Abstract

We extend the folk theorem of repeated games to two settings in which players' information about others' play arrives with stochastic lags. In our first model, signals are almost-perfect if and when they do arrive, that is, each player either observes an almost-perfect signal of period-t play with some lag or else never sees a signal of period-t play. The second model has the same lag structure, but the information structure corresponds to a lagged form of imperfect public monitoring, and players are allowed to communicate via cheap-talk messages at the end of each period. In each case, we construct equilibria in “delayed-response strategies,” which ensure that players wait long enough to respond to signals that with high probability all relevant signals are received before players respond. To do so, we extend past work on private monitoring to obtain folk theorems despite the small residual amount of private information.

Highlights

  • Understanding when and why individuals cooperate in social dilemmas is a key issue not just for economics but for all of the social sciences,1 and the theory of repeated games is the workhorse model of how and when concern for the future can lead to cooperation even if all agents care only about their own payoffs

  • We show that the HO2006 approach to repeated games with almost-perfect monitoring can be extended to lagged repeated games with almost-perfect monitoring, so long as positive lags are sufficiently rare, and use this to obtain a folk theorem in the auxiliary game

  • As we argued in the introduction, the key role of the repeated games model makes it important to understand which of its many simplifications are essential for the folk theorem

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Summary

Introduction

Understanding when and why individuals cooperate in social dilemmas is a key issue not just for economics but for all of the social sciences, and the theory of repeated games is the workhorse model of how and when concern for the future can lead to cooperation even if all agents care only about their own payoffs. The rest of the paper allows the lag distribution to have unbounded support, and allows for a small probability that some signals never arrive at all (corresponding to an infinite observation lag) In these cases the use of delay strategies reduces but does not eliminate the impact of lags, and the game played in each thread has some additional decision-relevant private information. 5 In each case, players do not know whether and when other players observe the signals associated with each period’s play, so there is a special but natural form of private information For both of our main results, we use a similar proof technique: First, we consider an auxiliary game with “rare” lags in which each player sees a private signal immediately with probability close to (but not equal to) 1. We prove a sort of folk theorem here using the techniques of Fudenberg, Levine, and Maskin (1994) ( FLM) and again use threads and delayed responses to extend this to a proof for the original game

Related Work
General Model
Bounded Lags
Lagged Almost-Perfect Monitoring with Two Players
The Folk Theorem
Auxiliary Repeated Game with Rare Observation Lags
Preliminaries
The Repeated Game with Frequent Observation Lags
Lagged Public Monitoring
Structure of the Observation Lags
Private Monitoring Game with Communication
Non-Communication Stages
The Repeated Game with Observation Lags
Discussion and Conclusion
A Definitions of sGi and sBi
C Conditions Guaranteeing Small γi
Full Text
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