Assortative Matching with Inequality in Voluntary Contribution Games

Abstract

Voluntary contribution games are a classic social dilemma in which the individually dominant strategies result in a poor performance of the population. However, the negative zero-contribution predictions from these types of social dilemma situations give way to more positive (near-)efficient ones when assortativity, instead of random mixing, governs the matching process in the population. Under assortative matching, agents contribute more than what would otherwise be strategically rational in order to be matched with others doing likewise. An open question has been the robustness of such predictions when heterogeneity in budgets amongst individuals is allowed. Here, we show analytically that the consequences of permitting heterogeneity depend crucially on the exact nature of the underlying public-good provision efficacy, but generally are rather devastating. Using computational methods, we quantify the loss resulting from heterogeneity vis-a-vis the homogeneous case as a function of (i) the public-good provision efficacy and (ii) the population inequality.

Contribution-Based Grouping under Noise

“…However, in [22], it is proven that, under heterogeneity, there cannot exist an equilibrium where two or more players contribute the same amount. The reason is that, if two or more players contribute the same amount, then the one with the highest endowment would have a profitable deviation in contributing slightly more due to the existence of the mixed group: contributing only slightly more, and he/she is guaranteed not to be placed in a worse group.…”

“…With heterogeneous endowments, symmetric mixed-strategy NE do not exist, either because the only existing equilibrium is non-contribution by all or because of the structure of the high-efficient equilibria (see Proposition A5 in Appendix A and Theorem 2 in [22]). …”

“…A key property of high-contribution equilibria in the baseline case is that there exists a mixed group where both defectors and contributors are placed. In these equilibria, a contributor has a high-enough probability to be placed in a group filled only with contributors and a low enough probability to be placed in the mixed group so that it is, in expectation, not beneficial to free-ride and be placed with certainty in the low group instead.However, in [22], it is proven that, under heterogeneity, there cannot exist an equilibrium where two or more players contribute the same amount. The reason is that, if two or more players contribute the same amount, then the one with the highest endowment would have a profitable deviation in contributing slightly more due to the existence of the mixed group: contributing only slightly more, and he/she is guaranteed not to be placed in a worse group.…”