Repeated game
Encyclopedia
In game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, a repeated game (supergame or iterated game) is an extensive form game
Extensive form game
An extensive-form game is a specification of a game in game theory, allowing explicit representation of a number of important aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when he...

 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. It captures the idea that a player will have to take into account the impact of his current action on the future actions of other players; this is sometimes called his reputation. The presence of different equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

 properties is because the threat of retaliation is real, since one will play the game again with the same person. It can be proved that every strategy that has a payoff greater than the minmax payoff can be a Nash Equilibrium, which is a very large set of strategies. Single stage game or single shot game are names for non-repeated games.

Finitely vs infinitely repeated games

Repeated games may be broadly divided into two classes, depending on whether the horizon is finite or infinite. The results in these two cases are very different. Even finitely repeated games are not necessarily finite horizon, the player may just perceive a probability of another cycle and act accordingly. For example, the fact that everyone has a fixed lifetime doesn't mean that all games should be finite horizon. Also, players might act differently when the horizon is far away as opposed to when it is close by, which can probably be thought of as a time modifier function applied to the payoff. The difference in strategies for finite versus infinite horizon games is a hotly debated topic, and many game theorists have differing views regarding it.

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"
Folk theorem (game theory)
In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players' minimax conditions are satisfied. The minimax condition states that a player will minimize the maximum possible loss which he...

. 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
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...

's Blackmailer Paradox appears to be a repeated game in which the ultimatum game
Ultimatum game
The ultimatum game is a game often played in economic experiments in which two players interact to decide how to divide a sum of money that is given to them. The first player proposes how to divide the sum between the two players, and the second player can either accept or reject this proposal. ...

 is played many times by the same players for high stakes.

Finitely repeated games

As explained earlier, finite games can be divided into two broad classes. In the first class of finitely repeated games where the time period is fixed and known, it is optimal to play the Nash strategy in the last period. When the Nash Equilibrium payoff is equal to the minmax payoff, then the player has no reason to stick to a socially optimum strategy and is free to play a selfish strategy throughout, since the punishment cannot affect him (being equal to the minmax payoff). This deviation to a selfish Nash Equilibrium strategy is explained by the Chainstore paradox
Chainstore paradox
Chainstore paradox is a concept that purports to refute standard game theory reasoning.-The chain store game:A monopolist has branches in 20 towns. He faces 20 potential competitors, one in each town, who will be able to choose IN or OUT. They do so in sequential order and one at a time...

. The second class of finitely repeated games are usually thought of as infinitely repeated games.

Repeated prisoner's dilemma

Although the Prisoner's dilemma
Prisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...

 has only one Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

 (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
Trigger strategy
In game theory, a trigger strategy is any of a class of strategies employed in a repeated non-cooperative game. A player using a trigger strategy initially cooperates but punishes the opponent if a certain level of defection is observed...

 comprise Nash equilibria of the repeated Prisoner's dilemma. However, Prisoner's dilemma is one where the minmax value is equal to the Nash Equilbrium payoff. This means that a player who knows the exact horizon may just decide to switch to Defect without fear of punishment.

An example of repeated prisoner's dilemma is the WWI 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 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
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 and the concepts expressed in fictitious play
Fictitious play
In game theory, fictitious play is a learning rule first introduced by G.W. Brown . In it, each player presumes that the opponents are playing stationary strategies. At each round, each player thus best responds to the empirical frequency of play of his opponent...

.

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