Nash equilibrium

In game theory, the Nash equilibrium, named after a mathematician John Forbes Nash Jr., is the near common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, used to refer to every one of two or more people or matters player is assumed to know the equilibrium strategies of the other players, as well as no one has anything to draw by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs.

If regarded and forwarded separately. player has chosen a strategy – an action plan based on what has happened so far in the game – together with no one can put one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current mark of strategy choices constitutes a Nash equilibrium.

If two players Alice and Bobstrategies A and B, A, B is a Nash equilibrium whether Alice has no other strategy usable that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy usable that does better than B at maximizing his payoff in response to Alice choosing A. In a game in which Carol and Dan are also players, A, B, C, D is a Nash equilibrium if A is Alice's best response to B, C, D, B is Bob's best response to A, C, D, and so forth.

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

Definitions

A strategy outline is a bracket of strategies, one for each player. Informally, a strategy sorting is a Nash equilibrium if no player can make better by unilaterally changing his strategy. To see what this means, imagine that used to refer to every one of two or more people or things player is told the strategies of the others. Suppose then that each player asks himself: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I return by changing my strategy?"

If all player could"Yes", then that set of strategies is non a Nash equilibrium. But if every player prefers non to switch or is indifferent between switching and not then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to the other players' strategies in that equilibrium.

Formally, allow be the set of any possible strategies for player , where . let be a strategy profile, a set consisting of one strategy for each player, where denotes the strategies of all the players apart from . Let be player i's payoff as a function of the strategies. The strategy profile is a Nash equilibrium if

A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak: a player might be indifferent among several strategies condition the other players' choices. this is the unique and called a strict Nash equilibrium if the inequality is strict so one strategy is the unique best response:

Note that the strategy set can be different for different players, and its elements can be a variety of mathematical objects. most simply, a player might choose between two strategies, e.g. Or, the strategy set might be a finite set of conditional strategies responding to other players, e.g. Or, it might be an infinite set, a continuum or unbounded, e.g. such(a) that is a non-negative real number. Nash's existence proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it.

The Nash equilibrium may sometimesnon-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal.

Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out. For such(a) games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?"

If every player'sis "Yes", then the equilibrium is classified as a strict Nash equilibrium.

If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that enable exactly the same payout i.e. this player is indifferent between switching and not, then the equilibrium is classified as a weak Nash equilibrium.

A game can have a pure-strategy or a mixed-strategy Nash equilibrium. In the latter a pure strategy is chosen stochastically with a constant probability.

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

Nash equilibria need not represent if the set of choices is infinite and non-compact. An example is a game where two players simultaneously name a number and the player naming the larger number wins. Another example is where each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists if the number could exist 5, the Nash equilibrium would have both players choosing 5 and tying the game. 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.