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Article Excerpt
1. Introduction
Cooperation is a widely observed feature of human and animal societies (Ostrom et al. 1999, Fehr and Fischbacher 2003), arising even in populations of nonsentient organisms (Boorman and Levitt 1980). Although it manifests itself in many versions, the crux of all cooperation problems is the notion of a social dilemma: Individuals in a pair, group, community, organization, or society are faced with a choice between two alternative courses of action, one of which is prosocial (e.g., "cooperation") and the other selfish (e.g., "defection"), where the former imposes a greater direct cost or confers less benefit on the individual than the latter. The dilemma arises because each individual is by definition always better off behaving selfishly, but when all individuals do so, the collective outcome is worse for everyone than if prosocial behavior had prevailed. Several decades of mathematical modeling and analysis, laboratory experiments, field studies, and philosophical debates have yielded a variety of mechanisms by which prosocial behavior can arise, of which the following is but a partial list: reciprocity over repeated interactions (Axelrod 1984); group selection (Boyd et al. 2003) and so-called "strong reciprocity" (Bowles et al. 2003, Bowles and Gintis 2004); altruism as an observable signal of (unobservable) fitness (Gintis et al. 2001); reinforcement via stochastic learning (Kim and Bearman 1997, Macy and Flache 2002), social networks (Coleman 1988), or formal organizations (Bendor and Mookherjee 1987); and in public goods games, the specific shape of the production function (Oliver and Marwell (1985). All these proposals, however, focus exclusively on an individual's choice of actions with respect to their interaction partners, treating the choice of partners--the individual's social network--as exogenous.
The main contributions of this paper are (1) to extend the standard modeling framework to include partner choice (what we call interaction dynamics) as well as the usual action choice (behavioral dynamics) in an individual's repertoire of decisions; and, in particular, (2) to examine the effect of a triadic closure bias (Rapoport 1963)--the tendency of an individual to connect to a "friend of a friend"--on both interaction dynamics and behavioral dynamics. Specifically, we introduce and study a model of a multiperson prisoners' dilemma game in which agents interact locally with a small subset of partners defined by a sparse network. Agents not only learn from the behavior of others by imitating the behavior of the best-performing player they observe, but also create and sever relationships over time based on myopic cost-benefit comparisons. We show that the combination of network dynamics and learning can, under some circumstances, resolve what we call the scalability problem. The problem is that although decentralized cooperation may be possible in small groups, it becomes increasingly difficult to sustain in the face of free-riding (Boyd and Richerson 1988) as the group size increases. Thus, mechanisms that bring about cooperation will not necessarily scale with group size (1) (Boyd et al. 2003). By contrast, we show that when players' interaction partners evolve over time in a manner we define below, the fraction of cooperating players in the population tends to be higher in large networks.
We also present several other findings that suggest new, and in some cases counterintuitive, results affecting the fraction of cooperating players in large populations; in particular, that randomness in network dynamics and the lack of information regarding potential partners can have a positive impact on the level of cooperation. Detailed examination shows that there is an important trade-off between local reinforcement and global expansion in achieving cooperation in dynamic networks. Specifically, while unilateral tie severance and consensual tie creation strengthen the local reinforcement of cooperation, triadic closure bias hinders global expansion. These results may help us to understand phenomena such as the rapid growth of online markets like eBay. On the one hand, the feedback mechanism provided by eBay, as well as members' freedom to choose with whom to trade, function as local reinforcement mechanisms that promote cooperative members to interact among themselves. On the other hand, the existing members' apparent willingness to trade with new entrants whose previous, and therefore likely future behavior are unknown to them, acts as a global expansion mechanism. The remainder of this paper is organized as follows: [section]2 motivates the model, distinguishing it from related prior work; [section]3 describes the technical details of the model; [section]4 presents the results of our analysis; [section]5 discusses our findings; and [section]6 concludes.
2. A Network Model of Cooperation
In this paper, we investigate the implications of the following two straightforward observations on individuals' behavior: (1) While cooperation necessarily entails interaction between individuals, all individuals do not interact equally with all others; rather, they can be thought of as interacting via a sparse network of social relationships. (2) Just as individuals alter their behavior subject to the logic of self-interest, they may also alter their choice of interaction partners over time.
