Let there be a large (continuum) population, composed of males
and females, who are involved in the so-called sex-ratio game. The females are
the genuine players of the game in that they are the ones who determine the sex
probabilities among their offspring. Suppose there are two options (or possible
strategies) in this respect. The first one, strategy s1, involves a probability of 0.1
that any given offspring be a male (thus 0.9 that it be a female) whereas in the
second one, s2, this probability is 0.6. To produce offspring, every female needs
the concourse of a male. However, independently of the characteristics of the male,
all females bear the same number of offspring (independently of the latter’s sex).
For each female, her payoffs are taken to be proportional to the number of grandchildren
they breed. (Of course, these grandchildren may be obtained either as the
offspring of their sons or the offspring of their daughters.) Thus, in contrast with the
leading context studied throughout this chapter (cf. Subsection 10.2.1), the players
in the present evolutionary game (i.e., the females) play the field. Their interaction
is not conducted through random matching in terms of a fixed bilateral game
and, therefore, their expected payoffs are not given by a linear combination of the
1. Define precisely the payoff function of the game, π : S × S → R, where π(s, s_) specifies the number of grandchildren bred by a female when her strategy is s and that of the rest of the population is, monomorphically, s_.
2. Redefine the notion of ESS presented in Definition 10.1 for the present context.
3. Show that no pure strategy can be an ESS.
4. Find the unique (mixed-strategy) ESS. Show that it induces an equal sex ratio (i.e., half males and half females) over the whole population.