On the size and structure of group cooperation

When time preferences are heterogeneous and bounded away from one, how “much” cooperation can be achieved by an ongoing group? How does group cooperation vary with the group's size and structure? This paper examines characteristics of cooperative behavior in the class of symmetric, repeated games of collective action. These are games characterized by “free rider problems” in the level of cooperation achieved. The Repeated Prisoner's Dilemma games is a special case.We characterize the level of maximal average cooperation (MAC), the highest average level of cooperation, over all stationary subgame perfect equilibrium paths, that the group can achieve. The MAC is shown to be increasing in monotone shifts, and decreasing in mean preserving spreads of the distribution of discount factors. The latter suggests that more heterogeneous groups are less cooperative on average. Finally, in a class of Prisoner's Dilemma games, we show under weak conditions that the MAC exhibits increasing returns to scale in a range of heterogeneous discount factors. That is, larger groups are more cooperative, on average, than smaller ones. By contrast, when the group has a common discount factor, the MAC is invariant to group size.     

Hope springs eternal: learning and the stability of cooperation in short horizon repeated games

This paper considers learning rates in finitely repeated prisoners’ dilemmas. If players think their opponents might be relatively cooperative (e.g., tit-for-tat or grim types), they will cooperate in finitely repeated prisoners’ dilemmas (see Kreps et al., J. Econom. Theory 27 (1982) 245). However, if there are actually no cooperative types, players will eventually learn this and cooperation will break down. This paper shows that this learning is extremely slow, so it will take an extremely long time for cooperation to break down.Thus, suppose the world is either “good” or “bad.” The probability of a grim type is δ>0 if the world is good, and zero if the world is bad. Successive generations pair up to play finitely repeated prisoners’ dilemmas. Players observe play in previous generations and use Bayes’ rule to update their prior, π, that the world is good. We show that, if the world is really bad, then π falls per generation on average. Thus, if δ is small, there is less cooperation if the world is good, but cooperation may become more stable. For a representative 19 period repeated prisoners’ dilemma, beliefs fall one percentage point on average after a thousand generations.To derive these learning rates, we must refine existing results on the sensitivity of repeated games to Kreps et al. (1982) type perturbations. Specifically, we show cooperation is possible in perturbed prisoners’ dilemmas repeated O(log(1/δ)) times. This improves significantly on the O(1/δ) results in previous work. The paper thus provides two new reasons why cooperation tends to be stable, even in short horizon repeated games.     

Asymmetric technologies in contests

I consider a contest between two risk-neutral players over a common-value prize, in which one player has a linear cost-of-effort function and the other a strictly convex cost-of-effort function f. I show that if the value of the prize is above (below) a certain threshold level, then the equilibrium aggregate effort in this contest is larger (smaller) than in a contest in which both players are characterized by the strictly convex cost-of-effort function f, and smaller (larger) than in a contest in which both players are characterized by a linear cost-of-effort function. Therefore, in contrast to the general result in the literature, asymmetry in contests can increase competition.     

Credible communication and cooperation: Experimental evidence from multi-stage Games

It is well known that communication often serves as a facilitator for cooperation in static games. Yet, communication can serve entirely different purposes in dynamic settings as communication during the game may work as a means for renegotiation, potentially undermining the credibility of cooperative strategies. To explore this issue, this paper experimentally investigates cooperation and non-binding communication in a two-stage game. More specifically, two treatments are considered: one with only pre-play communication and one where subjects can also communicate intra-play between the stages of the game. The results highlight a nontrivial difference concerning the effects of pre-play communication between the two treatments. Sending or receiving pre-play messages has a positive and significant effect on cooperation if there is no possibility of intra-play communication. However, this effect is significantly reduced when when intra-play communication is allowed. The results suggest that the credibility of pre-play messages may depend crucially on future communication opportunities.     

Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

Group decision-making in the shadow of disagreement

A model of group decision-making is studied, in which one of two alternatives must be chosen. While agents differ in their preferences over alternatives, everybody prefers agreement to disagreement. Our model is distinguished by three features: private information regarding valuations, differing intensities in preferences, and the option to declare neutrality to avoid disagreement. There is always an equilibrium in which the majority is more aggressive in pushing its alternative, thus enforcing their will via both numbers and voice. However, under general conditions an aggressive minority equilibrium inevitably makes an appearance, provided that the group is large enough. Such equilibria invariably display a “tyranny of the minority”: the increased aggression of the minority always outweighs their smaller number, leading to the minority outcome being implemented with larger probability than the majority alternative. We fully characterize the asymptotic behavior of this model as group size becomes large, and show that all equilibria must converge to one of three possible limit outcomes.     

A belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring

This paper extends the belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring. We first find that the previous belief-based techniques [T. Sekiguchi, Efficiency in repeated prisoners' dilemma with private monitoring, J. Econ. Theory 76 (1997) 345-361; V. Bhaskar, I. Obara, Belief-based equilibria in the repeated prisoners' dilemma with private monitoring, J. Econ. Theory 102 (2002) 40-69] cannot succeed when players' private monitoring technologies are sufficiently different. We then modify the previous belief-based approach by letting the player with smaller observation errors always randomize between cooperate and defect along the cooperative path of the play. We show that with vanishing observation errors, efficiency and a folk theorem can be approximated using our modified belief-based strategies.     

Contest success functions

Summary Tournaments, conflict, and rent-seeking have been modelled as contests in which participants exert effort to increase their probability of winning a prize. A Contest Success Function (CSF) provides each player's probability of winning as a function of all players' efforts. In this paper the additive CSF employed in most contests is axiomatized, with an independence from irrelevant alternatives property as the key axiom. Two frequently used functional forms are also axiomatized: one in which winning probabilities depend on the ratio of players' efforts and the other in which winning probabilities depend on the difference in efforts.I am grateful to three anonymous referees, Jack Hirshleifer, Guillermo Owen, and especially Duncan Luce for their helpful comments, and to the UC Irvine Academic Senate and the NSF for financial support.     

Social optimality and cooperation in nonatomic congestion games

Congestion externalities may result in nonoptimal equilibria. For these to occur, it suffices that facilities differ in their fixed utilities or costs. As this paper shows, the only case in which equilibria are always socially optimal, regardless of the fixed components, in that in which the costs increase logarithmically with the size of the set of users. Therefore, achieving a socially optimal choice of facilities generally requires some form of external intervention or cooperation. For heterogeneous populations (in which the fixed utilities or costs vary across users as well as across facilities), this raises the question of utility or cost sharing. The sharing rule proposed in this paper is the Harsanyi transferable-utility value of the game—which is based on the users’ marginal contributions to the bargaining power of coalitions.     

Cooperation and Institution in Games

Based on recent developments in non‐cooperative coalitional bargaining theory, I review game theoretical analyses of cooperation and institution. First, I present basic results of the random‐proposer model and apply them to the problem of involuntary unemployment in a labour market. I discuss extensions to cooperative games with externalities and incomplete information. Next, I consider the enforceability of an agreement as an institutional foundation of cooperation. I re‐examine the contractarian approach to the problem of cooperation from the viewpoint that individuals may voluntarily create an enforcement institution.     

Bargaining and markets: complexity and the competitive outcome

Rubinstein and Wolinsky (Rev. Econ. Stud. 57 (1990) 63-78) consider a simple decentralised market game in which agents meet randomly or voluntarily and bargain over the terms of trade. They show that any individually rational price can be sustained as a sequential equilibrium even though the model has a unique competitive outcome. Here, I consider Rubinstein and Wolinsky's model and show that if complexity costs of implementing strategies enter players’ preferences, together with the standard payoff in the game, then every equilibrium is stationary/Markov and induces the unique competitive price. Thus, I demonstrate that aversion to complexity may provide a justification for the competitive outcome.     

THE EXPECTED NUMBER OF COOPERATORS IN THE N‐PERSON PRISONERS’ DILEMMA WITH RANDOM ORDER MOVES*

The present paper examines the possibility of cooperation occurring in the N‐person prisoners’ dilemma, played with randomly ordered moves and imperfect information. To take into account imperfect cooperation, the degree of cooperation is evaluated in terms of the expected number of cooperators. It is shown that: (i) the expected number of cooperators is maximized at an equilibrium, a combination of strategies each of which is a type of trigger strategy; (ii) the maximum expected number of cooperators is attained in a state of imperfect cooperation for a range of payoff functions; and (iii) the maximum expected number of cooperators is non‐decreasing when the size of players’ group increases.     

