Infinitely Repeated Games
The most widely studied repeated games are games that are repeated a possibly infinite number of times. On many occasions, it is found that the optimal method of playing a repeated game is not to repeatedly play a Nash strategy of the constituent game (look at the Repeated prisoner's dilemma example), but to cooperate and play a socially optimum strategy. This can be interpreted as a "social norm" and one essential part of infinitely repeated games is punishing players who deviate from this cooperative strategy. The punishment may be something like playing a strategy which leads to reduced payoff to both players for the rest of the game (called a trigger strategy). There are many results in theorems which deal with how to achieve and maintain a socially optimal equilibrium in repeated games. These results are collectively called "Folk Theorems". An important feature of a repeated game is the way in which a player's preferences may be modeled. There are many different ways in which a preference relation may be modeled in an infinitely repeated game, the main ones are :
- Discounting - valuation of the game diminishes with time depending on the discount parameter
- Limit of means - can be thought of as an average over T periods as T approaches infinity.
- Overtaking - Sequence is superior to sequence if
Robert Aumann's Blackmailer Paradox appears to be a repeated game in which the ultimatum game is played many times by the same players for high stakes.
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