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 valations of clients are anonymous.
Dubey, Pradeep; Garg, Rahul; and De Meyer, Bernard, "Competing for Customers in a Social Network (R)" (2012). Cowles Foundation Discussion Papers. 2223.