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.     

A One-Stage Model of Link Formation and Payoff Division

In this paper we introduce a strategic form model in which cooperation structures and divisions of the payoffs are determined simultaneously. We analyze the cooperation structures and payoff divisions that result according to Nash equilibria, strong Nash equilibria, and coalition proof Nash equilibria. We find that no cycle will be formed if a player claims a positive amount for the formation of one of its links and that a player does not necessarily profit from a central position in a cooperation structure. Journal of Economic Literature Classification Numbers: C71, C72.     

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.     

Moral Hazard and Private Monitoring

We clarify the role of mixed strategies and public randomization (sunspots) in sustaining near-efficient outcomes in repeated games with private monitoring. We study a finitely repeated game, where the stage game has multiple equilibria and show that mixed strategies can support partial cooperation, but cannot approximate full cooperation even if monitoring is “almost perfect.” Efficiency requires extensive form correlation, where strategies can condition upon a sunspot at the end of each period. For any finite number of repetitions, we approximate the best equilibrium payoff under perfect monitoring, assuming that monitoring is sufficiently accurate and sunspots are available. Journal of Economic Literature Classification Numbers: C73, D82.     

Cooperation in international environmental negotiations due to a preference for equity

This paper demonstrates that cooperation in international environmental negotiations can be explained by preferences for equity. Within a N-country prisoner’s dilemma in which agents can either cooperate or defect, in addition to the standard non-cooperative equilibrium, cooperation of a large fraction or even of all countries can establish a Nash equilibrium. In an emission game, however, where countries can choose their abatement level continuously, equity preferences cannot improve upon the standard inefficient Nash equilibrium. Finally, in a two stage game on coalition formation, the presence of equity-interested countries increases the coalition size and leads to efficiency gains. Here, even a stable agreement with full cooperation can be reached.     

The role of social and technical excludability for the success of impure public good and common pool agreements: The case of international fisheries

We argue that international fisheries are a prime example to study the incentive structure of forming impure public good and common pool agreements. We consider a fully integrated multiple zone model, in which zones are linked through density-dependent migration. The incentive to accede to Regional Fishery Management Organizations (RFMOs) is related to multiple characteristics. Firstly, the relative patch sizes of the high seas, which is the internationally (publicly) accessible domain, compared to exclusive economic zones, which are state-owned (privately owned). This can be related to the degree of socially constructed excludability. Secondly, the intensity of fish migration between various zones, which can be related to the degree of technical excludability. Thirdly, the growth rate of the resource, which can be interpreted as the degree of rivalry, with a low (high) degree of rivalry approximating public good (common pool) features. We show that, generally, excludability reduces free-riding incentives but also the need for cooperation, a variant of the “paradox of cooperation”. Moreover, we show that the benefit-cost duality between public goods and common pool resources generally holds except for some extreme parameter values for which a low degree of rivalry fosters the success of cooperation. Finally, through a variation of the diffusion matrix, we can also analyze a closed as well as a sink-source system.     

The collective action problem: Within-group cooperation and between-group competition in a repeated rent-seeking game

This paper analyzes the ability of group members to cooperate in rent-seeking activities in a context of between-group competition. For this purpose, we develop an infinitely repeated rent-seeking game between two groups of different size. We first investigate Nash reversion strategies to support cooperative behavior in both groups, before analyzing double-edge trigger strategies which support cooperation in one group only. These last strategies have the property that cheating on the agreement in the cooperative group is followed by non-cooperation in this group and cooperation in the rival group. The main conclusion is that the set of parameters for which cooperation can be sustained within the larger group as a subgame perfect outcome is as large as that for which cooperation can be sustained in the smaller group. Hence, in contrast to Olson?s (1965) celebrated thesis, but in accordance with many informal and formal observations, larger groups can be as effective as smaller groups in furthering their interests.     

