Cooperation in public goods game does not require assortment and depends on population density

The threshold public goods game is one of the best-known models of
nonlinear public goods dilemmas. Cooperators and defectors typically
coexist in this game when the population is assumed to follow the
so-called structured deme model. In this paper we develop a dynamical
model of a general N-player game in which there is no deme structure:
individuals interact with randomly chosen neighbours and selection occurs
between randomly chosen pairs of indi- viduals. We show that in the
deterministic limit the dynamics in this model leads to the same
replicator dynamics as in the structured deme model, i.e. coexistence of
cooperators and defectors is typical in threshold public goods game even
when the population is completely well-mixed. We extend the model to study
the effect of density dependence and density fluctuation on the dynamics. We
show analytically and numerically that decreasing population density
increases the equilibrium frequency of cooperators till the fixation of
this strategy, but below a critical density coop- erators abruptly
disappear from the population. Our numerical investigations show that weak
density fluctuations enhance cooperation, while strong fluctuations suppress
it.