Competing for customers in a social network

Customers' proclivities to buy products often depend heavily on who else is buying the same product. This gives rise to non-cooperative games in which firms sell to customers located in a ``social network''. Nash Equilibrium (NE) in pure strategies exist in general. In the quasi-linear case, NE are unique. If there are no a priori biases between customers and firms, 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. The connectivity of a customer is related to the money firms spend on him. This 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 cost functions of firms are 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.