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Extensive research in recent years has shown that dynamic spectrum sharing is a promising approach to address the artificial spectrum scarcity problem by improving spectrum utilization. This new communication paradigm, however, requires a well-designed spectrum allocation mechanism. This paper designs a double spectrum auction framework that allows unlicensed secondary users to obtain selected idle spectra assigned to licensed primary users. This is a win-win game because primary users can earn extra revenue and secondary users can obtain spectra they desperately need. The competition among primary users in the auction framework is studied combining game theory with a double spectrum auction in a non-cooperative game with the Nash Equilibrium (NE) as the best solution. Primary users use the prices obtained from the NE as their bid strategies to participate in the auction. In this auction sellers and buyers bid privately and confidentially, which means that the secondary users do not actually know the price and the spectrum size offered by the primary users, then a new net utility function was developed for the primary users with an iterative algorithm to find the Nash equilibrium point. Simulations demonstrate that this design effectively improves spectrum utilization.