bestresponse

In game theory, a best response for a player is a strategy that yields the highest possible payoff given the strategies chosen by the other players. A best response depends on the opponents' actions and may vary across different strategy profiles. The concept is central to analyzing strategic interaction.

In a normal-form game with players i = 1, ..., n and strategy sets S_i, the best response of player i to a profile s_-i of the others is BR_i(s_-i) = argmax_{s_i in S_i} u_i(s_i, s_-i). When players can mix strategies, the best response to opponents' mixed strategy σ_-i is the set of (possibly mixed) strategies that maximize expected utility: BR_i(σ_-i) = argmax_{τ_i in Δ(S_i)} E[u_i(τ_i, σ_-i)].

A best-response correspondence BR_i maps opponents' strategies to sets of best responses, and the joint best-response correspondences define the set of all compatible strategy profiles.

A strategy profile s* is a Nash equilibrium if each player's strategy is a best response to the others, i.e., s*_i ∈ BR_i(s*_-i) for all i. Thus, equilibria are fixed points of the best-response mappings.

Examples illustrate the concept. In the prisoner's dilemma, defecting is a dominant best response to any action of the other, making (Defect, Defect) a Nash equilibrium. In matching pennies, no pure-strategy best response exists to the other's mixed strategy; the unique Nash equilibrium is for both players to randomize equally.

Computing best responses is a standard step in equilibrium analysis. In simple games, it involves comparing payoffs; in larger games, it may require solving optimization problems or using iterative best-response dynamics, which may converge to equilibrium or cycle without convergence.