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This paper investigates the radio resource allocation for the OFDMA systems. Two kinds of share auction mechanism are proposed, where the base station possesses of the subcarriers and plays as auctioneer, while the users play as bidders completing for them. These two kinds of auction are designed as first-price seal-bid, in which, the SINR auction aims at the desire for high throughput, and the fairness auction pay more attention to the fairness among users in the system. The auction-based resource allocation algorithm can reduce the burden of base station with a kind of distributed computation manner. Furthermore, the fairness auction only needs partial channel quality information with less overhead of signaling. Simulation results evaluate the performance of our proposed scheme.