Everything about Best Response totally explained
In
game theory, the
best response is the
strategy (or strategies) which produces the most favorable immediate
outcome for the current player, taking other players' strategies as given (; ). The concept of a best response is central to
John Nash's most well-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 .
Best response correspondence
Reaction correspondences, also known as best response correspondences, are used in the proof of the existence of
mixed strategy Nash equilibria (Section 1.3.B;, 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, for example 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
where
represents the expected payoff of action
, and
is a parameter that determines the degree to which the function deviates from the true best response (a larger
implies that the player is more likely to make 'mistakes').
There are several advantages to using smoothed best response, both theoretical and empirical. First, it's consistent with psychological experiments; when individuals are roughly indifferent between two actions they appear to choose more or less at random. Second, the play of individuals is uniquely determined in all cases, since the it's a
function and not a
correspondence. Finally, using smoothed best response with some learning rules (as in
Fictitious play) can result in players learning to play
mixed strategy Nash equilibria .
Further Information
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