Competing for customers in a social network

Pradeep Dubey, Rahul Garg, Bernard De Meyer
2014 Journal of Dynamics & Games  
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
more » ... 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. * pradeepkdubey@yahoo.com, Center for Game Theory, Stony Brook University and Cowles Foundation, Yale University, USA † rahul.garg@operasolutions.com, Opera Solutions, INDIA ‡ demeyer@univ-paris1.fr, PSE-Univesité Paris 1, Paris, FRANCE § This paper is an extension of [5]. The authors would like to thank two anonymous referees for extremely valuable suggestions. 1 clients that are of value to α. This in turn might instigate rival firms to spend further on i, since they wish to wean i away from an excessive tilt toward α; causing α to increase its outlay on i even more, unleashing yet another round of incremental expenditures on i. The scenario invites us to model it as a non-cooperative game between the firms. We take our cue from [3, 12] which explore the optimal marketing strategy of a single firm, based on the "network value" of the customers. (See also [9] for extensions and variants of [3, 12] , developed from an algorithmic standpoint.) Our innovation is to introduce competition between several firms in this setting. The model we present is more general than that of [3, 12] , though inspired by it. As in [3, 12] , the social network, specifying the field of influence of each customer, is taken to be exogenous. Rival firms choose how much money to spend on each customer. For any profile of firms' strategies, the externality effect stabilizes over the social network and leads to unambiguous customer-purchases. This defines the non-cooperative game between firms that we explore here. A prominent instance of our game arises when firms compete for advertisement space on different web-pages in the Internet (see Section 2.2). More generally our game pertains to a burgeoning class of industries in which firms compete for market share via strategies other than price variation. Think of new releases of movies or music albums. Here the prices are more or less the same across different products, and of little consequence for the consumer. The determining factor for him is his exposure to the product through advertisements; and, more importantly, the choices made by his "peer group" with whom he is eager to conform, including the critics whose reviews he values. Another example comes from telephony, where again the prices of different service providers are about the same (sometimes even controlled by a regulatory authority, as in India). But if most of the people that i calls (i.e., i's neighbors in the social network) subscribe to service provider α, and if α-to-α calls have superior connectivity compared to α-to-β calls, then i will have higher incentive to subscribe to α than to β. The reader can no doubt think of other products which are near-substitutes and whose sale prices have settled down to a small margin above cutthroat competitive costs; leaving firms to compete for clientele not by cutting prices, but via enhanced marketing strategies. Indeed, the growth of internet has engendered fierce competition in many product markets, so that only those firms survive which quote rockbottom prices. The competition between the survivors then happens mostly on the marketing plane. Of course this is not to say that there are not other important industries in which the competition between firms is centered on setting prices, taking the social network into account. There is an important strand of literature dealing with this (see in particular [6, 13, 2] for monopoly pricing, [1, 2, 8] for oligopolistic price competition, and the references therein). The model we present here should be viewed as complementary to that strand. Our main interest is in understanding the structure of the NE of the game between the firms. Will they end up as regional monopolies, operating in separate parts of the network? Or will they compete fiercely throughout? Which firms will enter the fray, and which will be blockaded? And how will the money spent on a customer depend on his connectivity in the social network? Section 2 introduces the quasi-linear model (and includes those in [3], [12] by setting # firms = 1). We show that NE are unique and can be easily computed in polynomial time via closed-form expressions involving matrix inverses. It turns out that, provided that there are no a priori biases between firms and customers, any NE has a cut-off cost: all firms whose costs are above the cut-off are blockaded, and the rest enter the fray. Moreover there is no 1 "regionalization" of firms in an NE: each active firm spends money on every customer-node of the social network. The money spent on node i is related to the connectivity of i, but the relation is somewhat subtle, 1 We also show that a priori biases can lead to regional monopolies among the firms.(See Section 3.3.)
doi:10.3934/jdg.2014.1.377 fatcat:atgwdtpkgncc3abluhdnlf7qrq