Nash equilibrium

In game theory, defines the Nash equilibrium (by John Forbes Nash) as a way to get an optimal strategy for games involving two or more players. If there is a set of strategies such that no player benefits by changing its strategy while the other will not change theirs, then that set of strategies and corresponding gains constitute Nash equilibrium.

The concept of Nash equilibrium first appeared in his dissertation Non-cooperative games (1950). John Forbes Nash showed that the different solutions that had previously been proposed to offer the games to produce Nash equilibrium.

A game may have no Nash equilibrium, or have more than one. Nash was able to show that if we allow mixed strategies (in which players may choose strategies at random, with a default probability), then every game of n players in which each player can choose between a finite number of strategies have at least a Nash equilibrium with mixed strategies.

If a game has a unique Nash equilibrium and the players are fully rational players will choose strategies that make up the balance.

Examples

Competitive Gaming

Consider the following game for two players:

"The players simultaneously choose a whole number between zero (0) and ten (10). The two players earn less in dollar value, but also, if the numbers are different, which has chosen the more you must pay another $ 2 andalusia.

This game has a unique Nash equilibrium: both players must choose zero (0). Any other strategy may be a disadvantage if another player chooses a number lower.

. If you change the game so that the two players earn the number chosen if both are equal, and otherwise would not win anything, there are 11 different Nash equilibrium.

Coordination games

This game is a game of coordination when driving. The options are driving or driving on the right or the left: 100 means that there is a crash and 0 means yes. The first number in each cell indicates the gain of the first player (whose options are shown at left) and the second gain of the second player (whose options are shown above).

. In this case there are two Nash equilibrium in pure strategies, where both lead to the right or both on the left lead. There is also Nash equilibrium with mixed strategies, where each player chooses randomly with a probability of 50% which of the two strategies applied.

Prisoner's Dilemma

The prisoner’s dilemma is Nash equilibrium: occurs when both players confess. Despite this, both confess is worse than "both cooperating" in the sense that the total time in prison to be met is higher. However, the strategy "both cooperating" is unstable because a player can improve your score if your opponent keeps deserting the cooperation strategy. Thus, "both cooperating" is not a Nash equilibrium, but a Paretian optimum. One way to arrive at this result is achieved through collusion and the promise of each player to "punish" the other if he breaks the agreement. It could also lead to a solution outside the Nash equilibrium if the game is repeated countless times, is achieved when the "eye for an eye."