Folk theorem (game theory)
| Folk theorem | |
|---|---|
| A solution concept in game theory | |
| Relationship | |
| Subset of | Minimax, Nash Equilibrium |
| Significance | |
| Proposed by | various, notably Ariel Rubinstein |
| Used for | repeated games |
| Example | Repeated prisoner's dilemma |
In game theory, folk theorems are a class of theorems about possible Nash equilibrium payoff profiles in repeated games (Friedman 1971). 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 equilibrium.
The Folk Theorem suggests that if the player is patient enough and far-sighted (i.e. if discount factor ) then not only can repeated interaction allow many SPE outcomes, but actually SPE can allow virtually any outcome in the sense of average payoffs. Put more simply, the theorem suggests that anything that is feasible and individually rational is possible.
For example, in the one-shot Prisoner's Dilemma, if both players cooperate that is not a Nash equilibrium. The only Nash equilibrium 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 in finitely repeated game by using backward induction it can be determined that players play Nash equilibrium in last period of the game which is defecting.
Any Nash equilibrium payoff in a repeated game must satisfy two properties:
1. Individual rationality (IR): the payoff must weakly dominate the minmax payoff profile of the constituent stage game. I.e, the equilibrium payoff of each player must be at least as large as the minmax payoff of that player. This is because a player achieving less than his minmax payoff always has incentive to deviate by simply playing his minmax strategy at every history.
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