Best response dynamics and socially stable strategies

An evolutionarily stable strategy (Maynard Smith and Price, Nature (London) 246 (1973), 15–18) is a strategy which is robust against a tiny invasion of mutants. Best response dynamics is a dynamic process in which the frequency of a strategy increases only if it is a best response to the present strategy distribution. Gilboa and Matsui (Econometrica 59 (1991), 859–867) proposed a stability concept directly derived from this dynamic process. Modifying the above two stability concepts, this paper shows the equivalence between the static concept and the dynamic one. Their set-valued versions always exist. Examples are given to see their usefulness in analyzing forward induction and preplay communication.