Competing for Customers in a Social Network (R)

There are many situations in which a customer’s proclivity to buy the product of any firm depends heavily on who else is buying the same product. We model these situations as non-cooperative games in which firms market their products to customers located in a “social network”. 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 firms could end up as regional monopolies. 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. When we allow for cost functions of firms to be convex, instead of just linear, NE need no longer 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. Finally we develop a general model of nonlinear externalities and show that existence of NE remains intact.