Evolutionarily Stable Strategies with Two Types of Player

A definition of ESS (evolutionarily stable strategy) is suggested for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole. We check that the standard ESS results hold for this definition and discuss a mate-desertion model which has appeared in the literature in which the two types are male and female. ESS; PLAYERS OF TWO TYPES; MATE DESERTION A number of papers have recently appeared (Dawkins and Carlisle (1976), Maynard Smith (1977), Grafen and Sibly (1978)), which discuss parental investment models (male versus female) using the concept of an evolutionarily stable strategy or ESS. The game being considered in these models has the following form. There are two types of player, male and female, each with its own set of possible strategies. The fitness of a strategy adopted by either type of player depends upon the strategy mix, of both types, in the population as a whole. However, it does not seem to be made clear just what the appropriate definition of ESS should be, for games of this general nature (more than one type of player). In this paper we propose a general definition of ESS for games with more than one type of player. We present versions of the basic ESs results which hold in this more general situation. In particular we observe that our ESSs are dynamically stable (Taylor and Jonker (1978)). Finally we discuss an example of mate desertion of Grafen and Sibly (1978) and find all the ESSS of this model. A model of this type was also considered by Maynard Smith (1977). The above authors restricted themselves to pure state ESSs, but we show that it is possible that one sex might play a mixed strategy at an Ess: some players deserting immediately, and others staying for a suitable common time.