Costly Participation and The Evolution of Cooperation in the Repeated Public Goods Game

In real life, individuals often need to pay some costs to build and maintain long-termed relationships as well as interactions among them. However, previous studies of the repeated public goods game have focused almost exclusively on the cost-free participation. Here we introduce costly participation to the repeated public goods game in a conditional manner, and study the evolution of cooperation in both deterministic and stochastic dynamics for well-mixed populations. In the limit of an infinite population size, the deterministic dynamics can lead to either a stable coexistence between cooperators and defectors or even a complete dominance of cooperators over defectors if the initial frequency of cooperators is larger than some invasion barrier. In general, defectors are always able to resist invasion by cooperators. However, in finite populations, we show that natural selection can favor the emergence of cooperation in the stochastic dynamics. In the limit of weak selection and large populations, we derive a critical condition required for a cooperator to replace a population of defectors with a selective advantage by using several approximation techniques. Theoretical analysis of the critical condition reveals that participation cost determines the impacts of the number of game round in the emergence of cooperation. Interestingly, there exists an intermediate value of the participation threshold of cooperators leading to the optimal condition for a cooperator to invade and fixate in a population of defectors if the participation cost is smaller than a critical value. Numerical calculations confirm that the validity of the analytical approximations extends to much wider ranges of the selection strength as well as of the population size.