Games on Social Networks: On a Problem Posed by Goyal

TL;DR

This paper investigates the existence of pure Nash equilibria in network games, demonstrating that for certain topologies, such equilibria with utility positively related to neighbor count do not exist.

Contribution

It introduces a class of network topologies and game settings where pure Nash equilibria with utility-neighbor count correlation are proven not to exist.

Findings

Pure Nash equilibria with positive neighbor-utility relation do not always exist.

Certain network topologies prevent the formation of such equilibria.

Abstract

Within the context of games on networks S. Goyal (Goya (2007), pg. 39) posed the following problem. Under any arbitrary but fixed topology, does there exist at least one pure Nash equilibrium that exhibits a positive relation between the cardinality of a player's set of neighbors and its utility payoff? In this paper we present a class of topologies/games in which pure Nash equilibria with the above property do not exist.