Nash bargaining and proportional fairness for log-convex utility sets

For comprehensive convex compact positive utility sets, the Nash bargaining solution (NBS) is obtained by maximizing a product of utilities, a strategy which is also known as "proportional fairness". However, the standard assumption of convexity may not be fulfilled. This is especially true for wireless communication systems, where interference and adaptive techniques can lead to complicated non-convex utility sets (e.g. the 2-user SIR region with linear receivers). In this paper, we show that the Nash bargaining framework can be extended to certain non-convex utility sets, whose logarithmic transformation is strictly convex comprehensive. As application examples, we consider feasible sets of signal-to-interference ratios (SIR), based on axiomatic log-convex interference functions. The resulting SIR region is known to be log-convex. However, strict log-convexity and compactness is required here. We derive conditions under which this is fulfilled. In this case, there is a single-valued Nash bargaining solution, which is equivalent to the proportionally fair operating point. The results are shown for a total power constraint, as well as for individual power constraints.