A generalized multi-person game: Evolutionary stability and cooperative behavior in finite populations

We introduce and study a generalized multi-person game that incorporates competitions among N players in groups formed temporarily for every round of s consecutive rounds followed by a round of birth–death process in a finite population of size N. While previously studied N-person public goods and snowdrift games were defined under different contexts, our model defines N-person prisoner’s dilemma, N-person snowdrift game, N-person stag hunt game, and N-person harmony game in a unified manner via the relative alignments of the payoff parameters and consistent with the corresponding 2-person games. The Nash equilibria are analyzed for each of the four games. The model is tractable analytically. We derive analytic expressions for calculating the fixation probability of cooperation Pmi for any initial number of cooperators mi to take over the population. The theory is validated by its complete agreement with results obtained by numerical simulations. It is then applied to investigate the fixation probability of a single cooperator invading a finite population consisting otherwise of defectors, and the opposite scenario of a single defector taking over an otherwise cooperative population. The key features of spanning different N-person games in a unified way and being tractable analytically provide convenience in tuning the model for specific applications in the future as well as insight into interpreting the results.