to waste one’s vote. One outcome of this situation is what we call bandwagon effects. Here, we consider an organization of agents G = { Ai ;1 ≤ i ≤ N } , in which each agent has the following two strategies: S 1 : votes for, S 2 : votes against
p(t ) S1
1 − p(t ) S2
S1
U i1
U i2
S2
U i3
U i4
The other agents Agent Ai
Table 1. The payoff matrix The proportion of agents in the organization who vote for at the time t is denoted by p (t ), (0 ≤ p (t ) ≤ 1) and the proportion of agents who vote against is then denoted by1 − p(t ) . The associated
,U i3 = 0 ,U i4 = β i
(2.5)
where α i represents the relative preference of Agent Ai over the strategy S 1 . If α i > 0 , Agent Ai prefers to vote for, and the value of α i represents the preference level. On the other hand, if α i < 0 , he prefers to vote against. The parameter β i (> 0) represents Agent Ai ’s preference so that his decision is consistent with the group decision. Using the relation in (2.5), the value of the right hand side of (2.3) can be given as (2.6) ( β i − α i ) /(α i + β i + α i − β i ) = (1 − α i / β i ) / 2 ≡ θ i We define the parameter θ i as the threshold of Agent Ai . Each agent’s rational decision rule given in (2.4) is then described as follows: (I) p (t ) ≥ θ i : votes for (II) p (t ) < θ i : votes against (2.7) These observations imply that each agent is decision can be determined by not only by his preference (his own internal attributes), but also by the common information p(t ) . The threshold is simply that point where the perceived benefits to an agent of choosing the alternatives in question exceed treat of the other. 3. A Dynamic Model of Collective Decision Threshold models share with game-theoretic models the assumption of rational agents with complete information. Much recent literature on collective behavior has reacted against the older
notion that irrationality is the key to explanation; agree that collective behavior often results from rational, sometimes calculated action. A formal account of rational action in situations of mutual interdependence is given by game theory. Actions of a large number of people into the analysis of a two-person game in which each person plays against all the others taken collectively. Threshold models take the elements of collective behavior which game theory handles only with difficulty and makes them central: substantial heterogeneity of preferences and inter dependence of decisions over time. This is possible because the n-dimensional payoff matrix of game theory is replaced by a one-dimensional vector of threshold, one for each agent. This allows enormous simplification in the ensuring analysis. Like all simplifications, this one carries a cost. The payoff matrix of game theory allows us to investigate, for any particular agent. Which outcome maximizes his utility and whether outcomes are Pareto optimal for the whole set of agents? The threshold model does not permit this. When an individual is activated because his threshold is exceeding, he acts so as to maximize his utility under existing conditions. The resulting equilibrium may or may not maximize anyone’s overall utility. From the distribution of thresholds alone, nothing can be said about this. Thresholds do not give information about the utility to an individual of each possible equilibrium outcome. Schelling describes models of binary choice situation where full utility functions are given explicitly. Most of his analysis requires every agent to have the same function, but see the brief discussion of how one might relax this limitation. The time sequence of choices is not treated. We especially observe the phenomena such as how each agent’s rational decision combines with the decisions of the others to produce the macro behavior and some unanticipated results. We especially investigate the long-run behavior of collective decisions where a finite number of agents are repeatedly matched to play a stage game by adjusting their decision over time. Beginning with a frequency distribution of thresholds, the models allow the calculation of the ultimate or equilibrium number choosing each strategy. The stability if equilibrium results against various possible changes in threshold distribution is then considered as follows. We denote the number of agents who have the threshold value of θ i in the organization G by n(θ i ) . We also define the accumulated distribution pattern of the threshold as follows; f (θ ) = ∑ n(θ i ) / N θ i ≤θ
(3.1)
The portion of the agents whose thresholds are less than θ is then given as follows; F (θ ) = f (θ ) / N (3.2) We now describe the dynamic process of collective decision that emerged from the interactions among agents. The proportion
of agents who vote for ( S 2 ) at the time period t is given as p(t ) . By the definition of the rational decision rule in (2.7) and the accumulated function F (θ ) in (3.2), and the proportion of agents who vote for at the next time period is given by F ( p(t )) , and the dynamic change of p(t ) is given by the following dynamics: p (t + 1) = F ( p (t )) (3.3) The aim of the model presented here is to predict, from the initial distribution of threshold, the ultimate number or proportion making each of the two strategies. Mathematically, the question is one of finding equilibrium in a process occurring over time. The collective decision at equilibrium is characterized by the fixed point of the accumulated distribution function, which is given by p* = F ( p* ) (3.4) E
