Folk theorem

A folk theorem describes 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 credit agreements, partnership agreements, (implicit) employment contracts, behavior in marriage or other social ties ...).

Naming

The theorem probably got its name from the fact that it was already considered evident in the minds of game theorists before it was formulated and that it is not due to a single scientist, but only to the "scientific people" as a whole. According to another view, the statement of the folk theorem is so understandable that it was implicit in the collective knowledge of humans, the people, long before it was written down scientifically.

Regardless of the intuitiveness of the statements, the mathematical proof is anything but trivial and should not be done here. The theorem is just explained in words.

For naming cf. Wrong Zipf's law .

Statement of the theorem

In an infinitely repeated game with actors and a finite amount of actions, any combination of individually rational, achievable payouts can be supported as a subgame-perfect Nash equilibrium. ${\ displaystyle n}$

Explanation

A Nash equilibrium in a repeated game is a combination of strategies in which neither player, given the other's strategies, can improve by deviating in any period. Each player discounts the payments from the future rounds with an (individual) discount factor . The present value of the game to the player is therefore given by ${\ displaystyle x}$ ${\ displaystyle \ delta}$

{\ displaystyle V = x_ {t} + \ delta x_ {t + 1} + \ delta ^ {2} x_ {t + 2} + \ delta ^ {3} x_ {t + 3} + \ cdots = \ sum _ {i = 0} ^ {\ infty} \ delta ^ {i} x_ {t + i}}

If the discount factor is high (close to 1), the future is only discounted slightly, i.e. H. the future is important and the player is patient. The future payments are very important. The player will not "jeopardize" future payouts because of a one-time variance win. ${\ displaystyle \ delta}$

In a Nash equilibrium, each player must, on average, get at least one payout equal to their Maximin payout. Payouts that are at least as large as the Maximin payout are called individually rational . An individually rational payout has an advantage over the Maximin payout.

A long-term equilibrium is achieved in that the players mutually threaten each other with punishment if they deviate from the “desired” behavior. Such a punishment is the one-time withdrawal of the advantage in the following period ("tit for tat") or even the dissolution of the game and thus the loss of the advantages in all subsequent periods ("grim strategy").

"Partial Perfect" means that these threats are believable; This means that the punishing player does not harm himself through the punishment (if he takes the behavior of his opponents for granted).

Payout profiles that are not individually rational for each player cannot be implemented as a long-term balance (as each player can always secure the Maximin payout without the involvement of the other players).

Deviating from equilibrium

According to the Folk Theorem, there are two ways a player might deviate from balance despite being punished. This is a low or even zero value - the future is not important to the player. On the other hand, the player will deviate if the probability that the game will end is very high and, despite a possibly high probability, he expects that he will not be able to draw any further advantages from the game. ${\ displaystyle \ delta}$ ${\ displaystyle \ delta}$

literature

Manfred Holler and Gerhard Illing: Introduction to game theory . 6th, revised edition, Springer Verlag, Berlin and Heidelberg, 2005. p. 143.
Norman Braun and Thomas Gautschi: Rational Choice Theory . Juventa Verlag, Munich 2011. ISBN 978-3-7799-1490-7 . Cape. 7.1.2 and 7.1.3