Externalities in appropriation: responses to probabilistic losses

Using an appropriation game setting, we examine individual responses to changes in a groups’ vulnerability to a probabilistic loss (L) of a public good. The probabilistic loss parameter entails losing 10, 50 or 90% of the value of the public good that is maintained through cooperation, where the likelihood of the loss decreases in total group cooperation. By design, the expected marginal net benefits to an individual and the expected harm to others depends endogenously on the individuals’ expectations of group cooperation and exogenously on the magnitude of the loss parameter. We find that individual cooperation is greater when forecasts of total group cooperation are greater and where the magnitude of the probabilistic loss is larger. There is, however, an interesting asymmetry in responses by two subgroups. Subjects who are pessimistic regarding total group cooperation decrease cooperation the higher the magnitude of the probabilistic loss and their decisions are tied systematically to changes in their expectations of other’s cooperation. On the other hand, subjects who are optimistic regarding total group cooperation are found to be more cooperative, but their decisions are not systematically tied to changes in expectations of others’ cooperation.     

Finite rationalizability and cooperation in the finitely repeated Prisoners' Dilemma

Using the concept of k-point rationalizability, it is shown that cooperation in the finitely repeated Prisoners' Dilemma can be explained as the outcome of rational behavior if the beliefs of players are limited in terms of their complexity.     

Not just like starting over - Leadership and revivification of cooperation in groups

We conduct a laboratory experiment to study how, after a history of decay, cooperation in a repeated voluntary contribution game can be revived in an enduring way. Simply starting the repeated game over—a simple fresh start—leads to an initial increase of cooperation, but to a subsequent new decay. Motivated by cooperation decay in organizations we study the potential of three interventions of triggering higher and sustained cooperation, which take place at the same time as a restart. Surprisingly, we find that the detailed explanation of the causes of the decay in cooperation of Fischbacher and Gächter (Am Econ Rev 100:541–556, 2010) combined with an advice on how to prevent decay do not have an effect beyond that of just starting over. In contrast, a one-way free form communication message sent by the leader to the followers strongly revives cooperation. We find evidence that repeated free form communication by the leader further strengthens the reviving effect on cooperation. Combining the two previous interventions does not outperform the pure effect of communication. Our content analysis reveals that leader communication is more people oriented than the expert advice.     

Introduction to Repeated Games with Private Monitoring

We present a brief overview of recent developments in discounted repeated games with (imperfect) private monitoring. The literature explores the possibility of cooperation in a long-term relationship, where each agent receives imperfect private information about the opponents' actions. Although this class of games admits a wide range of applications such as collusion under secret price-cutting, exchange of goods with uncertain quality, and observation errors, it has fairly complex mathematical structure due to the lack of common information shared by players. This is in sharp contrast to the well-explored case of repeated games under public information (with the celebrated Folk Theorems), and until recently little had been known about the private monitoring case. However, rapid developments in the past few years have revealed the possibility of cooperation under private monitoring for some class of games. Journal of Economic Literature Classification Numbers: C72, C73, D43, D82, L13, L41.     

On enhanced cooperation

We analyze the issues relative to the formation of sub-unions in a federation, called enhanced cooperation agreements in the European Union. When centralization is not politically feasible, an agreement among a subset of countries may allow such countries to exploit benefits from coordination that would otherwise be lost. Other countries in the federation may object to the sub-union because it changes the status quo; if cooperation at the federal level becomes convenient in the future, the change in the status quo may adversely affect the countries which remained initially outside the sub-union. We show that as long as the countries can commit to coordinate on a policy which takes into account the utility of the excluded countries, sub-union formation may be optimal. The relative advantage of a sub-union towards immediate centralization increases when transfers are costly. On the other hand, if commitment is not possible then the excluded countries may be penalized. We use the results to discuss the newly introduced rules for enhanced cooperation agreements in the European Union.     

Conditional cooperation in a sequential move game

Cooperation is a pervasive social phenomenon, but more often than not economic theories have little to say about its causes and consequences. In this paper, we explore the hypothesis that cooperative behaviour might be motivated by pure self-interest when the “social” payoff in a game is increasing. We report the results of a series of experiments on the Centipede game. The experiments are organised in two subsequent steps. Subjects first participate in a 2-period trust game, randomly matched with unknown partners. We apply the strategy method in order to elicit their social preferences. On the basis of their pre-game behaviour, individuals are divided into three main social groups: selfish individuals, pure altruists and reciprocators. At the second step of the experiment, subjects play a repeated 6-move Centipede game with an increasing final payoff. Each subject plays twice, in a low and in a high-stake Centipede game, and he/she is informed about his/her co-player social preferences. We provide statistical evidence to identify the origin of cooperation within homogeneous and heterogeneous social groups. We construct a Poisson regression model to assess the determinants of the duration of conditional cooperation in the Centipede.     

