Coevolutionary dynamics of initial investment and reciprocation rate in continuous public goods games

Direct reciprocity is a mechanism for the evolution of cooperation. Iterated games provide a game theoretic framework for studying the evolution of cooperation through direct reciprocity. Here, we study the iterated two-player continuous public goods game in finite populations by adaptive dynamics. We consider that cooperative investments are determined by the initial investment and the reciprocation rate which describes the relation between a player's current investment and its opponent's last investment. Many previous models neglect the interaction between the initial investment and the reciprocation rate and investigate them separately. Thus, it is still unclear how the coevolution of these two traits influences the game dynamics. Here, we study their coevolution. Mean strategies and generous ones are proposed. Players who adopt the former (latter) always invest no more (less) than what their opponents did in the last round. We find that the evolutionary fate of a mean population is entirely determined by the reciprocation rate: if it exceeds a threshold, the population evolves to full cooperation; otherwise, it evolves to full defection. For a generous population, full cooperation is always achieved.