scholarly journals The Folk Theorem in Repeated Games With Anonymous Random Matching

Econometrica ◽  
2020 ◽  
Vol 88 (3) ◽  
pp. 917-964 ◽  
Author(s):  
Joyee Deb ◽  
Takuo Sugaya ◽  
Alexander Wolitzky
Keyword(s):  
Repeated Games ◽  
Cheap Talk ◽  
Folk Theorem ◽  
Random Matching ◽  
Record Keeping ◽  
Communication Devices ◽  
Stage Game

We prove the folk theorem for discounted repeated games with anonymous random matching. We allow non‐uniform matching, include asymmetric payoffs, and place no restrictions on the stage game other than full dimensionality. No record‐keeping or communication devices—including cheap talk communication and public randomization—are necessary.

Record-Keeping and Cooperation in Large Societies

2021 ◽  
Author(s):  
Daniel Clark ◽  
Drew Fudenberg ◽  
Alexander Wolitzky
Keyword(s):  
Steady State ◽  
Repeated Games ◽  
Overlapping Generations ◽  
Rational Action ◽  
Random Matching ◽  
Record Keeping ◽  
Large Populations ◽  
Stage Game ◽  
The One ◽  
Full Cooperation

Abstract We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”

A Few Bad Apples Spoil the Barrel: An Anti-Folk Theorem for Anonymous Repeated Games with Incomplete Information

2020 ◽  
Vol 110 (12) ◽  
pp. 3817-3835
Author(s):  
Takuo Sugaya ◽  
Alexander Wolitzky
Keyword(s):  
Incomplete Information ◽  
Repeated Games ◽  
Prisoner’S Dilemma ◽  
Folk Theorem ◽  
Smoothness Condition ◽  
Cournot Oligopoly ◽  
Random Matching ◽  
Games With Incomplete Information ◽  
Linear Demand ◽  
Number Of Firms

We study anonymous repeated games where players may be “commitment types” who always take the same action. We establish a stark anti-folk theorem: if the distribution of the number of commitment types satisfies a smoothness condition and the game has a “pairwise dominant” action, this action is almost always taken. This implies that cooperation is impossible in the repeated prisoner's dilemma with anonymous random matching. We also bound equilibrium payoffs for general games. Our bound implies that industry profits converge to zero in linear-demand Cournot oligopoly as the number of firms increases. (JEL C72, C73, D83)

scholarly journals Reactive Strategies: An Inch of Memory, a Mile of Equilibria

Games ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 42
Author(s):  
Artem Baklanov
Keyword(s):  
Nash Equilibrium ◽  
Repeated Games ◽  
Folk Theorem ◽  
Pareto Improvement ◽  
Symmetric Equilibrium ◽  
Equilibrium Payoff ◽  
Incremental Change ◽  
Symmetric Games ◽  
Stage Game

We explore how an incremental change in complexity of strategies (“an inch of memory”) in repeated interactions influences the sets of Nash Equilibrium (NE) strategy and payoff profiles. For this, we introduce the two most basic setups of repeated games, where players are allowed to use only reactive strategies for which a probability of players’ actions depends only on the opponent’s preceding move. The first game is trivial and inherits equilibria of the stage game since players have only unconditional (memory-less) Reactive Strategies (RSs); in the second one, players also have conditional stochastic RSs. This extension of the strategy sets can be understood as a result of evolution or learning that increases the complexity of strategies. For the game with conditional RSs, we characterize all possible NE profiles in stochastic RSs and find all possible symmetric games admitting these equilibria. By setting the unconditional benchmark as the least symmetric equilibrium payoff profile in memory-less RSs, we demonstrate that for most classes of symmetric stage games, infinitely many equilibria in conditional stochastic RSs (“a mile of equilibria”) Pareto dominate the benchmark. Since there is no folk theorem for RSs, Pareto improvement over the benchmark is the best one can gain with an inch of memory.

Cooperative Homo economicus

2011 ◽  
Author(s):  
Samuel Bowles ◽  
Herbert Gintis
Keyword(s):  
Private Information ◽  
Repeated Games ◽  
Evolutionary Dynamics ◽  
Public Information ◽  
Folk Theorem ◽  
The Self ◽  
Correlated Equilibrium ◽  
Homo Economicus ◽  
Self Interest ◽  
Game Theoretic

This chapter examines whether recent advances in the theory of repeated games, as exemplified by the so-called folk theorem and related models, address the shortcomings of the self-interest based models in explaining human cooperation. It first provides an overview of folk theorems and their account of evolutionary dynamics before discussing the folk theorem with either imperfect public information or private information. It then considers evolutionarily irrelevant equilibrium as well as the link between social norms and the notion of correlated equilibrium. While the insight that repeated interactions provide opportunities for cooperative individuals to discipline defectors is correct, the chapter argues that none of the game-theoretic models mentioned above is successful. Except under implausible conditions, the cooperative outcomes identified by these models are neither accessible nor persistent, and are thus labeled evolutionarily irrelevant Nash equilibria.

