The dynamics of cooperation in asymmetric sub-populations

Sacrificing personal benefits for a common good is at odds with the fundamental principle of Darwinian evolution: if only the fittest survives, then there should be no place for cooperation. But cooperative behavior actually abounds, and constitutes one of the most persistent and fascinating puzzles of nature. One solution to this puzzle is network reciprocity, where the collective dynamics of cooperators affords them protection against invading defectors. Commonly, however, such a competition does not unfold in isolation. Populations are often divided into sub-populations, with different evolutionary rules describing the interactions between them. Here we propose and study a paradigmatic model that captures the essence of this setup. Specifically, if two players belong to the same sub-population, they play the prisoner's dilemma game. If not, they play either the harmony game, the snowdrift game, the stag-hunt game, or the prisoner's dilemma game. Due to such an asymmetry in the interactions across sub-populations, a fascinating evolutionary dynamics sets up that greatly expands the survivability of cooperators. For instance, when the harmony game applies, cyclic dominance spontaneously emerges, wherein cooperators in one sub-population become predators of defectors in the other sub-population. One also may observe self-organized segregation, wherein both sub-populations maintain a mixed state of cooperators and defectors. As a general rule, we show that the lower the dilemma strength between sub-populations, the more abundant the cooperative strategy in the entire population. Results are confirmed by means of Monte Carlo simulations with pair approximation method, which reveals a rich plethora of novel and generally valid paths to cooperation.