A Generalization of Nash Bargaining and Proportional Fairness to Log-Convex Utility Sets With Power Constraints

  • Authors:

  • H. Boche;M. Schubert

  • Affiliations:

  • Fraunhofer Inst. for Telecommun., Tech. Univ. of Berlin, Berlin, Germany;-

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2011

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Abstract

Many solutions and concepts in resource allocation and game theory rely on the assumption of a convex utility set. In this paper, we show that the less restrictive assumption of a logarithmic “hidden” convexity is sometimes sufficient. We consider the problems of Nash bargaining and proportional fairness, which are closely related. We extend the Nash bargaining framework to a broader family of log-convex sets. We then focus on the set of feasible signal-to-interference-plus-noise ratios (SINRs), for the cases of individual power constraints and a sum power constraint. Under the assumption of log-convex interference functions, we show how Pareto optimality of boundary points depends on the interference coupling between the users. Finally, we provide necessary and sufficient conditions for strict log-convexity of the feasible SINR region.