Power allocation game in a four node relay network: A lower bound on the price of anarchy

In this paper, we propose an idea on how game and information theoretic results can be combined to analyze the performance of wireless cooperative networks. More precisely, we consider a four node wireless network, where the transmit nodes help each other acting as relays during the periods in which they do not transmit their own information. In order to help the other node, each node has to use a part of its available power to relay the signal of the other transmitter. The network is modeled as a non-cooperative game in which each player (node) maximizes its own utility function (information rate). The goal of the game designer (network provider) is to maximize the objective function (in this case the sum rate) in order to get better network efficiency. Here, we analyze the so called equilibrium efficiency, as the ratio between the objective function at the worst Nash equilibrium and the optimal objective function. Using game theoretical language, it is the price of anarchy of the proposed game. In this scenario, the Nash equilibrium is achieved by selfish (non-cooperative) behavior between the players. In other words, in order to maximize its own utility function each node chooses a strategy to use its available power only for itself, and not helping the other node. Earlier, we derived an upper bound for the worst case equilibrium efficiency and in this paper we present a lower bound. From the comparisons, we conclude that for path loss coefficients that are of practical importance the proposed bounds are tight. Our results show that the worst case equilibrium efficiency for the proposed simple network is very small (below 10%). Hence, there is a large possibility for improvements if the network nodes are encouraged to cooperate by designing certain mechanisms.