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From Discrete to Continuous Binary Best-Response Dynamics: Discrete Fluctuations Almost Surely Vanish with Population Size

In binary decision-making, individuals often choose either the rare or the common action. In the framework of evolutionary game theory, the best-response update rule can be used to model this dichotomy. Those who prefer the common action are called \emph{coordinators}, and those who prefer the rare one are called \emph{anticoordinators}. A finite mixed population of the two types may undergo perpetual fluctuations, the characterization of which appears to be challenging. It is particularly unknown whether the fluctuations persist as population size grows. To fill this gap, we approximate the discrete finite population dynamics of coordinators and anticoordinators with the associated mean dynamics in the form of differential inclusions. We show that the family of the state sequences of the discrete dynamics for increasing population sizes forms a generalized stochastic approximation process for the differential inclusion. On the other hand, we show that the differential inclusions always converge to an equilibrium. This implies that the reported perpetual fluctuations in the finite discrete dynamics of coordinators and anticoordinators almost surely vanish with population size. The results encourage to first analyze the often simpler {continuous-time} mean dynamics of the discrete population dynamics as the continuous-time dynamics partly reveal the asymptotic behavior of the discrete dynamics.

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