  We model evolution according to an asymmetric game as occurring in multiple
finite populations, one for each role in the game, and study the effect of
subjecting individuals to stochastic strategy mutations. We show that, when
these mutations occur sufficiently infrequently, the dynamics over all
population states simplify to an ergodic Markov chain over just the pure
population states (where each population is monomorphic). This makes
calculation of the stationary distribution computationally feasible. The
transition probabilities of this embedded Markov chain involve fixation
probabilities of mutants in single populations. The asymmetry of the underlying
game leads to fixation probabilities that are derived from
frequency-independent selection, in contrast to the analogous single-population
symmetric-game case (Fudenberg and Imhof 2006). This frequency independence is
useful in that it allows us to employ results from the population genetics
literature to calculate the stationary distribution of the evolutionary
process, giving sharper, and sometimes even analytic, results. We demonstrate
the utility of this approach by applying it to a battle-of-the-sexes game, a
Crawford-Sobel signalling game, and the beer-quiche game of Cho and Kreps
(1987).
