A power allocation game in a four node relay network: An upper bound on the worst-case equilibrium efficiency

We introduce a power allocation game in a four node relay network which consists of two source and two destination nodes. The sources employ a time sharing protocol such that in each discrete time instance one of the sources communicates with its destination while the other source aids this communication by acting as a relay. Each source uses some portion of its limited power for its own transmission and uses the remaining portion to aid the other source. The noncooperative solution, which is the Nash equilibrium of the game where each source tries to maximize its own rate, dictates each source to use all of its power for its own use, i.e., no relaying. This results in an inferior sum rate with respect to the optimum sum rate jointly maximized over all possible power allocations. The main contribution of this paper is to establish an upper bound on the worst-case equilibrium efficiency (a.k.a. the price of anarchy), defined as the ratio of the equilibrium sum rate to the optimal sum rate for the worst channel conditions. More specifically, we show that if the path loss coefficient is beta > 0 and the received signals are corrupted by additive white Gaussian noise, then the worst case equilibrium efficiency is upper bounded by (1/2)beta. We also note that this upper bound can be extended to relay networks with more than two sources.