17 Facts About Nash Equilibria

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players.

Nash Equilibria showed that there is a Nash Equilibria equilibrium for every finite game: see further the article on strategy.

The simple insight underlying Nash Equilibria's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation.

Nash Equilibria equilibrium requires that one's choices be consistent: no players wish to undo their decision given what the others are deciding.

Nash Equilibria equilibrium is named after American mathematician John Forbes Nash Equilibria Jr.

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Game theorists have discovered that in some circumstances Nash Equilibria equilibrium makes invalid predictions or fails to make a unique prediction.

However, subsequent refinements and extensions of Nash Equilibria equilibrium share the main insight on which Nash Equilibria's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others.

Nash Equilibria proved that if mixed strategies are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash Equilibria equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each player.

However, a Nash Equilibria equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players.

An application of Nash equilibria is in determining the expected flow of traffic in a network.

Nash Equilibria equilibrium defines stability only in terms of unilateral deviations.

Strong Nash Equilibria equilibrium allows for deviations by every conceivable coalition.

Formally, a strong Nash Equilibria equilibrium is a Nash Equilibria equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members.

However, the strong Nash Equilibria concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication.

Second interpretation, that Nash Equilibria referred to by the mass action interpretation, is less demanding on players:.

Nash Equilibria equilibrium is a superset of the subgame perfect Nash Equilibria equilibrium.

The subgame perfect equilibrium in addition to the Nash Equilibria equilibrium requires that the strategy is a Nash Equilibria equilibrium in every subgame of that game.