Power bargaining for amplify and forward relay channel

We consider a system of two user terminals, both of which are capable of acting as relay stations for each others. We consider a simple scenario where the user equipment that acts as an amplify-and-forward relay station tries to sell its data rate at some fixed price to the user equipment that acts as a source station. The benefit the source station receives is equal to the difference between the maximal ratio combining rate and the data rate it buys from the relay station, converted into a monetary value. The cost of cooperation for both the stations is reflected by their power consumption, which both of them want to reduce. This conflicting goal of increasing the revenue, while at the same time, decreasing the cost, by both the users, is studied using game theoretical modeling. The situation is modeled as a symmetric, two person bargaining game. The solution of such a game is unique and Pareto optimal. We derive the necessary and sufficient conditions for the bargaining solution, and obtain explicit solutions for the source and the relay transmit power. Lastly, we plot a graph of the dependence of transmit power on the pricing factor.