  In binary decision-makings, individuals often go for a common or rare action.
In the framework of evolutionary game theory, the best-response update rule can
be used to model this dichotomy. Those who prefer a common action are called
coordinators and those who prefer a rare one are called 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 scale with the population size. To fill this
gap, we approximate the discrete finite population dynamics of coordinators and
anticoordinators with the associated mean dynamics in the form of
semicontinuous 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 do not scale as
the population size do. The results encourage to first analyze the often
simpler semicontinuous mean dynamics of the discrete population dynamics as the
semicontinuous dynamics partly reveal the asymptotic behaviour of the discrete
dynamics.
