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Cognitive radio is emerging as a promising technique to improve the utilization of the radio frequency spectrum. In this paper, we consider the problem of spectrum sharing among primary (or ??licensed??) users (PUs) and secondary (or ??unlicensed??) users (SUs). We formulate the problem based on bandwidth auction, in which each SU makes a bid for the amount of spectrum and each PU may assign the spectrum among the SUs by itself according to the information from the SUs without degrading its own performance. We show that the auction is a noncooperative game and that Nash equilibrium (NE) can be its solution. We first consider a single-PU network to investigate the existence and uniqueness of the NE and further discuss the fairness among the SUs under given conditions. Then, we present a dynamic updating algorithm in which each SU achieves NE in a distributed manner. The stability condition of the dynamic behavior for this spectrum-sharing scheme is studied. The discussion is generalized to the case in which there are multiple PUs in the network, where the properties of the NE are shown under appropriate conditions. Simulations were used to evaluate the system performance and verify the effectiveness of the proposed algorithm.