Evolutionary dynamics of cooperation in the N-person stag hunt game

In this paper, we consider the N-person stag hunt game based on the two-person stag hunt game and assume that the payoff of successful stag hunters is larger than that of hare hunters, which is an important feature of the game, but is often ignored in previous works. We first study the evolutionary dynamics of cooperation for the game in infinite well-mixed populations by using the replicator equation, and find that there always exists only one interior equilibrium which is unstable. We then investigate the game in finite well-mixed populations by applying the Markov process, and observe that the equation of gradient of selection always has a unique interior root, which is consistent with the finding in infinite populations. We finally consider the game in structured populations by means of the pair approximation approach. We accordingly obtain the dynamical equation for weak selection to depict the evolutionary dynamics of cooperation in structured populations, and find that there still exists the case in which there is only one interior unstable equilibrium. Our work unveils the universal characteristics of cooperative dynamics in different scenarios for the N-person stag hunt game.