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Cooperation in anonymous dynamic social networks

In the study of social networks, the interplay between network games and network formation is significant yet not well understood. Research in network games seeks to explain strategic interactions between neighbors, whereas research in network formation explores the evolution of link patterns. Our work combines these approaches. We show how cooperative behavior in prisoners' dilemma (PD) interactions can be sustained via the endogenous structure of the social network, demonstrating that the co-evolution of network games and network formation results in new phenomena.
 Early research explained cooperation in static infinitely repeated settings via an application of the Folk Theorem where agents use the threat of defection to sustain cooperation [2]. When agents change partners over time, community enforcement procedures can sustain cooperation through public reputations [4, 6]. If agents are anonymous (no reputation), the community can enforce cooperation by defecting with all partners as soon as any defection is observed [1], thereby punishing defection through a costly contagion.
 In these models, partnerships are determined exogenously. More recent literature explores the effect of allowing agents to choose partners via buildup of trust [3, 5]. We also allow discretion over partners. The novel feature of our model is that it sustains cooperation through the emergence of social capital and holds when agents are anonymous, without the use of potentially costly social enforcement protocols.
 Model: There is a countable set of agents interacting through a directed network. The network is dynamic: nodes are removed at a fixed rate and replaced by new agents. Each agent sponsors a finite number of connections to others, each of which persists until one partner dies or chooses to break it. When a connection is broken, the agent who sponsored it randomly re-matches with another agent at the next time period. In each round, an agent plays a PD with each neighbor, choosing the same action for all neighbors.
 Result: We exploit three assumptions (relaxed in the full version). Agents are: 1) unforgiving: they sever relationships with defectors; 2) consistent: they commit to a strategy at birth; 3) trusting: they always accept proposed links. Under these assumptions, we derive an exact characterization of equilibrium actions. Since relationships to defectors are broken, an agent can build up social capital (a network of long-term relationships) only by committing to cooperation. The rate at which this happens depends on the proportion of cooperators in the network, and the rate at which such agents are looking for new partners. If an agent's expected lifetime is sufficiently high, the promise of social capital - high payoffs achieved by maintaining many relationships may induce (some) agents to commit to cooperation in spite of the short-run temptation to defect. Notably, the model supports the stable co-existence of cooperators and defectors; this occurs when (i) defection is optimal in a world dominated by cooperation, and (ii) the addition of defectors in such a world hurts the payoff to defectors more than the payoff to cooperators. In turn, (ii) requires that the cost to a cooperator of meeting a defector is not too large, and (i) requires either that defecting on a cooperator brings a large gain, or that lifetimes are not too long.