Purely competitive evolutionary dynamics for games

We introduce and analyze a purely competitive dynamics for the evolution of an infinite population subject to a three-strategy game. We argue that this dynamics represents a characterization of how certain systems, both natural and artificial, are governed. In each period, the population is randomly sorted into pairs, which engage in a once-off play of the game; the probability that a member propagates its type to its offspring is proportional only to its payoff within the pair. We show that if a type is dominant (obtains higher payoffs in games with both other types), its "pure" population state, comprising only members of that type, is globally attracting. If there is no dominant type, there is an unstable "mixed" fixed point; the population state eventually oscillates between the three near-pure states. We then allow for mutations, where offspring have a nonzero probability of randomly changing their type. In this case, the existence of a dominant type renders a point near its pure state globally attracting. If no dominant type exists, a supercritical Hopf bifurcation occurs at the unique mixed fixed point, and above a critical (typically low) mutation rate, this fixed point becomes globally attracting: the implication is that even very low mutation rates can stabilize a system that would, in the absence of mutations, be unstable.