Public Goods Provision in Directed Networks: A Kernel Approach

This paper investigates the decentralized provision of public goods in directed networks. We establish a correspondence between kernels in graph theory and specialized equilibria in which players either contribute a fixed threshold amount or free-ride entirely. Leveraging this relationship, we derive sufficient conditions for the existence and uniqueness of specialized equilibria in deterministic networks and prove that specialized equilibria exist almost surely in large random networks. We further demonstrate that enhancing network reciprocity weakly expands the set of specialized equilibria without destroying existing ones. Moreover, we propose an iterative elimination algorithm that simplifies the network while preserving equilibrium properties. Finally, we show that a Nash equilibrium is stable only if it is specialized, thereby providing dynamic justification for our focus on this equilibrium class.