1 .0 0 .3 0 .8 E
2
0 .6
0 .2
F (p (t) ) n (θ )/N
0 .4 0 .1 0 .2
0
E
1
0 0
0 .2
0 .4
θ
0 .6
0 .8
1 .0
0
0 .2
0 .4
0 .6
0 .8
1 .0
p (t)
Fig.3 (a): The distribution pattern of the thresholdθ Fig.3 (b): The convergence and the properties of collective decision 4. The Complexity of the Collective Decision In this section, we characterize the collective decision of with different distribution patterns of thresholds. In the previous section, we showed that heterogeneous agents with different thresholds could be classified into three types, the hard core agent of type1, the hard core agent of type2, and the opportunistic agent of type3. Agents of the type1 choose the strategy S1 without regard to the
other agents’ decisions. Agents of the type2, on the other hands, always choose the strategy S 2 , without regard to the other agents’ decisions. The opportunistic agents, on the other hand, change their decision depending on how the other agents may decide. We consider the distribution pattern as shown in Fig.4 (a). The organization with this distribution pattern of the threshold consists of the hard core of type1 and type2. Its accumulated function is obtained as shown in Fig.4 (b). There are one stable equilibrium E2 and two unstable equilibria E1 and E3. In this case, starting from any initial position, the collective decision always converges to the equilibrium E2. By comparing the collective decision in Fig.3 (b), we can characterize the role of the hard core as follows: If some hard core of type1 or type2 join the organization with the distribution pattern of the threshold as given in Fig.3 (a) their decision becomes decisive, and it does not depend on the initial condition. From this, an agent of the hard core is said to be stabilizing agent. Let consider the case where some opportunistic agents join the organization with the distribution function of threshold given in
3
Fig.4 (a). the new distribution pattern of thresholds is then given in Fig.5 (a). There exist many equilibria as shown in Fig.5 (b), and this organization becomes to be uncertain, since which equilibria can be realized is uncertain. By comparing the Fig.4 (b), with joining some opportunistic agents, the collective decision becomes unstable. From this, an opportunistic agent is said to be a destabilizing agent. E
1 .0
3
0 .3 0 .8
0 .6
0 .2
F (p ( t) ) E
n (θ )/N
0 .4
2
0 .1 0 .2 E
1
0
0 0
0 .2
0 .4
θ
0 .6
0 .8
1 .0
0
0 .2
0 .4
0 .6
0 .8
1 .0
p (t)
Fig.4 (a): The distribution pattern of an organization with hard cores
Fig.4 (b) The convergence to the collective decision
E3 E1
E2
Fig.5 (a) The distribution pattern of Threshold Fig.5 (b) The convergence to the collective decision 5.Conclusion In this paper, we investigated the collective behavior of an organization that consists of many heterogeneous agents. Each agent was modeled to be rational in that he sought to optimize his own interest. We obtained the relation between each agent's rational decision (micro behavior) and the collective decision (macro behavior) as the nonlinear dynamics. We provided a methodology for characterizing the nonlinear dynamics between micro and macro behavior. Sometimes the analysis of the collective behavior is difficult, and it is inconclusive. But even an inconclusive analysis warned against jumping to conclusions about the behavior of aggregates from what one knows or can guess about agent interests or motivations, or jumping to conclusions about agent intentions from the observations of aggregates. By explaining paradoxical outcomes as the result of aggregation process, our model take the strangeness often associated with collective behavior out of the heads of agents and put it into the dynamics of situations. The proposed model may be useful in large numbers of heterogeneous agents. Their greatest promise lies in analysis of situations where many agents behave in ways contingent on one another, where there are few institutionalized precedents and little preexisting structure. Providing tools for analyzing them is part of the important task of linking micro to macro levels of adaptive behavior. In general, results are surprising; how well each agent
does in adapting to his social environment is not the same as how to satisfy a social environment, therefore we also need to discuss how to manipulate the collective decision as a future research. References [1] Arthur R., Increasing Returns and Path Dependence in the Economy, Michigan University Press, 1994. [2] Frank R., Passions with Reason, Norton Publishing Company, 1988. [3] Fudenberg D., Game Theory, The MIT Press, 1991. [4] Grosz. K. & Kraus S., Collaborative plans for complex action, Artificial Intelligence, Vol.86, pp195-244, 1996. [5] Holland G., Hidden Order, Addison-Wisely Publishing, 1995. [6] Paul D., Clio and Economics of QWERTY, Amer.Econ.Rev.Vol.75, pp332-337, 1985. [7] Rosenschein J S. & Zlotkin G., Rules of Encounter, Designing Conventions for Automated Negotiation among Computers, The MIT Press, 1994. [8] Schelling T., Micromotives and Macrobehavior, Norton, 1978. [9] Weibull J., Evolutionary Game Theory, MIT Press, 1996. Mark Granovetter., Threshold Models of Collective Behavior, AJS Vol83 No6, 1978.
