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Decision theory

In game theory, a **repeated game** (or **iterated game**) is an extensive form game which consists in some number of repetitions of some base game (called a **stage game**). The stage game is usually one of the well studied 2 person games. The repeated game can have different equilibrium properties because the threat of retaliation is real, since one will play the game again with the same person. **Single stage game** or **single shot game** are names for non-repeated games.

## Infinitely repeated games[]

Repeated games may be repeated finitely or infinitely many times, the latter are called supergames. The most widely studied repeated games are games that are repeated a possibly infinite number of times. These games are modeled by applying a discount factor to each future stage. This discount factor has two primary interpretations. First, it might be that at each stage there is some finite probability that the game ends. Second, it might be that each individual cares slightly less about each successive future stage.

## Repeated prisoner's dilemma[]

Although the Prisoner's dilemma has only one Nash equilibrium (everyone defect), cooperation can be sustained in the repeated Prisoner's dilemma if the discount factor is not too low, that is if the players are interested enough in future outcomes of the game. Strategies known as trigger strategies comprise Nash equilibria of the repeated Prisoner's dilemma. This result is part of a larger class of results known as the folk theorem. Many authors believe that this constitutes the explanation for social cooperation.

An example of repeated prisonner's dilemma is the WW1 trench warfare. Here, though initially it was best to cause as much damage to the other party as possible, as time passed and the opposing parties got to 'know' each other, they realised that causing as much damage as possible to the other by, e.g. artillery will only prompt a similar response: e.g. blowing up the foodstock of the other (through bombardment) will only leave both battalions hungry. After some time, the opposing battalions learned that it is sufficient enough to *show* what they are capable of, instead of actually carrying out the act.

## Solving repeated games[]

Complex repeated games can be solved using various techniques most of which rely heavily on linear algebra and the concepts expressed in fictitious play.

## References[]

- Fudenberg, Drew and Jean Tirole (1991)
*Game Theory*MIT Press.

## External links[]

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