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Nash equilibrium

The Nash equilibrium is a concept in game theory originated by John Nash, who was awarded The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, effectively the Nobel Prize in economics, for his work in the area. It serves to define a kind of "optimum" strategy for games where no such optimum was previously defined. A basic definition is this: If there is a set of strategies for a game with the property that no player can benefit by changing his strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute a Nash equilibrium.

This definition applies to games of two or more players, and Nash showed that the various definitions of "solutions" for games that had been given earlier all yield Nash equilibria.

As a simple example, consider the following two-player game: both players simultaneously choose a whole number between 0 and 10, inclusive. Both players then win the minimum of the two numbers in dollars. In addition, if one player chose a larger number than the other, then he has to pay $2 to the other. This game has a unique Nash equilibrium: both players have to choose 0. Any other choice of strategies can be improved if one of the players lowers his number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 11 Nash equilibria.

If a game has a unique Nash equilibrium and is played among completely rational players, then the players will choose the strategies that form the equilibrium.

A game may have many Nash equilibria, or none. Nash was able to prove that, if we allow mixed strategies (players choose strategies randomly according to preassigned probabilities), then every n-player game in which every player can chose from finitely many strategies admits at least one Nash equilibrium of mixed strategies.

Example: Prisoner's dilemma

The Prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is clearly inferior to "both cooperate". The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. This indicates one of the limitations of using the Nash equilibrium to analyze a game. As Ian Stewart put it, ‘sometimes rational decisions aren't sensible!’

See also