Refinement of the Equilibrium of Public Goods Games over Networks: Efficiency and Effort of Specialized Equilibria

TL;DR

This paper analyzes specialized equilibria in public goods games over networks, showing their efficiency and effort characteristics, and proposing them as a refinement of the equilibrium concept, with results dependent on network structure and benefit function concavity.

Contribution

It introduces a detailed analysis of specialized equilibria, demonstrating their efficiency, effort, and cost properties, and establishes conditions under which they approximate maximum welfare.

Findings

Specialized equilibria approach maximum welfare as benefit function concavity tends to unity.

In forest networks, specialized equilibria require maximum effort and incur minimal total cost.

All welfare-maximizing equilibria are specialized in well-covered forest networks.

Abstract

Recently Bramoulle and Kranton presented a model for the provision of public goods over a network and showed the existence of a class of Nash equilibria called specialized equilibria wherein some agents exert maximum effort while other agents free ride. We examine the efficiency, effort and cost of specialized equilibria in comparison to other equilibria. Our main results show that the welfare of a particular specialized equilibrium approaches the maximum welfare amongst all equilibria as the concavity of the benefit function tends to unity. For forest networks a similar result also holds as the concavity approaches zero. Moreover, without any such concavity conditions, there exists for any network a specialized equilibrium that requires the maximum weighted effort amongst all equilibria. When the network is a forest, a specialized equilibrium also incurs the minimum total cost amongst all equilibria. For well-covered forest networks we show that all welfare maximizing equilibria are specialized and all equilibria incur the same total cost. Thus we argue that specialized equilibria may be considered as a refinement of the equilibrium of the public goods game. We show several results on the structure and efficiency of equilibria that highlight the role of dependants in the network.