Infinitely repeated games with public monitoring and monetary transfers

This paper studies infinitely repeated games with imperfect public monitoring and the possibility of monetary transfers. It is shown that all public perfect equilibrium payoffs can be implemented with a simple class of stationary equilibria that use stick-and-carrot punishments. A fast algorithm is developed that exactly computes the set of pure strategies equilibrium payoffs for all discount factors.     

Reputation with equal discounting in repeated games with strictly conflicting interests

We analyze reputation effects in two-player repeated games of strictly conflicting interests. In such games, player 1 has a commitment action such that a best reply to it gives player 1 the highest individually rational payoff and player 2 the minmax payoff. Players have equal discount factors. With positive probability player 1 is a type who chooses the commitment action after every history. We show that player 1's payoff converges to the maximally feasible payoff when the discount factor converges to one. This contrasts with failures of reputation effects for equal discount factors that have been demonstrated in the literature.     

Bounded rationality,neural network and folk theorem in repeated games with discounting

Summary The perfect folk theorem (Fudenberg and Maskin [1986]) need not rely on excessively complex strategies. We recover the perfect folk theorem for two person repeated games with discounting through neural networks (Hopfield [1982]) that have finitely many associative units. For any individually rational payoff vector, we need neural networks with at most 7 associative units, each of which can handle only elementary calculations such as maximum, minimum or threshold operation. The uniform upper bound of the complexity of equilibrium strategies differentiates this paer from Ben-Porath and Peleg [1987] in which we need to admit ever more complex strategies in order to expand the set of equilibrium outcomes.I would like to thank Hao Li and John Curran for excellent research assistance. Financial support from National Science Foundation (SES-9223483), Sloan Foundation and the Division of Social Sciences at the University of Chicago is gratefully acknowledged.     

Adaptation and complexity in repeated games

The paper presents a learning model for two-player infinitely repeated games. In an inference step players construct minimally complex inferences of strategies based on observed play, and in an adaptation step players choose minimally complex best responses to an inference. When players randomly select an inference from a probability distribution with full support the set of steady states is a subset of the set of Nash equilibria in which only stage game Nash equilibria are played. When players make ‘cautious’ inferences the set of steady states is the subset of self-confirming equilibria with Nash outcome paths. When players use different inference rules, the set of steady states can lie between the previous two cases.     

Dismissals and quits in repeated games

This paper studies repeated games, where player 1 can decide to let the opponent continue or replace her by a new player. We also allow for the possibility of player 2 quitting the game. When only layoffs can occur, a folk theorem for finite horizons obtains due to the threat that termination of the relationship imposes on player 2. However, quits limit this result to those cases in which the outside option for player 2 is small (lower than some Nash equilibrium payoff of the stage game).     

Transparency and cooperation in repeated dilemma games: a meta study

We use data from experiments on finitely repeated dilemma games with fixed matching to investigate the effect of different types of information on cooperation. The data come from 71 studies using the voluntary contributions paradigm, covering 122 data points, and from 18 studies on decision-making in oligopoly, covering another 50 data points. We find similar effects in the two sets of experimental games. We find that transparency about what everyone in a group earns reduces contributions to the public good, as well as the degree of collusion in oligopoly markets. In contrast, transparency about choices tends to lead to an increase in contributions and collusion, although the size of this effect varies somewhat between the two settings. Our results are potentially useful for policy making, because they provide guidance on the type of information to target in order to stimulate or limit cooperation.     

Individual characteristics and behavior in repeated games: an experimental study

Using a laboratory experiment, we investigate whether a variety of behaviors in repeated games are related to an array of individual characteristics that are popular in economics: risk attitude, time preference, trust, trustworthiness, altruism, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, ability to plan ahead, and gender. We do find some systematic relationships. A subject’s compliance with first-order stochastic dominance as well as, possibly, patience, gender, and altruism have some systematic effects on her behavior in repeated games. At the level of a pair of subjects who are playing a repeated game, each subject’s gender as well as, possibly, patience and ability to choose an available dominant strategy in a one-shot matrix game systematically affect the frequency of the cooperate–cooperate outcome. However, overall, the number of systematic relationships is surprisingly small.     

Equilibrium selection in similar repeated games: experimental evidence on the role of precedents

We report on an experiment examining behavior and equilibrium selection in two similar, infinitely repeated games, Stag Hunt and Prisoner’s Dilemma under anonymous random matching. We are interested in the role that historical precedents may play for equilibrium selection between these two repeated games. We find that a precedent for efficient play in the repeated Stag Hunt game does not carry over to the repeated Prisoner’s Dilemma game despite the possibility that efficient play can be sustained as an equilibrium of the indefinitely repeated game. Similarly, a precedent for inefficient play in the repeated Prisoner’s Dilemma game does not extend to the repeated Stag Hunt game. We conclude that equilibrium selection between similar repeated games may have less to do with historical precedents and might instead depend more on strategic considerations associated with the different payoffs of these similar repeated games.     

