8.4 Power Allocation Games in Parallel Interference Relay Channels 231 This theorem shows that, for the pathloss channel model where h k > 0, (k, ) {1, 2, r} 2 , there always exists an equilibrium. As a consequence, if some relays are added into the network, the transmitters will adapt their PA policies accordingly and, whatever the locations of the relays, an equilibrium will be observed. This is a nice property for the system under investigation. Since the PA game with DF is concave, it is tempting to try to verify whether a sufficient condition for uniqueness of Rosen (1965) is met here. It turns out that the diagonally strict concavity condition of Rosen (1965) is not trivial to confirm. Additionally, it is possible that the game has several equilibria, as is proven to be the case for the AF protocol. In a context of decentralized networks, each source S k has to optimize the param- eter k in order to maximize its transmission rate R k . In the rate region above, one can observe that this choice is not independent of the choice of the other source. Therefore, each source finds its optimal strategy by optimizing its rate w.r.t. k ( ). In order to do that, each source has to make some assumptions about the value used by the other source. This is in fact a non-cooperative game where each player makes some assumptions about the other player's behavior and maximizes its own utility. Interestingly, we see that, even in the single-band case, the DF protocol introduces a power allocation game through the parameter k representing the cooperation degree