17 Facts About Nash equilibrium

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.

Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers.

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

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

The concept of a mixed-strategy Nash equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games.

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

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

However, a Nash 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 equilibrium equilibria is in determining the expected flow of traffic in a network.

Nash equilibrium defines stability only in terms of unilateral deviations.

Strong Nash equilibrium allows for deviations by every conceivable coalition.

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

Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k CPNE is related to the theory of the core.

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

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

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