Pricing in Heterogeneous Wireless Networks: Hierarchical Games and Dynamics

In this paper, a novel game-theoretic model of the complex interactions between network service providers (NSPs) and users in heterogeneous small-cell networks is investigated. In this game, the NSPs selfishly aim at maximizing their profit while, simultaneously, the users seek to optimize their chosen service's quality-price tradeoff. A Stackelberg formulation in which the NSPs act as leaders and the users as followers is proposed. The users' interactions are modeled as a general nonatomic game. The existence of a Wardrop equilibrium (WE) in the users' game is proven, and its expression as a solution of a fixed-point equation is provided (irrespective of the number of NSPs, services offered, pricing policies, and QoS functions). Moreover, a set of sufficient conditions that ensure the uniqueness of the WE is provided. Notably, the uniqueness of the equilibrium for the particular case of congestion games is shown. An algorithm approximating these equilibria is provided and its convergence to an ε-WE is proven. The existence of Nash equilibria for the leaders' game is shown and illustrated via numerical simulations.