Consider two distinct populations with the same number of individuals
. At each iteration (of time length
), all individuals are randomly matched in pairs made up of one individual from each population to play a symmetric 2×2 game. The two possible actions (or pure strategies) in the game are labeled
. Thus, each individual (regardless of the population to which it belongs) is either an
-strategist or a
-strategist. The payoffs of the game are
(parameters), where, for instance,
denotes the payoff obtained by an
-strategist when he plays with a
At the end of each iteration, after all individuals have played the game, one randomly selected player from each population revises her strategy—
—according to the following rule: "I look at another (randomly selected) individual in my population; if and only if she got a payoff higher than mine, I adopt her strategy". Thus, the game is played between individuals of different populations, but imitation takes place within each population.
The figure shows a simulation of the proportion of
-strategists in each population (in white), its expected dynamics (in dashed red), and the phase plane of the expected dynamics (mean field) in the background.