Optimal play

In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given (Fudenberg & Tirole 1991, p. 29; Gibbons 1992, pp. 33–49). The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response (or one of the best responses) to the other players' strategies (Nash 1950).

Reaction correspondences, also known as best response correspondences, are used in the proof of the existence of mixed strategy Nash equilibria (Fudenberg & Tirole 1991, Section 1.3.B; Osborne & Rubinstein 1994, Section 2.2). Reaction correspondences are not "reaction functions" since functions must only have one value per argument, and many reaction correspondences will be undefined, i.e. a vertical line, for some opponent strategy choice. One constructs a correspondence ${\displaystyle b(\cdot )}$, for each player from the set of opponent strategy profiles into the set of the player's strategies. So, for any given set of opponent's strategies ${\displaystyle \sigma _{-i}}$, ${\displaystyle b_{i}(\sigma _{-i})}$ represents player i 's best responses to ${\displaystyle \sigma _{-i}}$.

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