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Distributed and efficient resource allocation is critical for fully realizing the benefits of cooperative communications in large scale communication networks. This paper proposes two auction mechanisms, the SNR auction and the power auction, that determine relay selection and relay power allocation in a distributed fashion. A single-relay network is considered first, and the existence and uniqueness of the Nash Equilibrium (i.e., the auction's outcome) are proved. It is shown that the power auction achieves the efficient allocation by maximizing the total rate increase, and the SNR auction is flexible in trading off fairness and efficiency. For both auctions, the distributed best response bid updates globally converge to the unique Nash Equilibrium in a completely asynchronous manner. The analysis is then generalized to networks with multiple relays, and the existence of the Nash Equilibrium is shown under appropriate conditions. Simulation results verify the effectiveness and robustness of the proposed algorithms.