Adoption and Cooperation in Communities: Mixed aquilibrium in polymorphic populations

Basic evidences on non profi t making and other forms of benevolent-based organizations reveal a rough partition within these groups between some pure consumers of the public good (free-riders) and benevolent individuals (cooperators). This polymorphic con figuration seems to be a stable form of organization in a variety of situations. We study the relationship between the community size and the level of cooperation. We assume that each individual has his own willingness to join the community. Members of the community have an additive utility proportional to the fraction of members in his neighbourhood. Cooperators bear a fixed cost, while free-riders bear a cost idiosyncratically proportional to the fraction of cooperators. Both kinds of members earn a bene fit proportional to the fraction of cooperators in their neighborhood. We consider simultaneously the two inter-linked decisions of the agents: whether to join or not the community and, in the former case, whether to contribute or not to the public good. We study the equilibrium regimes of the polymorphic population using a statistical mechanics approach in which the three possible decisions, to join and cooperate, to join and free-ride and not to join, are considered as Potts variables in a quenched random field, with interactions. We show that there is a threshold to the fraction of cooperators below which free-riders and cooperators cannot co-exist in equilibrium, and we analyze whether a polymorphic equilibrium with a large proportion of free-riders may be better, from a collective point of view, than an equilibrium with fewer members but mostly being cooperators. We provide results in the case of uniform probability distribution of the idiosyncratic weight of the free-riding cost, and di fferent distributions of the adhesion willingness.