Cooperation in public goods games: Stay, but not for too long

Cooperation in repeated public goods game is hardly achieved, unless contingent behavior is present. Surely, if mechanisms promoting positive assortment between cooperators are present, then cooperators may beat defectors, because cooperators would collect greater payoffs. In the context of evolutionary game theory, individuals that always cooperate cannot win the competition against defectors in well-mixed populations. Here, we study the evolution of a population where fitness is obtained in repeated public goods games and players have a fixed probability of playing the next round. As a result, the group size decreases during the game. The population is well-mixed and there are only two available strategies: always cooperate (ALLC) or always defect (ALLD). Through numerical calculation and analytical approximations we show that cooperation can emerge if the players stay playing the game, but not for too long. The essential mechanism is the interaction between the transition from strong to weak altruism, as the group size decreases, and the existence of an upper limit to the number of rounds representing limited time availability.