Stackelberg Game on the Boundary of Coexistence

This paper combines Convex analysis and Game theory to investigate the problem of maximizing the sum rate of transceivers operating in the same frequency band. In our earlier work, we proved that for transceivers operating under a total power constraint, the power distribution that maximizes the sum rate lies on the boundary of the feasible set formed by the power constraint. In this paper, we first prove that for two users, the sum rate is convex on the boundary formed by the line segment representing the power constraint, and the maximum sum rate is achieved when all the power is allocated to one of the users. Obviously, such a power allocation is unfair to the other user. We consider a scenario in which the first user is willing to sell some of the power allocated to him to the second user. We use Stackelberg Game theory to analyze this scenario and prove the existence of a unique competitive equilibrium. We derive the best response functions, and determine the optimum price the first user must charge and the optimum amount of power the second user should buy at this price. We also use simulation to obtain the best response functions and the equilibrium point, and demonstrate their agreement with our analytical results.