In this paper, a supermodular game theoretic approach is investigated for spectrum sharing in cognitive radio networks. We consider a Bertrand competition model, in which primary service providers compete to sell their spare spectrum and then to maximize their individual profits. We demonstrate that the Bertrand competition is a smooth supermodular game, and a round-robin optimization algorithm is developed to obtain the optimal price solutions. Simulation results verify that the algorithm approximately converges to an equilibrium point, and the influence of the exogenous variable on the equilibrium point is analyzed.