Folk theorem (game theory) – Wikipedia

Folk theorem (game theory)

In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971).[1] The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman’s (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.[2]The Folk Theorem suggests that if the players are patient enough and far-sighted (i.e. if the discount factor {\displaystyle \delta \to 1}), then repeated interaction can result in virtually any average payoff in an SPE equilibrium.[3] “Virtually any” is here technically defined as “feasible” and “individually rational”.For example, in the one-shot Prisoner’s Dilemma, both players cooperating is not a Nash equilibrium. The only Nash equilibrium is that both players defect, which is also a mutual minmax profile. One folk theorem says that, in the infinitely repeated version of the game, provided players are sufficiently patient, there is a Nash equilibrium such that both players cooperate on the equilibrium path. But if the game is repeated a known finite number of times, backward induction shows that both players will play the one-shot Nash equilibrium in each period, i.e. they will defect each time.

Folk theorem (game theory) – Wikipedia