Instead, one must ask what each player would do taking into account what she/he expects the others to do. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers. 8.2 Alternate proof using the Brouwer fixed-point theorem.8.1 Proof using the Kakutani fixed-point theorem.Nash showed that there is a Nash equilibrium for every finite game: see further the article on strategy. 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. 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. If each player has chosen a strategy – an action plan choosing their own actions based on what has happened so far in the game – and no 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. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only their own strategy. In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is the most common way to define the solution of a non-cooperative game involving two or more players. Rationalizability, Epsilon-equilibrium, Correlated equilibriumĮvolutionarily stable strategy, Subgame perfect equilibrium, Perfect Bayesian equilibrium, Trembling hand perfect equilibrium, Stable Nash equilibrium, Strong Nash equilibrium, Cournot equilibrium It has been suggested that portions of this article be split out into another article titled Existence proofs for Nash equilibrium.