A Robust Folk Theorem for the Prisoner's Dilemma

Abstract

We prove the folk theorem for the Prisoner's dilemma using strategies that are robust to private monitoring. From this follows a limit folk theorem: when players are patient and monitoring is sufficiently accurate, (but private and possibly independent) any feasible individually rational payoff can be obtained in sequential equilibrium. The strategies used can be implemented by finite (randomizing) automata. * Thanks to Görkem Celik for valuable research assistance.

Modeling cooperation among self-interested agents: a critique

“…The second major thrust involved the development of sophisticated repeated game-theoretic models of strategic interaction, which began with Shubik (1959), culminating in the major contributions of Fudenberg and Maskin (1986), Abreu et al (1990), Fudenberg et al (1994), Piccione (2002), Ely and Välimäki (2002), Bhaskar and Obara (2002) and others. While there is no question but these models have strongly advanced our understanding of the theory of social cooperation, the very fact these contributions prove "folk theorems" that sustain full-dimensional open sets of sequential equilibria virtually assure that these models will have poor dynamic qualities.…”

“…Important contributions to this research agenda include Sekiguchi (1997), who was the first to propose a Nash equilibrium that approximately sustains the cooperative payoff in the two-person prisoner's dilemma, assuming that private monitoring is nearly perfect. Following this, contributions by Piccione (2002), Ely and Välimäki (2002), and Bhaskar and Obara (2002), and Matsushima (2000) considerably deepened the approach.…”

Robustness of public equilibria in repeated games with private monitoring

“…9 This is always close to 2 in Example 1 if β is large and is always equal to 0 in Example 2. We use this variation of player i's beliefs to induce her to report her private signals truthfully.…”

Optimal collusion with internal contracting

“…The first assumption is that there is no side-contracting collusion between agents; agents across firms cannot form a cartel to make collusive reports using side-payments. 13 The second assumption is that the firm can make an ex ante commitment to production schedules. Under ρ < C H , before the agent's report, the virtual cost of producing q 1 HH + q 2 HH = 1 is C H , which is too high.…”

“…13 Our paper does not allow any form of side-payment across firms or across agents. Martimort (1997, 2000) characterize optimal collusion-proof mechanisms when privately-informed agents are collusive in their sidecontracting games.…”