What pair formation can do to the battle of the sexes: towards more realistic game dynamics.

In the various dynamic models of Dawkin's Battle of the Sexes, payoff matrices serve as the basic ingredients for the specification of a game-dynamic model. Here I model the sex war mechanistically, by expressing the costs of raising the offspring and performing a prolonged courtship via a time delay for the corresponding individuals, instead of via payoff matrices. During such a time delay an individual is not able to have new matings. Only after the delay has occurred, an individual (and its offspring) appears on the mating market again. From these assumptions I derive a pair-formation submodel, and a system of delay-differential equations describing the dynamics of the game. By a time-scale argument, I obtain an approximation of this system by means of a much simpler system of ordinary differential equations. Analysis of this simplified system shows that the model can give rise to two non-trivial asymptotically stable equilibrium points: an interior equilibrium where both female strategies and both male strategies are present, and a boundary equilibrium where only one of the female strategies and both male strategies are present. This behaviour is qualitatively different from that of models of the battle of the sexes formulated in the traditional framework of game-dynamic equations. In other words, the addition of a most elementary further assumption about individual life history fundamentally changes the model predictions. These results show that in analysing evolutionary games one should pay careful attention to the specific mechanisms involved in the conflict. In general, I advocate deriving simple models for evolutionary games, starting from more complex, mechanistic building blocks. The wide-spread method of modelling games at a high phenomenological level, through payoff matrices, can be misleading.