  We study the population genetics of Evolution in the important special case
of weak selection, in which all fitness values are assumed to be close to one
another. We show that in this regime natural selection is tantamount to the
multiplicative updates game dynamics in a coordination game between genes.
Importantly, the utility maximized in this game, as well as the amount by which
each allele is boosted, is precisely the allele's mixability, or average
fitness, a quantity recently proposed in [1] as a novel concept that is crucial
in understanding natural selection under sex, thus providing a rigorous
demonstration of that insight. We also prove that the equilibria in two-person
coordination games can have large supports, and thus genetic diversity does not
suffer much at equilibrium. Establishing large supports involves answering
through a novel technique the following question: what is the probability that
for a random square matrix A both systems Ax = 1 and A^T y = 1 have positive
solutions? Both the question and the technique may be of broader interest.
  [1] A. Livnat, C. Papadimitriou, J. Dushoff, and M.W. Feldman. A mixability
theory for the role of sex in evolution. Proceedings of the National Academy of
Sciences, 105(50):19803-19808, 2008.
