Cooperation and free riding with moral cost

We study social organizations withpossible coexistence at equilibrium ofcooperating individuals and pure consumers(free-riders). We investigate this polymorphicequilibrium using a game-theoretic approach anda statistical physics analysis of a simple model.The agents face a binary decision problem:whether to contribute or not to the public good,through the maximization of an additive utilitythat has two competing terms, a fixed cost forcooperating and an idiosyncratic moral cost forfree-riding proportional to the fraction ofcooperators. We study the equilibria regimes ofthis model. We show that there is a fraction ofexpected cooperators below which cooperationfails to emerge. Besides the homogeneous stableequilibria (everybody cooperates or everybodyfree-rides), it exists a solution in whichcooperators coexist with free-riders. Thispolymorphic equilibrium is a consequence of theheterogeneous (idiosyncratic) perceptions of thesocial reproval by the different individuals. Weprovide analytic results in the case of a simpledistribution of the idiosyncratic moral weights,and discuss them on the basis of concepts ofgame theory