Cooperate More, Connect Less: The Network Consequences of Zero-Sum Environments

We study how the degree of zero-sumness in a social environment shapes cooperation and network formation. When individual gains partly come at others’ expense, agents face a trade-off between contributing to a public good and extracting rents from their network position. We model this as a public good game on a directed, weighted network with a parameter capturing the intensity of zero-sum conflict. Our Zero-Sum Theorem shows that greater zero-sumness increases contributions while reducing social connectedness: agents sever links to limit exposure to extraction and increase self-provision to compensate for the diminished free-riding opportunities associated with sparser networks. The unique Nash equilibrium has contributions equal to weighted sign-alternating Bonacich centralities, reflecting how network position shapes behavior through alternating spillovers while incorporating extraction incentives. With endogenous links, equilibrium total connectivity is uniform across homogeneous agents, and heterogeneity in extraction incentives generates an inverse relationship between rent-extraction benefits and network centrality.