Competing for Customers in a Social Network (R)

31 Pages Posted: 10 May 2012

See all articles by Pradeep K. Dubey

Pradeep K. Dubey

SUNY Stony Brook - Center for Game Theory in Economics

Rahul Garg

IBM India Research Lab

Bernard De Meyer

University of Paris I Pantheon-Sorbonne

Date Written: May 9, 2012

Abstract

There are many situations in which a customer’s proclivity to buy the product of any firm depends not only on the classical attributes of the product such as its price and quality, but also on who else is buying the same product. Under quite general circumstances, it turns out that customers’ influence on each other dynamically converges to a steady state. Thus we can model these situations as games in which firms compete for customers located in a “social network”. A canonical example is provided by competition for advertisement on the web.

Nash Equilibrium (NE) in pure strategies exist in general. In the quasi-linear version of the model, NE turn out to be unique and can be precisely characterized. If there are no a priori biases between customers and firms, then there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. Otherwise there is a tendency towards regionalization, with firms dominating disjoint territories.

We also explore the relation between the connectivity of a customer and the money firms spend on him. This relation becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network. Finally we consider convex (instead of linear) cost functions for the firms. Here NE need not be unique as we show via an example. But uniqueness is restored if there is enough competition between firms or if their valuations of clients are anonymous.

Keywords: Nash equilibrium, Social network, Advertisement on the web

JEL Classification: C7, D2, D4, L1

Suggested Citation

Dubey, Pradeep K. and Garg, Rahul and De Meyer, Bernard, Competing for Customers in a Social Network (R) (May 9, 2012). Cowles Foundation Discussion Paper No. 1862, Available at SSRN: https://ssrn.com/abstract=2055215 or http://dx.doi.org/10.2139/ssrn.2055215

Pradeep K. Dubey (Contact Author)

SUNY Stony Brook - Center for Game Theory in Economics ( email )

Stony Brook, NY 11794
United States
631-632-7555 (Phone)
631-632-7516 (Fax)

Rahul Garg

IBM India Research Lab ( email )

Block 1, IIT Campus
Hauz Khas
New Delhi, 110016
India

Bernard De Meyer

University of Paris I Pantheon-Sorbonne ( email )

12, place du Panthéon
Paris, IL
France

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