Axiomatic characterization of Nash networks

Abstract

This paper provides an axiomatic approach to characterizing the Nash architectures in directed networks. In a directed network (also called one-way flow networks) when player i establishes a link with player j, only player i is able to access player j's information. Player j must establish a separate link with i to gain access to her information. The common example of such a phenomenon would be visiting webpages. Following their introduction in the economics literature by Bala and Goyal (2000) there is a small but growing body of literature on directed networks. In such a network formation model, directed links are costly but provide benefits to those who establish them. The original Bala and Goyal model assumes that all model parameters (costs and benefits) are homogeneous. Galeotti (2006) introduces a type of heterogeneity into this set up by making cost and benefits depend on the identity of the player under consideration. Billand, Bravard and Sarangi (2009) consider a situation where costs and benefits in the network depend on the identity of the person with whom the link is being formed. Billand, Bravard and Sarangi (2008) examines the issue of existence of equilibrium in directed networks, while directed spillovers are examined by the same authors in another paper (2009). Our goal in this paper is to develop a set of properties of the payoff function under which the equilibria of different models can be easily obtained. The first of these axioms is about the profitability of individual players. It says that if player i is willing to connect to player j then a player to whom j is worth more should also be willing to connect to j. Hence it is about the attractiveness of partners in the network. The second one suggests a player will form fewer links in a network that gives her access to fewer resources. The third axiom called monotonicity with respect to players utilizes the same concept as the second axiom but for players instead of resources. The fourth axiom penalizes players for creating redundant links. We find that under monotonicity with respect to resources wheel type architectures predominate, though with more specific assumptions minimally connected networks can also arise. With player monotonicity, flower networks are the predominant strict Nash architecture. Examples in the paper demonstrate the independence of these axioms.