Public Goods Provision in Directed Networks: A Kernel Approach

TL;DR

This paper explores how public goods are provided in directed networks by linking graph kernels to specialized equilibria, establishing existence conditions, and proposing algorithms to analyze stability and network simplification.

Contribution

It introduces a novel connection between graph kernels and specialized equilibria in public goods provision, with new existence, uniqueness, and stability results.

Findings

Specialized equilibria exist almost surely in large random networks

Enhancing reciprocity slightly increases the set of specialized equilibria

A Nash equilibrium is stable only if it is specialized

Abstract

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.