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Expected Dynamics of an Intra-Population Imitation Model for Inter-Population 2x2 Symmetric Games

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 and . Thus, each individual (regardless of the population to which it belongs) is either an -strategist or a -strategist. The payoffs of the game are , , , and (parameters), where, for instance, denotes the payoff obtained by an -strategist when he plays with a -strategist.
At the end of each iteration, after all individuals have played the game, one randomly selected player from each population revises her strategy— or —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.

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Reference
[1] S. S. Izquierdo and L. R. Izquierdo, "Stochastic Approximation to Understand Simple Simulation Models," Journal of Statistical Physics, Dec 2012. dx.doi.org/10.1007/s10955-012-0654-z

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