Game theory III: Folk theorem

Folk theorems are used in Economics specially in the field of game theory and specifically to repeated games. This theorem is said to be satisfactorily fulfilled when the equilibrium outcome in a game that is repeated an infinity number of times, is the same as the feasible and strongly individually rational outcome in the one-shot game. The outcome is said to be in equilibrium, because any attempt of trying to increase a player’s individual outcome will imply a decrease in at least another player’s individual outcome.

The origin of this theorem is unknown but it appeared in the late fifties, and became quickly known and spread within the game theory academia, and thus the reason why it became known as the folk theorem.  James W. Friedman in his article “A Non-cooperative Equilibrium for Supergames”, 1971, was the first to write and publish a paper in which he thoroughly deals with the theorem.

The main issue that this theorem explains, is the strong relationship between repeated and cooperative games, revealing the enforcement mechanism characteristic of repetition. One of the first applications of the theorem was to collude when dealing with a Cournot duopoly:
 

The left hand side represents the payoff derived from collusion, which can be held infinitely over time, with δ being the discount factor to bring future benefits forward to the present. For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel. The higher δ is, the higher the value assigned to future benefits and therefore the greater the chances of collusion. It is worth reminding, that fair competition is regulated in almost all countries, with cartelsbeing banned, and so most markets that lend themselves to reduced competition and price fixing, are closely monitored.