  The evolution of cooperation has been a perennial problem for evolutionary
biology because cooperation is undermined by selfish cheaters (or "free
riders") that profit from cooperators but do not invest any resources
themselves. In a purely "selfish" view of evolution, those cheaters should be
favored. Evolutionary game theory has been able to show that under certain
conditions, cooperation nonetheless evolves stably. One of these scenarios
utilizes the power of punishment to suppress free riders, but only if players
interact in a structured population where cooperators are likely to be
surrounded by other cooperators. Here we show that cooperation via punishment
can evolve even in well-mixed populations that play the "public goods" game, if
the synergy effect of cooperation is high enough. As the synergy is increased,
populations transition from defection to cooperation in a manner reminiscent of
a phase transition. If punishment is turned off, the critical synergy is
significantly higher, illustrating that (as shown before) punishment aids in
establishing cooperation. We also show that the critical point depends on the
mutation rate so that higher mutation rates discourage cooperation, as has been
observed before in the Prisoner's Dilemma.

Populations and Evolution;Quantitative Methods