Folk theorem (game theory)

A folk theorem describes the possible equilibria in repeated games. The field of application of the Folk Theorem is the modeling of long-term contracts and interactions of people (for example, loan agreements, partnership agreements, (implicit) contracts, behavior in marriage or any other social bond, ...).

Naming

The name was given the theorem probably out of the fact that it was before its formulation in the minds of game theorists as evident and it is due to any individual scientist, but only the " scientific people" as a whole. According to another opinion, the statement of the folk theorem is understood that they implicitly long before the scientific writing in the collective knowledge of the people, of the people, was nonexistent.

Despite the intuitiveness of the statements of mathematical proof is not trivial and should not be done here. The theorem is only explained in words.

To see naming wrong Zipfsches law.

Statement of the theorem

In an infinitely repeated game with players and a finite set of actions can be any combination of individually rational, achievable disbursements be supported as a subgame perfect Nash equilibrium.

Explanation

A Nash equilibrium in a repeated game is a strategy combination in which no player - for given strategies of the other - can improve by deviating in any period. Each player discounted payments from future rounds with a (individual) discount factor. The current value of the game for the player is therefore given by

If the discount factor is high ( close to 1), the future is only slightly discounted, ie the future is important, and the player is patient. Future payments are heavy. The player will not put the future payments due to a unique Abweichgewinns " at risk ".

In a Nash equilibrium each player at least a payoff equal to his maximin payoff must get on average. Payouts that are at least as large as the maximin payoff, hot individually rational. An individually rational payoff is connected to an advantage over the maximin payoff.

A long-term balance is achieved by the players alternately threaten at deviation from the " desired " behavior punishment. Such punishment is the one-time withdrawal of the benefit in the next period ( " tit for tat " ) or even the resolution of the game and the loss of benefits in all subsequent periods ( "grim strategy" ).

" Subgame perfection " means that these threats are credible, ie that the punishing players by punishing non-self- harm ( if he accepts the behavior of his opponent as a given ).

Payoff profiles that are not individually rational for each player, can not as a long- term equilibrium implement ( since each player can always back up the maximin payoff without the participation of the other players).

Deviation from equilibrium

There are, according to the folk theorem in two ways, why a player deviates despite punishment from equilibrium. This is again a low or even zero-valent - the player is not the future important. On the other hand, the player will be different if the probability that the game ends, is very high and it has persisted despite a possibly high reckons that he can not draw further advantages from the game.