Nash equilibrium
In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players. 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.
The concept of the Nash equilibrium was originated by Nash in his dissertation, Non-cooperative games (1950). Nash showed that the various solutions for games that had been given earlier all yield Nash equilibria.
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 choose from finitely many strategies admits at least one Nash equilibrium of mixed strategies.
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.
Examples
Competition game
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 choses 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.
Coordination game
The coordination game is a classic two person game bi-matrix, person A is usually on the left (and corresponds to the first number in the pair) person B is usually listed on the top (and corresponds to the second number of the pair).
This game is a coordination game for driving. The choices are either to drive on the left or to drive on the right, with 100 meaning no crash and 0 meaning a crash.
| Drive on the Left: | Drive on the Right: | |
| Drive on the Left: | 100,100 | 0,0 |
| Drive on the Right: | 0,0 | 100,100 |
In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. There is one mixed-strategy Nash equilibrium, where each player randomizes between the two strategies with a probability of 50%.
Prisoner's dilemma
The Prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is inferior to "both cooperate", in the sense that the total jail time served by the two prisoners is greater if both defect. The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. Thus, "both cooperate" is not an equilibrium. As Ian Stewart put it, ‘sometimes rational decisions aren't sensible!’