Folk theorem (game theory)

In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players' minimax conditions are satisfied. The minimax condition states that a player will minimize the maximum possible loss which they could face in the game. An outcome is said to be feasible if it satisfies this condition for each player of the game. A repeated game is one in which there is not necessarily a final move, but rather, there is a sequence of rounds, during which the player may gather information and choose moves. An early published example is Friedman (1971).

In mathematics, the term folk theorem refers generally to a theorem which is believed and discussed, but has not been published. In order that the name of the theorem be more descriptive, Roger Myerson has recommended the phrase general feasibility theorem in the place of folk theorem for describing theorems which are of this class.

Sketch of proof

A commonly referenced proof of a folk theorem was published in 1979 by Ariel Rubinstein.

The method for proving folk theorems is actually quite simple. A grim trigger strategy is a strategy which punishes an opponent for any deviation from some certain behavior. So, all of the players of the game first must have a certain feasible outcome in mind. Then the players need only adhere to an almost grim trigger strategy under which any deviation from the strategy which will bring about the intended outcome is punished to a degree such that any gains made by the deviator on account of the deviation are exactly cancelled out. Thus, there is no advantage to any player for deviating from the course which will bring out the intended, and arbitrary outcome, and the game will proceed in exactly the manner to bring about that outcome.

Applications

It is possible to apply this class of theorems to a diverse number of fields. An application in anthropology, for example, would be that in a community where all behavior is well known, and where members of the community know that they will continue to have to deal with each other, then any pattern of behavior (traditions, taboos, etc) may be sustained by social norms so long as the individuals of the community are better off remaining in the community than they would be leaving the community (the minimax condition).

References

* Myerson, Roger B. Game Theory, Analysis of conflict, Cambridge, Harvard University Press (1991)
* Rubinstein, Ariel. "Equilibrium in Supergames with the Overtaking Criterion." Journal of Economic Theory 21:1-9
* Mas-Collel, A., Whinston, M and Green, J. (1995) Micreoconomic Theory, Oxford University Press, New York (readable; suitable for advanced undergraduates.)
* Tirole, J. (1988) The Theory of Industrial Organization, MIT Press, Cambridge MA (An organized introduction to industrial organization)

;Classical paper on this subject
* Friedman, J. (1971). A non-cooperative equilibrium for supergames, Review of EconomicStudies 38, 1-12.