Tit for tat: Foundations of preferences for reciprocity in strategic settings

This paper assumes that in addition to conventional preferences over outcomes, players in a strategic environment have preferences over strategies. It provides conditions under which a player's preferences over strategies can be represented as a weighted average of the utility from outcomes of the individual and his opponents. The weight one player places on an opponent's utility from outcomes depends on the players’ joint behavior. In this way, the framework is rich enough to describe the behavior of individuals who repay kindness with kindness and meanness with meanness. The paper identifies restrictions that the theory places on rational behavior.     

The swing voter's curse with adversarial preferences

We analyze voting behavior in a large electorate in which voters have adversarial state-contingent preferences with incomplete information about the state of the world. We show that one type of voter can suffer from the swing voter's curse à la Feddersen and Pesendorfer [The swing voter's curse, Amer. Econ. Rev. 86 (1996) 408-424], and go on to characterize the symmetric Nash equilibria of this model under different parameter values. We prove that unlike settings with nonadversarial preferences, there are equilibria in which in one state of the world, a minority-preferred candidate almost surely wins the election and thus the election may fail to correctly aggregate information. Indeed, we show that the fraction of the electorate dissatisfied with the result can be as large as .     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

Community enforcement when players observe partners' past play

I investigate whether a large community can sustain cooperation in the repeated prisoner's dilemma by having cheaters punished not by their victims but by third parties. In the setting where players can observe their partners' past play only, I show that cooperation can be sustained by an equilibrium that has the following two properties: players choose their actions independently of their own past play, and they are indifferent between cooperation and defection at all histories. This equilibrium carries over to the finite-population setting and is robust to noise in the process of choosing actions or of recording past play.     

Asymmetric first price auctions

A new approach to asymmetric first price auctions is proposed which circumvents the need to examine bidding strategies directly. Specifically, the ratio of bidders' (endogenous) payoffs is analyzed and compared to the ratio of the (exogenous) distribution functions that describe beliefs. Most of the results are inferred from this comparison. In the existing theoretical literature, assumptions of first order stochastic dominance or stronger imply that the latter ratio has very specific properties, but no such assumptions are imposed here. It is proven that first order stochastic dominance is necessary for bidding strategies not to cross. When this assumption is relaxed in the numerical literature it is done in a manner that leads to exactly one crossing. However, it is straightforward to construct examples with several crossings. Finally, bid distributions will cross in auctions with two bidders whenever second order (but not first order) stochastic dominance applies.     

Cooperation in Stochastic OLG games

This paper builds on Cremer's (Quart. J. Econ. 101 (1986) 33) seminal analysis which shows that (almost) complete cooperation can be achieved as an equilibrium in a game played by overlapping generations of players if the institution in which players cooperate is infinitely lived.We analyze a similar model in which the costs of cooperation are subject to random shocks. Even if these random shocks are very small, the range of parameters for which cooperation can be sustained is decreased considerably in comparison to the deterministic case. Furthermore, we show how the efficient outcome can be approximated if the level of cooperation can be varied continuously and the cooperation technology has decreasing or constant returns to scale, while this is not possible in the case of increasing returns to scale.     

Migration as an antidote to rent-seeking?

We develop a new mechanism through which skilled migration may influence economic performance in the sending country. If agents can choose between acting as rent-seekers and engaging in productive activities, and only productive skills are exportable, a positive probability of migration (to a more secure economy) reduces the relative expected returns from rent-seeking, thus decreasing the proportion of skilled workers who opt for “parasitic” activities. Such an improvement in the allocation of talent may prevail over the loss of skilled workers due to outmigration. However, we show that this result is not robust to the introduction of endogenous protection. If productive workers share their resources between accumulation of productive capital and investment in security, prospective migration may induce a weaker protection against rent-seeking, which in turn might depress average income in the source economy.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.