In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two firms in a market competing to maximize the consumption of their products. Each firm has a limited budget which can be invested on the quality of the product or spent on seeding key individuals in the network to incentivize others to consume more of their product. After the decision of firms, agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for their consumption decision. We characterize the unique Nash equilibrium of the game between firms and study the effect of the budgets as well as the network structure on the optimal allocation. We show that at the equilibrium, firms invest more budget on quality when their budgets are close to each other. However, as the gap between budgets widens, competition in qualities becomes less effective and firms spend more of their budget on seeding. We also show that given equal budget of firms, if the seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if the seeding budget is zero for a star graph it will be zero for any other graph as well. As a practical extension, we then consider a similar setup for two products that are already being offered with some preset qualities in a network. At some point in time, firms learn about the network structure and decide to utilize a limited budget to mount their market share by either improving the quality or new seeding some agents to incline consumers toward their products. We show that the optimal budget allocation in this case simplifies to a threshold strategy. Interestingly, we derive similar results to those of the original problem, in which preset qualities simulate the role that budgets had in the original setup. Index Terms-Competitive diffusion, game theory, network externalities, social networks. upper bound for the proportion of agents adopting a new product where agents follow a threshold model. Montanari and Saberi in [19] studied the spread of innovation in social networks which are based on the dynamics of coordination games. They showed that innovation spreads much more slowly on well-connected network structures dominated by long-range links than in low-dimensional ones dominated, for example, by geographic proximity. This result differs from the ones studied in epidemic models [23], [24] . Recently, a game-theoretic model of competition and product adoption was proposed by Goyal and Kearns in [25] . They computed upper bounds of the price of anarchy and showed how network structure may amplify the initial budget differences. Similarly, in [26] , Bimpikis, Ozdaglar, and Yildiz proposed a game-theoretic model of competition between firms which can target their marketing budgets to individuals embedded in a social network. They provided conditions under which it is optimal for the firms to asymmetrically target a subset of the individuals. Also, Chasparis and Shamma assumed a dynamical model of preferences in [11] and computed optimal policies for finite and infinite horizon where endogenous network influences, competition between two firms, and uncertainties in the network model were studied. Moreover, the equilibria of network games with linear best response functions have been completely characterized in [27] . Similarly, optimal pricing in a monopoly was recently characterized in [28] . These results are based on the assumption of a quadratic payoff function for agents which is assumed in [6] and [29] as well. Arastoo Fazeli received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, the M.A. in statistics and the Ph.D. degree in electrical and system engineering from Wharton School, University of Pennsylvania, Philadelphia, PA. USA. Currently, he is a Data Scientist at Google, where he works on modeling and analysis of auctions pricing. He has conducted research on option trading as well as marketing and pricing of products in social networks and consensus in random networks. His research interests include stochastic processes, game theory, and optimization with applications to operations research and finance problems.
The goal of REF was to investigate whether a firm with a limited budget should spend on the quality of the product or on initial seeding in the network.
Competitive Diffusion in Social Networks: Quality or Seeding?
IEEE Transactions on Control of Network Systems
IEEE Transactions on Control of Network Systems
Mathematics Computer Science