A note on evolutionary stable strategies and game dynamics.

In 1974 J. Maynard Smith introduced the fundamental notion of an evolutionarily stable strategy (ESS) in order to explain the evolution of genetically determined social behaviour within a single animal species. If the possible pure strategies for contests within a species are 1,2,. . ., ~1, and if A = (aij) is the payoff matrix, then aij is the payoff for the pure strategy i played against the pure strategy j; c aijqj is the payoff for the pure strategy i against the mixed strategy given by $ probability vector q = be the simplex of all possible strategies. Dl : (Maynard Smith, 1974) A state p E S, is called an ESS if for all q # p either pAp > qAp or pAp = qAp and pAq > qAq. In Taylor & Jonker (1978) the authors used the fact that the payoff, in animal contests, corresponds " by definition " to the rate of increase. This suggests for the investigation of the evolution of behaviour the dynamical model given by ~i/Xi = C aijxj. j With this equation, however, the strategies (xi,. . ., x,) don't remain on the simplex. But since only the differences in payoff are relevant for the game, one may consider ii/xi = c aijxj-A, j where the function A is chosen in such a way that 1 ii = 0 whenever 609 0022%5193/79/230609+04 $02.00/O 0 1979 Academic Press Inc. (London) Ltd.