Coyness, Philandering and Stable Strategies

An example put forward by Dawkins to describe the evolution of strategies in the conflict of the sexes over parental investment is discussed by means of a simple dynamic system. It is shown that the equilibrium of the strategies is not evolutionarily stable, but equal to the time average of the endlessly oscillating strategies. The theory of games and the notion of evolutionarily stable strategies, as introduced by Maynard Smith & Price (1973), has proved to be very helpful for the understanding of the evolution of animal conflicts. Recently, Zeeman (1979), Taylor & Jonker (1978) and Hofbauer et al. (1979) have used a class of ordinary differential equations which makes the underlying dynamics of this approach more explicit. In this paper we show how to use this method to investigate situations leading to oscillations of genetically influenced social behaviour. Within the context of Trivers's (1972) theory of parental investment, we shall discuss an example introduced by Dawkins (1976) and also described by Wickler & Seibt (1977). Suppose that the pure strategies for contests within one species are labelled 1 , . . . , n and that aij is the payoff for the player using the pure strategy i when his adversary uses the pure strategy j. Then ~ a~jq~ is the payoff for the pure J strategy i against the mixed strategy given by the probability vector q = (q~ . . . . . qn), and algpiq~ is the payoff for the strategy p = t , j (P 1 . . . . . pn) played against q. Let A denote the payoff matrix (aig) and let S , = (x = (Xl . . . . . x , ) ~ rR n : Zxj = 1,