Price 2.0
From Evolution and Games
A model with sexual reproduction
If we add sexual reproduction, everything gets slightly more complicated, the proper statistics as well as the Price equation. This makes the statistics much more fun, and it makes the failure of the Price equation to be of any help even more salient. We like to keep it relatively simple though, so for the moment we do not assume actual different sexes, but we do assume that every individual is diploid. That means that individuals can be AA, Aa or aa. The frequency of the gene is then, resp., 1, 1/2 or 0. Individuals are hermaphrodites for the sake of simplicity.
Suppose that we have a population of 4 individuals, q1 = 1, q2 = q3 = 1 / 2 and q4 = 0. The next generation of 4 individuals is drawn as follows. First we draw a father for the first individual. From the father we draw one of the two loci, which gives us an A or an a. Then we draw a mother, and one of her loci. These two loci go into the gametes for reproduction, that together give us the first individual of the offspring generation. Three repetitions of this procedure give us a whole new generation.
We will make two crucial assumptions here.
1) A is not dominant nor recessive in fitness terms
2) Fair meosis
Both assumptions concern probabilities in this update step. The first concerns the probabilities with which an individual is chosen for parenthood. We assume that the probability with which individual i is chosen is
