The Folk Theorem in Repeated Games with Discounting or with Incomplete Information

Abstract

When either there are only two players or a full dimensionality condition holds, any individually rational payoff vector of a one-shot game of complete information can arise in a equilibrium of the infinitely-repeated game if players are sufficiently patient. In contrast to earlier work, mixed strategies are allowed in determining the individually rational payoffs (even when only realized actions are observable). Any individually rational payoffs of a one-shot game can be approximated by sequential equilibrium payoffs of a long but finite game of incomplete information, where players' payoffs are almost certainly as in the one-shot game. THAT STRATEGIC RIVALRY in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other's actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated for repeated games. An outcome that Pareto dominates the minimax point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a equilibrium in infinitely repeated games with sufficiently little discounting. As Aumann and Shapley [3] and Rubinstein [20] have shown, the same result is true when we replace the word Nash by (subgame) perfect and assume no discounting at all.…