page_title: Best response
section_title: Correspondence
hierarchical_section_title: Best response / Correspondence
caption_attribution_description: English: Reaction correspondence for player Y in the Stag Hunt game, image creasted by Pete.Hurd
context_page_description: In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. 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 to the other players' strategies.
context_section_description: 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 , 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 ,  represents player i 's best responses to .
  Response correspondences for all 2x2 normal form games can be drawn with a line for each player in a unit square strategy space. Figures 1 to 3 graphs the best response correspondences for the stag hunt game. The dotted line in Figure 1 shows the optimal probability that player Y plays 'Stag' (in the y-axis), as a function of the probability that player X plays Stag (shown in the x-axis). In Figure 2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x-axis), as a function of the probability that player Y plays Stag (shown in the y-axis). Note that Figure 2 plots the independent and response variables in the opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the Nash equilibria at the points where the two player's best responses agree in Figure 3.
  There are three distinctive reaction correspondence shapes, one for each of the three types of symmetric 2x2 games: coordination games, discoordination games and games with dominated strategies
  (the trivial fourth case in which payoffs are always equal for both moves is not really a game theoretical problem). Any payoff symmetric 2x2 game will take one of these three forms.
caption: Figure 1. Reaction correspondence for player Y in the Stag Hunt game.
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