Cooperation in public goods game in populations of dynamic groups with varying evolution time scales

The cooperation in a large population engaging in the Public Goods Game (PGG) is investigated. Agents have two strategies to choose, i.e., to cooperate or to defect. These agents are temporally formed into groups for playing the PGG. The group sizes are randomly distributed and the payoffs obtained by the two-type agents after playing the games are related to the strategy selection implemented by the death–birth process. The time scales between the interactions of the agents and the strategy update are also considered. We find that the frequency of cooperation as a function of the game parameter is greatly affected by the group size distributions and the time scales of the evolutionary dynamics. At some cases, the cooperative behavior can be analytically solved. In this model, a mixed state with coexistence of cooperators and defectors and a totally cooperative state are observed. The critical game parameter, above which a totally cooperative state will occur, is obtained by an analytical treatment or analyzed by the last-surviving pattern. When the process of the strategy selection is fast, the critical game parameter for the emergence of an AllC state depends on the size scope of the group size distribution and the time scale of the interactions via grouping dynamics. At slow strategy selection, the critical game parameter approaches to rely on the mean group size of the dynamically formed groups. All the theoretical results are verified by computer simulations and they are in a good agreement. •Studied Public Goods Game in dynamic groups for a large population.•Time scales of the evolutionary dynamics greatly affect cooperation.•Derived analytic expression for the frequency of cooperation.•Derived analytic expression for the critical multiplicative factor rc.•Cooperative population emerges for r>r c.