Repeated game

In game theory, a repeated game (supergame 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. 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 properties is present 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 (see the repeated prisoner's dilemma example below), 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 a discount factor $\delta <1$ : $V=\sum _{t=1}^{\infty }\delta ^{t}v_{t}$
Limit of means - can be thought of as an average over T periods as T approaches infinity: $V=\lim _{T\to \infty }{\frac {1}{T}}\sum _{t=1}^{T}v_{t}$
Overtaking - Sequence $v_{t}$ is superior to sequence $w_{t}$

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

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 (which is always between 0 and 1) is sufficiently large, while if the discount factor is close to 0 players do not care about the future and have no incentive to cooperate; but if the discount factor is close to 1 players are better off if they cooperate.^[1] Strategies known as trigger strategies comprise Nash equilibria of the repeated prsoner's dilemma. However, the prisoner's dilemma is one where the minmax value is equal to the Nash Equilibrium 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 World War I 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 and the concepts expressed in fictitious play.

Incomplete information

Repeated games can include incomplete information. Repeated games with incomplete information were pioneered by Aumann and Maschler.^[2] While it is easier to treat a situation where one player is informed and the other not, and when information received by each player is independent, it is possible to deal with zero-sum games with incomplete information on both sides and signals that are not independent.^[3]

References

↑ Osborne, Martin J.; Rubinstein, Ariel (1994). A Course in Game Theory. Cambridge: MIT Press. ISBN 0-262-15041-7.
↑ Aumann, R. J.; Maschler, M. (1995). Repeated Games with Incomplete Information. Cambridge London: MIT Press.
↑ Mertens, J.-F. (1987). "Repeated Games". Proceedings of the International Congress of Mathematicians, Berkeley 1986. Providence: American Mathematical Society. pp. 1528–1577. ISBN 0-8218-0110-4.

Fudenberg, Drew; Tirole, Jean (1991). Game Theory. Cambridge: MIT Press. ISBN 0-262-06141-4.
Mailath, G. & Samuelson, L. (2006). Repeated games and reputations: long-run relationships. New York: Oxford University Press. ISBN 0-19-530079-3.
Osborne, Martin J.; Rubinstein, Ariel (1994). A Course in Game Theory. Cambridge: MIT Press. ISBN 0-262-15041-7.
Sorin, Sylvain (2002). A First Course on Zero-Sum Repeated Games. Berlin: Springer. ISBN 3-540-43028-8.

External links

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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