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Repeated game
In game theory, a repeated game (or iterated 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 their current action on the future actions of other players; this impact is sometimes called their reputation. Single stage game or single shot game are names for non-repeated games.
Consider two gas stations that are adjacent to one another. They compete by publicly posting pricing, and have the same and constant marginal cost c (the wholesale price of gasoline). Assume that when they both charge p = 10, their joint profit is maximized, resulting in a high profit for everyone.
Despite the fact that this is the best outcome for them, they are motivated to deviate. By modestly lowering the price, either can steal all of their competitors' customers, nearly doubling their revenues. p = c, where their profit is zero, is the only price without this profit deviation. In other words, in the pricing competition game, the only Nash equilibrium is inefficient (for gas stations) that both charge p = c.
This is more of a rule than an exception: in a staged game, the Nash equilibrium is the only result that an agent can consistently acquire in an interaction, and it is usually inefficient for them. This is because the agents are just concerned with their own personal interests, and do not care about the benefits or costs that their actions bring to competitors. On the other hand, gas stations make a profit even if there is another gas station adjacent. One of the most crucial reasons is that their interaction is not one-off. This condition is portrayed by repeated games, in which two gas stations compete for pricing (stage games) across an indefinite time range t = 0, 1, 2,....
Repeated games may be broadly divided into two classes, finite and infinite, depending on how long the game is being played for.
Even if the game being played in each round is identical, repeating that game a finite or an infinite number of times can, in general, lead to very different outcomes (equilibria), as well as very different optimal strategies.
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.
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 modelled. There are many different ways in which a preference relation may be modelled in an infinitely repeated game, but two key ones are :
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Repeated game AI simulator
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Repeated game
In game theory, a repeated game (or iterated 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 their current action on the future actions of other players; this impact is sometimes called their reputation. Single stage game or single shot game are names for non-repeated games.
Consider two gas stations that are adjacent to one another. They compete by publicly posting pricing, and have the same and constant marginal cost c (the wholesale price of gasoline). Assume that when they both charge p = 10, their joint profit is maximized, resulting in a high profit for everyone.
Despite the fact that this is the best outcome for them, they are motivated to deviate. By modestly lowering the price, either can steal all of their competitors' customers, nearly doubling their revenues. p = c, where their profit is zero, is the only price without this profit deviation. In other words, in the pricing competition game, the only Nash equilibrium is inefficient (for gas stations) that both charge p = c.
This is more of a rule than an exception: in a staged game, the Nash equilibrium is the only result that an agent can consistently acquire in an interaction, and it is usually inefficient for them. This is because the agents are just concerned with their own personal interests, and do not care about the benefits or costs that their actions bring to competitors. On the other hand, gas stations make a profit even if there is another gas station adjacent. One of the most crucial reasons is that their interaction is not one-off. This condition is portrayed by repeated games, in which two gas stations compete for pricing (stage games) across an indefinite time range t = 0, 1, 2,....
Repeated games may be broadly divided into two classes, finite and infinite, depending on how long the game is being played for.
Even if the game being played in each round is identical, repeating that game a finite or an infinite number of times can, in general, lead to very different outcomes (equilibria), as well as very different optimal strategies.
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.
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 modelled. There are many different ways in which a preference relation may be modelled in an infinitely repeated game, but two key ones are :