  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.
