Nash equilibrium

In game theory, a Nash equilibrium is a situation where no player could gain more by changing their own strategy (holding all other players' strategies fixed) in a game. A Nash equilibrium is the most commonly used solution concept for non-cooperative games. If each player has chosen a strategy — an action plan based on what has happened so far in the game — and no individual player can increase their own expected payoff by changing their strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available 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. The concept of a Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly. John Nash showed that there is a Nash equilibrium, possibly in mixed strategies, for every finite game.