Games on Social Networks: On a Problem Posed by Goyal

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.