Folk Theorem

Game Theory · Axiom Academy

Unit 4 - Repeated Games: Understanding the Multiplicity of Equilibria 1. Statement of the Folk Theorem The Folk Theorem states that in infinitely repeated games with sufficiently patient players, any feasible and individually rational payoff profile can be sustained as a subgame perfect equilibrium. Players must be sufficiently patient (discount factor δ close to 1) Payoffs must be feasible (achievable through some combination of actions) Payoffs must be individually rational (at least as good as the minmax payoff) This means the shadow of the future creates opportunities for cooperation that don't exist in one-shot interactions. 2. Feasible and Individually Rational Payoffs Feasible payoffs are those that can be achieved as convex combinations of pure strategy payoffs in the stage game. They form a region in payoff space bounded by the outcomes from all possible action profiles. Individually rational payoffs are those that give each player at least their minmax value - the lowest payoff that opponents can force them to receive. The Folk Theorem applies to the intersection of these sets - payoffs that are both feasible and give everyone at least their minmax value. 3. Minmax Payoffs as Credible Threats The minmax value serves as a threat point in repeated games. If a player deviates from the cooperative strategy, opponents can punish them by playing their minmax strategy, driving the deviator's payoff down to v i .

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