Most previous work that considers cooperation on sparse interaction networks treats the network as exogenous with respect to the game being played, representing the network either as some kind of spatial lattice (Nowak and May 1992, Bergstrom and Stark 1993, Eshel et al. 1998) or, more recently, as a partly ordered, partly random network (Watts 1999a, b). These studies assume that the network in question remains fixed for the duration of the game, and is unaffected by it. In many social situations, however, individuals choose not only how to interact with others, but also with whom they interact. Furthermore, these two processes--what we call behavioral dynamics and interaction dynamics, respectively--coevolve in the sense that an individual's behavior is conditioned on the behavior of those with whom he is interacting (i.e., interactions affect behavior), and in turn his choice of partners will be conditioned on some assessment of their past or anticipated behavior (i.e., behavior also affects interactions).
While some limited work has explored endogenously generated relationships in the context of multiplayer games (Skyrms and Permantle 2000, Jackson and Watts 2002), it differs from our own, principally in that it focuses on coordination games, not social dilemmas. Furthermore, to the extent that networks are relevant to the results, quite different aspects of the networks in question are emphasized. In the case of Skyrms and Permantle (2000), the results actually concern the dynamics of pair formation, not networks in the sense usually intended by sociologists and, increasingly, other disciplines as well (Watts 2004). Jackson and Watts (2002) propose a model of interaction dynamics that is similar to ours in the sense that decisions both to create and sever ties are based on myopic cost-benefit comparisons. Our model differs, however, from theirs in that whereas Jackson and Watts focus on tie formation and termination exclusively between randomly chosen pairs, we explicitly incorporate features that are thought to be important to the evolution of social networks. In particular, we introduce triadic closure bias (Rapoport 1963, Granovetter 1973, Watts 1999a among others)--that is, the tendency for individuals to meet a "friend of a friend"--into the interaction dynamics. This bias has two important implications: (1) it makes the evolution of a network nonrandom, and (2) information regarding potential partners is available when an individual meets a "friend of a friend," whereas such information may not be available when an individual meets a stranger.
Because both these works are concerned with notions of collective behavior and also of networks that are substantively different from the kind we consider here, the relevant analyses, although quite general, cannot easily be extended to our case. Nevertheless, our findings are in agreement with the above studies in the general sense that they highlight not only the importance of the coevolution of networks and behaviors in determining possible outcomes, but also the relative speeds of the two modes of evolution (Skyrms and Permantle 2000). Furthermore, we concur with Jackson and Watts (2002) when they note (echoing an earlier warning of Oliver and Marwell 2001) that generalizations regarding prosocial behavior must be carefully qualified because the outcome tends to depend on at least some of the details of both how network and behavior are updated and, of course, the game itself.
We explore the coevolution of networks and collective behavior using a stochastic learning approach (Kim and Bearman 1997, Macy and Flache 2002) in which individuals attempt to optimize their behavior based on some limited memory of their past experience, but are otherwise myopic. Unlike forward-looking models of rationality, which also work on the principle of utility optimization, stochastic learning is a backward-looking approach, and thus assumes much lighter cognitive capabilities on the part of individuals than does traditional rationality. Furthermore, stochastic learning also lends itself naturally to a decision framework in which individuals must choose both actions and interactions in a mutually interdependent manner. That is, each individual evaluates his performance not only relative to his past performance, but also relative to the performance of his neighbors, where performance is now a function both of the payoff derived from participating in dyadic interactions, and also the cost of maintaining the interactions themselves. In keeping with the spirit of stochastic learning, individuals can sever relationships, but they do so selfishly and myopically, based solely on the relative costs and payoffs of each relationship.
3. Model Description
We consider a population of N players, each of whom repeatedly engages in a multiperson prisoners' dilemma game with an evolving subset of other players. We denote by [[GAMMA].sub.i,t] the set of partners with whom player i interacts in period t (=1, 2,...
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