Optimal sharing in social dilemmas

Abstract

Public goods games are frequently used to model strategic aspects of social dilemmas and to understand the evolution of cooperative behaviour among members of a group. While providing a baseline case, a (local) public goods model implies an equal sharing of returns. This appears an unsatisfying modelling choice in contexts where contributors are heterogeneous and returns can be divided freely. Furthermore, it is intrinsically linked to the negative effect of inequality on cooperation, which is observed both theoretically and experimentally. To better understand the link between inequality and cooperation when returns can be shared flexibly, we characterise sharing rules that maximise contributions in an infinitely repeated voluntary contribution game, where players differ in both their endowments as well as the productivities of their contributions. In sharp contrast to egalitarian sharing, we find that such (unequal) sharing rules reduce the incentive to free ride among unequal individuals, and endowment inequality makes cooperation easier to sustain. Maybe surprisingly, this qualitative relation between endowment inequality and cooperation is independent of players' productivities. We derive a unique sharing rule as a function of productivities and endowments that is weakly superior to all other sharing rules. This rule generically departs from both equal as well as proportional sharing. If inequality is high, for example, individuals with the highest endowment need to be compensated more in absolute terms, but their relative share may be significantly less than their proportional contribution. Our analytical findings are qualitatively supported by numerical simulations of simple learning dynamics.