Replicator dynamics for public goods game with resource allocation in large populations

Costly punishment can promote human cooperation, but the effectiveness of punishment is reduced because of the existence of second-order free-rider problem. How to solve the problem remains a challenge for the emergence of costly punishment. Motivated by the regimes of resource allocation in human society, in this work we consider the resource allocation with threshold for the common pool in the public goods game with an additional strategy of peer punishment, and aim to explore whether such proposed resource allocation can solve the problem of second-order free-riders by using replicator equations in infinite well-mixed populations. We assume that if contributing resources in the common pool exceed the threshold, the contributing resources will be divided into two parts: the first part will be equally allocated by all the players, and the second part will be allocated by all the players based on their strategy choices. Otherwise all the contributing resources are equally allocated by all the players. We find that the second-order free-rider problem can be effectively solved by this regime of resource allocation even when most of contributing resources are equally allocated among individuals. In addition, we find that punishment is the dominant strategy in a broad region of allocation parameters. Our work may thus suggest an effective approach about resource allocation for resisting second-order free-riders in the public goods dilemma.