We consider a situation where wireless service providers compete for heterogenous wireless users. The users differ in their willingness to pay as well as in their individual channel gains. We prove existence and uniqueness of the Nash equilibrium for the competition of two service providers, for a generic channel model. Interestingly, the competition of two providers leads to a globally optimal outcome. We extend some of the results to the case where more than two providers are competing. Finally, we provide numerical examples that illustrate the effects of various parameters on the Nash equilibrium.
@cite_10: This paper addresses the behavior of the selfish service providers in the form of IP sinks providing high-speed IP access. Service providers compete for mobile users by adjusting the price they charge for their services. Their aim is to maximize the total collected profit. Mobile users are also selfish choosing the service provider offering the best quality of service and price combination. As the service providers come closer to each other, we show the existence of three critical phase transitions in their behavior. Depending on the separation between them, there may exists a unique Nash equilibrium, or a continuum of Nash equilibria, or no Nash equilibrium. We completely characterize the pricing strategies of service providers at Nash equilibria. We also prove that the total social welfare in the presence of selfish providers is close to the maximum social welfare that can reached through non- selfish optimization.
@cite_2: One of the most successful applications of the price of anarchy---the worst-case ratio between the objective function values of noncooperative equilibria and optima---is to "selfish routing", a classical model of how independent network users route traffic in a congested network. However, almost all existing work on this topic (e.g., [2, 5, 7]) assumes a large population of very small network users, so that the actions of a single individual have negligible impact on the cost incurred by others. This assumption---in game theory terminology, that the game is nonatomic---is obviously not justifiable in all applications.
Until recently, the heterogeneity of the users was largely ignored. The first work that explicitly takes into account the channel differences for different users is @cite_10 . To our knowledge, our work is the first one to consider the pricing competition of providers for users who are heterogenous in both willingness to pay and the channel quality for arbitrary channel coefficients. Also, most previous work considered nonatomic users, i.e., each user's influence on the network is small and negligible @cite_2 . This assumption may not be realistic in practice, which motivates us to study the atomic user case that involves resource splitting among networks (see Section for details).