infinitely repeated game

Nash Equilibrium

Nash equilibrium, named after the Nobel laureate John Nash, is the main solution concept used in game theory. Given a game between two or more players, the Nash equilibrium determines which strategies each player should select in order for all players to be in optimal positions. In most situations, the Nash equilibrium provides the best solution for all players involved since it stops all players from independently changing their strategies, despite knowing that by doing so they may get better results.

To better understand Nash equilibrium and its importance in game theory, one must first understand the Prisoners’ Dilemma. In this game, two prisoners must decide whether to cooperate or to defect while they are being held by the police. If both prisoners cooperate, they will each receive a light sentence. If both prisoners defect, they will each receive a medium sentence. However, if one prisoner defects and the other cooperates, the defector will get a much lighter sentence and the cooperator will get a much harsher sentence. The outcome of this game is best represented by a payoff matrix, which shows the rewards each prisoner receives depending on what the other prisoner decides.

If the prisoners do not have any way of communicating, the Nash equilibrium of this game would stipulate that both prisoners should defect. This is because the prisoners will not be able to benefit from cooperation and must assume that the other prisoner is defecting as well. Although both prisoners would be better off if they could trust each other to cooperate, the Nash equilibrium defines the best-case scenario and it is no surprise that both prisoners should defect in this case.

But this is where the Nash equilibrium comes into play; it allows the two prisoners to communicate beforehand and find a way to benefit from cooperation. If one prisoner agrees to cooperate, the other prisoner has an incentive to cooperate as well (since they will receive a lighter sentence than if they both defect). Likewise, if one prisoner agrees to defect, the other prisoner has an incentive to defect too (since they will still be better off than if both cooperate). Therefore, the Nash equilibrium for this game would be for both prisoners to cooperate; a result which benefits both players.

The Nash equilibrium is not only important in game theory, but also in everyday life. For example, when an employer is negotiating a salary with a job applicant, the Nash equilibrium would be to agree on a salary that justifies the work they will be doing, thus ensuring that both parties benefit. In this game, the employer would not have an incentive to offer more salary than the job applicant is asking for (since they will be overpaying); and the job applicant would no longer have an incentive to ask for more than what the employer is willing to pay (since they will end up with less money than they get if they accepted the employer’s offer).

In summary, the Nash equilibrium is a solution concept used in game theory to determine the best strategies for all players involved. It is based on the idea of mutual benefit and rationality—both parties must be better off if the other party chooses a certain strategy. The Prisoners’ Dilemma provides a simple example of the Nash equilibrium, which explains why the two prisoners should cooperate if they are able to communicate beforehand. Finally, the Nash equilibrium can be applied in real life, where rational decision-making can lead to mutually beneficial outcomes.

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game or base 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 feedback is studied in a sequence or a timeline of events.

In infinite repeated games, the stage game is repeated any number of times with full or perfect monitoring. Each repetition of the game is treated as a stage of the extensive form, thus all players are assumed to recall the events which have happened in all previous stages. The stage game may be a non-cooperative game composed of two or more players (or firms functioning as players).

In the theory of repeated games, a class of dynamic learning models are used in order to analyze the behavior of each player in such a situation. As learning models, they assume that players have a knowledge structure which is composed of beliefs, preferences, and strategies. The goal of these models is to understand and predict how players may modify their strategies as a function of their expectations about the behavior of other players throughout a game.

In the study of these games, the game is often modeled as a Markov chain, allowing the use of dynamic programming techniques to study the behavior of the players. Through this, researchers have been able to determine the Nash equilibrium of full information perfect monitoring repeated games, by which the optimal theoretical prediction of behavior can be formulated.