A Network Centrality Approach to Coalitional Stability

We study games in which each player can simultaneously exert costly effort that provides different benefits to some of the other players. The static analysis of the game yields a prediction of no cooperation, while a standard repeated games approach yields a folk theorem. To obtain more refined predictions about repeated play, we start from the observation that outcomes in such settings are typically negotiated in multilateral meetings involving various subsets of players. Thus, our goal is to find and describe effort profiles that can be sustained in equilibrium despite the possibility of coordinated coalitional deviations. In general, even the existence of such solutions is a difficult matter. This paper argues that there are simple, efficient, and coalition-proof equilibria that can be found by analyzing the setting as a network of marginal benefit flows among players. In these equilibria, each player’s effort is equal to a sum (appropriately weighted) of the efforts of those whose contributions help him at the margin; this is an eigenvector centrality condition in the network. The analysis is done without parametric assumptions on utility functions. To establish the main result, we study connections among three concepts: coalition-proof equilibria of a repeated game; Lindahl equilibria (which are “Walrasian” solutions in a static public goods environment); and effort profiles satisfying the centrality condition. We also find a simple spectral characterization of Pareto-efficient outcomes of our public goods environment: they are the ones where a certain marginal benefits matrix has a largest eigenvalue of 1.