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. 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 (price of anarchy), as the ratio between the objective function at the worst Nash equilibrium and the optimal objective function. In this scenario the Nash equilibrium is achieved by selfish (non-cooperative) behavior between the players. We derive upper and lower bounds on the worst case equilibrium efficiency. From the comparison of the bounds, we conclude that for path loss coefficients that are of practical interest 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 improvement if the network nodes are encouraged to cooperate by designing certain mechanisms.