Repeated game

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game (called a stage game). The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for non-repeated games.

Repeated games may be broadly divided into two classes, depending on the horizon. The horizon is the player's belief about the number of repetitions of the stage game, and may be finite or infinite. The horizon is infinite in infinitely repeated games, so players in this type of game can expect to play indefinitely. By contrast, the horizon is finite in finitely repeated games, so players expect the game to terminate after playing the stage game some number of times. Note that for a game with an infinite horizon, the stage game will not necessarily be repeated an infinite number of times. A game repeated a finite number of times may be regarded as having an infinite horizon if the players in the game do not know how many times the game will be repeated. The results in these two cases are very dissimilar. The difference in strategies for finite versus infinite horizon games is a hotly debated topic, and many game theorists have differing views regarding it.

The most widely studied repeated games are games that are repeated an infinite number of times. In iterated prisoner's dilemma games, it is found that the preferred strategy is not to play a Nash strategy of the stage game, but to cooperate and play a socially optimum strategy. An essential part of strategies in infinitely repeated game is punishing players who deviate from this cooperative strategy. The punishment may be playing a strategy which leads to reduced payoff to both players for the rest of the game (called a trigger strategy). A player may normally choose to act selfishly to increase their own reward rather than play the socially optimum strategy. However, if it is known that the other player is following a trigger strategy, then the player expects to receive reduced payoffs in the future if they deviate at this stage. An effective trigger strategy ensures that cooperating has more utility to the player than acting selfishly now and facing the other player’s punishment in the future.

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