How Robust Is the Folk Theorem?^*

Abstract

The folk theorem of repeated games has established that cooperative behavior can be sustained as an equilibrium in repeated settings. Early papers on private monitoring and a recent paper of Cole and Kocherlakota (Games and Economic Behavior, 53 [2005], 59-72) challenge the robustness of this result by providing examples in which cooperation breaks down when players observe only imperfect private signals about other players' actions, or when attention is restricted to strategies with finite memory. This paper shows that Cole and Kocherlakota's result is an artefact of a further restriction that they impose. We prove that the folk theorem with imperfect public monitoring holds with strategies with finite memory. As a corollary, we establish that the folk theorem extends to environments in which monitoring is close to public, yet private.

The folk theorem for irreducible stochastic games with imperfect public monitoring

“…One of them is whether the folk theorem for stochastic games holds when strategies are restricted to have finite memory, as it does in games with a single state (Hörner and Olszewski (2009)). A second is to suppose that the discount factor tends to one because time periods grow short, so that the state transition probabilities will vary with the discount factor; we explore this in ongoing research with Johannes Horner.…”

Instability of belief-free equilibria

“…Mailath and Morris (2002), Hörner and Olszewski (2009), and Mailath and Olszewski (2011 show that when the private signals are almost perfectly correlated conditional on the action profile (i.e., when there is almost public monitoring), then any sequential equilibrium of the nearby public monitoring game with bounded memory remains an equilibrium also with almost public monitoring. Some of these equilibria are evolutionarily stable, and, in particular, cooperation can be the outcome of an evolutionarily stable strategy.…”

Delayed-response strategies in repeated games with observation lags

“…The role of such messages has been studied in a number of subsequent papers, including Ben-Porath and Kahneman (2003), Fudenberg and Levine (2007b), and Escobar and Toikka (2011). Public communication has also been used as a stepping stone to results for games where communication is not allowed (Hörner and Olszewski (2006), Olszewski (2009), andSugaya (2011)). …”

“…Note that this definition of ε-closeness to a public monitoring structure is quite different from the one used by Hörner and Olszewski (2009).…”