The problem of distributed power allocation for an interference relay network is considered, where multiple source-relay-destination (S-R-D) links communicate concurrently in the same frequency and interfere over two hops, referred to as a two-hop interference channel. The direct source-destination connections are not considered in this work. Power allocation in this scenario is challenging as it necessitates not only performance tradeoff among multiple links but also power distribution for each link along spatial dimensions. We approach the problem from a game-theoretic perspective and propose a novel implicit-based approach to prove the uniqueness of the Nash equilibrium (NE). This technique complements the existing literature on equilibria analysis by revealing the benefits of expressing the best responses in implicit forms. To benchmark the performance of the NE, we have investigated the non-convex sum-utility maximization problem, and have 1) identified the optimal solution structures; 2) proved that linear pricing is locally optimal in maximizing the sum utilities. A simple and distributed pricing algorithm is then proposed. Simulation results show that for a two-user network, the proposed scheme closely approaches the optimal sum rate, while, for a ten-user network, the improvement is up to 340% compared to the non-cooperative game model without pricing.
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