Punishment strategies in repeated games: Evidence from experimental markets

An experiment is designed to provide a snapshot of the strategies used by players in a repeated price competition game with a random continuation rule. One hundred pairs of subjects played the game over the Internet, with subjects having a few days to make their decisions in each round. Occasionally subjects are asked to enter one-period-ahead pricing strategies instead of prices. According to the elicited strategies, between 90% and 95% of subjects punish less harshly (in their initial response to a deviation) than implied by the grim trigger strategy, and do so in a way that depends on the size of the other subjectʼs deviation. Future earnings are highest for subjects adopting the tit-for-tat strategy, even after controlling for a subjectʼs past earnings. Punishment strategies are generally softer and more graduated than implied by a grim trigger strategy, and do better as a result.     

Coalitional bargaining games with random proposers: Theory and application

We consider a non-cooperative coalitional bargaining game with random proposers in a general situation for which players differ in recognition probability and time preference. We characterize an efficient equilibrium as the generalized Nash bargaining solution that belongs to the core. The model is applied to wage bargaining between an employer and multiple workers. Although involuntary unemployment may occur in equilibrium, full employment emerges as players become sufficiently patient.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

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.     

Estimating the causal effect of beliefs on contributions in repeated public good games

We use instrumental variables for estimating the causal effect of beliefs on contributions in repeated public good games. The effect is about half as large as suggested by ordinary least squares. Thus, we present evidence that beliefs have a causal effect on contributions, but also that beliefs are endogenous. We compare the causal, belief-based model of contributions to alternative models based on matching the previous contributions of others and responding to one’s deviation from the average in the previous round. The causal, belief-based model performs well, indicating that beliefs have a central role in determining contributions.     

Edgar Allan Poe's riddle: Framing effects in repeated matching pennies games

Framing effects have a significant influence on the finitely repeated matching pennies game. The combination of being labelled “a guesser,” and having the objective of matching the opponent's action, appears to be advantageous. We find that being a player who aims to match the opponent's action is advantageous irrespective of whether the player moves first or second. We examine alternative explanations for our results and relate them to Edgar Allan Poe's “The Purloined Letter.” We propose a behavioral model which generates the observed asymmetry in the players' performance.     

Personality and cooperation in finitely repeated prisoner’s dilemma games

We investigate the role personality plays in Finitely Repeated Prisoner’s Dilemma (FRPD) games. Even after controlling for demographic factors such as race, course of study, and cognitive ability, we find that cooperative behavior is significantly related to the Big Five personality trait Agreeableness. A one standard deviation increase in agreeableness increases the predicted probability of cooperation by a subject with modal demographic characteristics from 67.9% to 80.6%.     

On the limit perfect public equilibrium payoff set in repeated and stochastic games

This paper provides a dual characterization of the existing ones for the limit set of perfect public equilibrium payoffs in a class of finite stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg et al. (1994), Kandori and Matsushima (1998) and Hörner et al. (2011) obtain. As a second corollary, it is shown that this limit set of payoffs is a convex polytope when attention is restricted to perfect public equilibria in pure strategies. This result fails for mixed strategies, even when attention is restricted to two-player repeated games.     

Separating the Hawks from the Doves: Evidence from continuous time laboratory games

Human players in our laboratory experiment received flow payoffs over 120 seconds each period from a standard Hawk–Dove bimatrix game played in continuous time. Play converged closely to the symmetric mixed Nash equilibrium under a one-population matching protocol. When the same players were matched in a two-population protocol with the same bimatrix, they showed clear movement towards an asymmetric (and very inequitable) pure Nash equilibrium of the game. These findings support distinctive predictions of evolutionary game theory.     

Intertemporal substitution and random changes in preferences: Time series evidence for four countries

This study attempts to estimate a model of intertemporal substitution taking into account serially correlated changes in preferences with quarterly aggregate time series data for the USA, Great Britain, West Germany and Switzerland. Thereby, necessary optimality conditions representing the trade-offs between present and future consumption, present and future leisure, as well as present consumption and leisure are estimated using the forward-filtered estimator of Hayashi and Sims. The estimation results obtained for the dynamic optimality conditions reject the model formulated. The analysis of the static optimality conditions gave more satisfactory results with respect to intertemporal substitution elasticity in consumption.     

Exploitable actions of believers in the “law of small numbers” in repeated constant-sum games

In repeated fixed-pair constant-sum games with unique equilibria in mixed strategies, such as matching pennies, the subgame perfect equilibrium is repeating the stage-game mixed-strategy equilibrium action. In such games rational players avoid strategies that are exploitable, in that current actions either deviate systematically from the equilibrium action probabilities or fail to be serially independent of past actions. I revisit classic experiments and find that subjects’ actions are sometimes exploitable because they are serially dependent. Subjects have difficulty in producing serially independent actions and in recognizing serially dependent sequences due to a bias called local representativeness.     

Hyperbolic discounting and the standard model: Eliciting discount functions

Experiments on static intertemporal choice find evidence of particularly extreme impatience toward immediate rewards. While this is often taken as support for hyperbolic discounting, it could also arise because the most likely participants in experiments may be those with the most immediate need for money. We conduct a calibration exercise and find that the extreme impatience observed in experiments can be accommodated by a standard exponential discounting model with no discounting and expectation of a ‘small’ increase in the base consumption level. The calibration uses existing estimates of curvature of utility.