Dynamic service selection games in heterogeneous small cell networks with multiple providers

The strategic deployment of small cells, overlaid on existing wireless infrastructure is foreseen as a key technologies enabling wireless operators to offer a plethora of innovative technologies to their customers. This, in turn, introduces novel technical challenges arising from the selfish behavior of the customers who ultimately seek to select their preferred wireless service while optimizing the quality-price trade-off of this choice. Consequently, wireless providers have to select the optimal pricing policies which maximize their revenues while anticipating the customers´ behavior. In this paper, we provide a game-theoretic model suitable for modeling and analyzing the complex interactions between providers and customers in heterogeneous wireless networks. The proposed game is based on a hybrid model that combines a Stackelberg formulation in which the service provider acts as a leader and the customers, acting as followers, engage in a non-cooperative Wardrop game. For the customers´ game, we show the existence and give the analytic expression of the Wardrop equilibria irrespective of the number of providers, services offered and quality of services functions. Then, we propose and study different best response based dynamics at the customers´ level. We show that these dynamics converge to the equilibrium if only a portion of the customers are allowed to change strategy at each instant. For characterizing the equilibrium of the Stackelberg game, we propose a novel distributed algorithm that can be adopted by the providers and customers in a practical network. Numerical results show that providers are able to reasonably predict the customers´ behavior by evaluating the outcome of the so called Krasnoselskij algorithm, and, subsequently optimize their price.