Pricing in heterogeneous wireless networks: Hierarchical games and dynamics

Luca Rose, E. Veronica Belmega, Walid Saad, Merouane Debbah

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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.

Original languageEnglish (US)
Article number6820798
Pages (from-to)4985-5001
Number of pages17
JournalIEEE Transactions on Wireless Communications
Issue number9
StatePublished - Sep 1 2014
Externally publishedYes


  • game theory
  • nonatomic game
  • Small-cell networks
  • Stackelberg game
  • Wardrop equilibrium (WE).

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Computer Science Applications
  • Applied Mathematics


Dive into the research topics of 'Pricing in heterogeneous wireless networks: Hierarchical games and dynamics'. Together they form a unique fingerprint.

Cite this