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Partner selections in public goods games with constant group size

Most of previous studies concerning the Public Goods Game assume either participation is unconditional or the number of actual participants in a competitive group changes over time. How the fixed group size, prescribed by social institutions, affects the evolution of cooperation is still unclear. We propose a model where individuals with heterogeneous social ties might well engage in differing numbers of Public Goods Games, yet with each Public Goods Game being constant size during the course of evolution. To do this, we assume that each focal individual unidirectionally selects a constant number of interaction partners from his immediate neighbors with probabilities proportional to the degrees or the reputations of these neighbors, corresponding to degree-based partner selection or reputation-based partner selection, respectively. Because of the stochasticity the group formation is dynamical. In both selection regimes, monotonical dependence of the stationary density of cooperators on the group size was found, the former over the whole range but the latter over a restricted range of the renormalized enhancement factor. Moreover, the reputation-based regime can substantially improve cooperation. To interpret these differences, the microscopic characteristics of individuals are probed. We later extend the degree-based partner selection to general cases where focal individuals have preferences towards their neighbors of varying social ties to form groups. As a comparison, we as well investigate the situation where individuals locating on the degree regular graphs choose their co-players at random. Our results may give some insights into better understanding the widespread teamwork and cooperation in the real world.