Nash equilibrium

In game theory, a Nash equilibrium, named after a mathematician John Forbes Nash Jr., is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, regarded and noted separately. player is assumed to know the equilibrium strategies of the other players, and 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 indicated separately. player has chosen a strategy – an action schedule based on what has happened so far in the game – & no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current bracket of strategy choices constitutes a Nash equilibrium.

If two players Alice and Bobstrategies A and B, A, B is a Nash equilibrium if 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.

Stability

The concept of stability, useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria.

A Nash equilibrium for a mixed-strategy game isif a small conform specifically, an infinitesimal conform in probabilities for one player leads to a situation where two conditions hold:

If these cases are both met, then a player with the small modify in their mixed strategy will service immediately to the Nash equilibrium. The equilibrium is said to be stable. If assumption one does not realise then the equilibrium is unstable. If only precondition one holds then there are likely to be an infinite number of optimal strategies for the player who changed.

In the "driving game" example above there are bothand unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player reform their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The 50%,50% equilibrium is unstable. If either player redesign their probabilities which would neither return or damage the expectation of the player who did the change, if the other player's mixed strategy is still 50%,50%, then the other player immediately has a better strategy at either 0%, 100% or 100%, 0%.

Stability is crucial in practical a formal request to be considered for a position or to be allowed to do or have something. of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to occur in practice, since all minute change in the proportions of used to refer to every one of two or more people or matters strategy seen will lead to a change in strategy and the breakdown of the equilibrium.

The Nash equilibrium defines stability only in terms of unilateral deviations. In cooperative games such a concept is non convincing enough. Strong Nash equilibrium ensures 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 concept is sometimes perceived as too "strong" in that the environment provides for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto efficient/a>. As a calculation of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.