On Failing to Cooperate When Monitoring Is Private

I consider a repeated prisoners' dilemma where in each period, each player receives an imperfect private signal about his opponent's current action. I show that when players are patient enough, any equilibrium where players use trigger strategies (i.e., do not revert to cooperation once they have started defecting) yields players a value arbitrarily close to the mutual minimax. I also examine the robustness of the result to perturbations of the game. Journal of Economic Literature Classification Numbers: C72.     

On the evolution of conditional cooperation

A long series of laboratory and field experiments, as well as conventional empirical studies, has established that (1) individuals voluntarily provide themselves with public goods at levels exceeding those predicted by the Nash voluntary contributions mechanism, and (2) agents reciprocate increases in the contributions of their counterparts in such settings (conditional cooperation). This paper presents a simple model of the evolution of preferences for conditional cooperation in the presence of a public good, which explains these two empirical findings without employing reputational or group selection arguments. In this model, individuals inherit preferences to match other agents' contributions to the provision of a public good, at some specified “matching rate.” Agents whose preferences induce them to be relatively successful – in material terms – increase in number, from one generation to the next. Under complete information and with randomly matched groups of N agents who have quasilinear preferences over the public good and a private good, the unique evolutionarily stable matching rate is 1, leading to Pareto optimal voluntary provision of the public good, regardless of group size N. The evolutionarily stable matching rate can be viewed as an endogenous social norm.     

Interbranch cooperation and the shadow of the future

Prisoners’ Dilemma (PD) experiments confirm and extend Axelrod’s (The evolution of cooperation. Basic Books, New York, 1984) Shadow of the Future hypothesis: subjects cooperate in infinitely repeated PD, but they also cooperate until near the end in finitely repeated PD. So the extended hypothesis is that cooperation depends on the probability of continued play. Observational tests of this hypothesis, or even applications, have been rare at best. Here we not only apply but test it for interbranch cooperation under separated-powers constitutions, specifically those of the American states, using the end of governors’ final terms as end points and the rate of overridden vetoes as the extreme case of a breakdown in interbranch cooperation. Controlling for a variety of factors, including divided government, we find support for the hypothesis, whose explanation of interbranch interaction fills a gap left open by Madison’s Federalist 51: how republican government can control itself when what is needed is “energy” more than safeguards.     

Religious rules as a means of strengthening family ties: Theory and evidence from the Amish

Religious institutions impose many rules on their members. I argue that a central function of these rules in many religious communities is to promote altruistic and cooperative behavior within families. My argument contrasts with the dominant view in the literature that the primary economic function of religious rules is to promote altruism and cooperation between unrelated religious community members. I develop my argument using a novel dataset on the Amish. My theory can explain selection patterns into the Amish church, Amish cultural persistence, persistent inequality between the Amish and non-Amish people, and high Amish fertility rates.     

Strategic complexity in repeated extensive games

This paper studies a machine (finite automaton) playing a two-player repeated game of a simple extensive-form game with perfect information. We introduce a new complexity measure called multiple complexity which incorporates a strategyʼs responsiveness to information in the stage game as well as the number of states of the machine. We completely characterize the Nash equilibrium of the machine game. In the sequential-move prisonerʼs dilemma, cooperation can be sustained as an equilibrium.     

Starting Small and Commitment

I study a model of a long-term partnership with two-sided incomplete information. The partners jointly determine the stakes of their relationship and individually decide whether to cooperate with or betray each other over time. I characterize the extremal—interim incentive efficient—equilibria. In these equilibria, the partners generally “start small,” and the level of interaction grows over time. The types of players separate quickly. Further, cooperation between “good” types is viable regardless of how pessimistic the players are about each other initially. The quick nature of separation in an extremal equilibrium contrasts with the outcome selected by a strong renegotiation criterion (as studied in Watson (1999, J. Econ. Theory85, 52–90). Journal of Economic Literature Classification Numbers: C72, C73, D74.