Consider a large population of individuals which can be in one of two distinct roles. The role of the individual is switched every now and then, and interactions occur between randomly paired individuals in different roles. These interactions are represented by a bimatrix game and individuals are modeled as boundedly rational expected utility maximizers who choose their actions according to a myopic best response rule. The resulting dynamics of the population state is given by a system of differential equations and differential inclusions. If the bimatrix game is zero-sum, the population state converges to a fixed point set corresponding to the set of Nash equilibria of this game. Moreover, if the zero-sum game has a unique Nash equilibrium, the global attractor of the population state is a unique and explicitly computable fixed point, even if the set of fixed points is a continuum (which is the case, if the unique Nash equilibrium is completely mixed). This global attractor does neither depend on the rates of role switching and strategy reviewing, nor on the relative size of the two subpopulations of individuals in different roles.
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