Dalarna University's logo and link to the university's website

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 cooper-ators 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 indeed punishment aids in establishing cooperation. We also show that the critical point depends on the mutation rate so that higher mutation rates actually promote cooperation, by ensuring that punishment never disappears.