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A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). [1] The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly. [2]
The solutions are normally based on the concept of Nash equilibrium, and these solutions are reached by using methods listed in Solution concept. Most solutions used in non-cooperative game are refinements developed from Nash equilibrium, including the minimax mixed-strategy proved by John von Neumann. [8] [13] [20]
A Nash equilibrium is a strategy profile (a strategy profile specifies a strategy for every player, e.g. in the above prisoners' dilemma game (cooperate, defect) specifies that prisoner 1 plays cooperate and prisoner 2 plays defect) in which every strategy played by every agent (agent i) is a best response to every other strategy played by all the other opponents (agents j for every j≠i) .
Nash equilibria Sequential Perfect information Zero sum Move by nature; Battle of the sexes: 2 2 2 No No No No Blotto games: 2 variable variable No No Yes No Cake cutting: N, usually 2 infinite variable [1] Yes Yes Yes No Centipede game: 2 variable 1 Yes Yes No No Chicken (aka hawk-dove) 2 2 2 No No No No Coordination game: N: variable >2 No No ...
The decision of each player can be viewed as determining two angles. Symmetric Nash equilibria that attain a payoff value of / for each player is shown, and each player volunteers at this Nash equilibrium. Furthermore, these Nash equilibria are Pareto optimal. It is shown that the payoff function of Nash equilibria in the quantum setting is ...
Strict stationary equilibria: [6] A Nash equilibrium is called strict if each player strictly prefers the infinite sequence of outcomes attained in equilibrium, over any other sequence he can deviate to. A Nash equilibrium is called stationary if the outcome is the same in each time-period. An outcome is attainable in strict-stationary ...
A Bayesian Nash Equilibrium (BNE) is a Nash equilibrium for a Bayesian game, which is derived from the ex-ante normal form game associated with the Bayesian framework. In a traditional (non-Bayesian) game, a strategy profile is a Nash equilibrium if every player's strategy is a best response to the other players' strategies.
Unlike the pure Nash equilibria, the mixed equilibrium is not an evolutionarily stable strategy (ESS). The mixed Nash equilibrium is also Pareto dominated by the two pure Nash equilibria (since the players will fail to coordinate with non-zero probability), a quandary that led Robert Aumann to propose the refinement of a correlated equilibrium.