Nash equilibria for spectrum sharing of two bands among two players

The spectrum sharing game and the quality of its equilibria have been widely studied in a variety of contexts. In this paper we consider two pairs of communicating users that share two bands of spectrum. Through the analysis of the Nash equilibria, we provide the conditions, with respect to the normalized signal and interference strengths, for the set of equilibria power allocations to coincide with the set of optimal allocations. In contrast, when these sets do not coincide, we characterize the quality of the equilibria using the price of stability and the price of anarchy measures. In the more general case of N pairs of transmit receive pairs in an ad hoc network, we provide simulation results of a simple distributed player power allocation update heuristic that improves the sum rate utility above that achieved by the equilibrium of splitting the power evenly between the two bands.