  We introduce and analyze a purely competitive dynamics for the evolution of
an infinite population subject to a 3-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 non-zero 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.
