Competitive Centrality: Equilibrium and Comparative Statics in Networks

We study equilibrium behavior and the propagation of local shocks in networked competition. We introduce competitive centrality, a nonlinear centrality measure that is uniquely determined by network primitives and provides a sufficient statistic for equilibrium behavior. It determines players’ effort allocations and winning probabilities in each bilateral competition. We then use competitive centrality to derive exact expressions for equilibrium responses to local shocks and characterize their signs. Local shocks propagate along network paths according to players’ relative competitive positions. Although these responses can be non-monotone, a simple sign rule determines whether a shock raises or lowers competitive centrality and effort along a propagation path. Together, these results show how network structure and relative competitive positions shape equilibrium responses to local shocks. Interventions that appear to reduce competition—including removing players or links and imposing effort caps—can therefore have unintended consequences elsewhere in the network.