Private Strategies in Finitely Repeated Games with Imperfect Public Monitoring

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
George J. Mailath ◽  
Steven A. Matthews ◽  
Tadashi Sekiguchi
Keyword(s):  
Nash Equilibrium ◽  
Repeated Games ◽  
Finitely Repeated Games ◽  
Game Equilibrium ◽  
Public Signals ◽  
Stage Game ◽  
Public Monitoring ◽  
Public Signal ◽  
Action Profile

We present three examples of finitely repeated games with public monitoring that have sequential equilibria in private strategies, i.e., strategies that depend on own past actions as well as public signals. Such private sequential equilibria can have features quite unlike those of the more familiar perfect public equilibria: (i) making a public signal less informative can create Pareto superior equilibrium outcomes; (ii) the equilibrium final-period action profile need not be a stage game equilibrium; and (iii) even if the stage game has a unique correlated (and hence Nash) equilibrium, the first-period action profile need not be a stage game equilibrium.

The Foole, the Shepherd, and the Knave

2018 ◽  
Author(s):  
Peter Vanderschraaf
Keyword(s):  
Repeated Games ◽  
Folk Theorem ◽  
Social Sanctions ◽  
Identity Concealment ◽  
Reconciliation Project ◽  
The Invisible

The Reconciliation Project, the attempt to show that justice is compatible with rational prudence, is evaluated in light of the classic challenges of Hobbes’ Foole, Plato’s Lydian Shepherd, and Hume’s Sensible Knave. Hobbes’ response to the Foole is justice-reciprocalist, emphasizing social sanctions, and is naturally interpreted in terms of folk theorem interactions of repeated games. Plato’s justice-Platonist response to the Shepherd, who has identity-concealing power, emphasizes goods allegedly inseparable from justice. A new Invisible Foole challenge is considered where an agent like the Foole who takes seriously only social sanctions acquires identity-concealment technology, and folk theorem responses are proposed for this challenge. The Invisible Foole challenge is similar to the most serious challenge, that of the Sensible Knave. The most compelling response to the Knave’s challenge combines elements of justice-reciprocalism and justice-Platonism.

The Folk Theorem for Repeated Games with Time-Dependent Discounting

2021 ◽  
Author(s):  
Daehyun Kim ◽  
Xiaoxi Li
Keyword(s):  
Repeated Games ◽  
Folk Theorem ◽  
The Self ◽  
Time Dependent ◽  
Constructive Proof ◽  
The Public ◽  
Discount Factors ◽  
Time Inconsistent ◽  
Public Monitoring ◽  
Perfect Monitoring

This paper defines a general framework to study infinitely repeated games with time-dependent discounting in which we distinguish and discuss both time-consistent and -inconsistent preferences. To study the long-term properties of repeated games, we introduce an asymptotic condition to characterize the fact that players become more and more patient; that is, the discount factors at all stages uniformly converge to one. Two types of folk theorems are proven without the public randomization assumption: the asymptotic one, that is, the equilibrium payoff set converges to the feasible and individual rational set as players become patient, and the uniform one, that is, any payoff in the feasible and individual rational set is sustained by a single strategy profile that is an approximate subgame perfect Nash equilibrium in all games with sufficiently patient discount factors. We use two methods for the study of asymptotic folk theorem: the self-generating approach and the constructive proof. We present the constructive proof in the perfect-monitoring case and show that it can be extended to time-inconsistent preferences. The self-generating approach applies to the public-monitoring case but may not extend to time-inconsistent preferences because of a nonmonotonicity result.

Repeated play: Cooperation and reputation

2020 ◽  
pp. 503-550
Author(s):  
David M. Kreps
Keyword(s):  
Repeated Games ◽  
Basic Structure ◽  
General Notion ◽  
Prisoners Dilemma ◽  
Stage Game

This chapter reviews the concepts of cooperation and reputation. It begins by looking at the game called the prisoners' dilemma. The basic structure of options and payoffs that characterize this game occur over and over in economics. In this basic structure, players can cooperate to greater or to lesser extent. If one player unilaterally decreases the lever of their cooperation, they benefit and their rival is made worse off. But if both decrease their level of cooperation equally, both are made worse off. The chapter then considers repeated games and reflects on variations of the general notion of a supergame, wherein a particular game called the stage game is played over and over by two (or more) players.

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