Nash Bargaining and Proportional Fairness for Wireless Systems

Nash bargaining and proportional fairness are popular strategies for distributing resources among competing users. Under the conventional assumption of a convex compact utility set, both techniques yield the same unique solution. In this paper, we show that uniqueness is preserved for a broader class of logarithmically convex sets. Then, we study a scenario where the performance of each user is measured by its signal-to-interference ratio (SIR). The SIR is modeled by an axiomatic framework of log-convex interference functions. No power constraints are assumed. It is shown how existence and uniqueness of a proportionally fair optimizer depends on the interference coupling among the users. Finally, we analyze the feasible SIR set. Conditions are derived under which the Nash bargaining strategy has a single-valued solution.