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In this paper, we study the problem of downlink bandwidth allocation in a non-cooperative IEEE 802.16 OFDM/TDMA based wireless mesh network, and propose an auction based framework in which the gateway, equipped with the precious high speed Internet connection, serves as the auctioneer while the first-level mesh routers (MRs) (i.e., those with direct wireless connections to the gateway) act as bidders competing resources among each other. We then present two novel relay auction (RA) approaches to allocate time-slots among MRs and analyze the bidding strategy using the solution concept of Nash equilibrium. Through simulations, we find that the proposed RA algorithms can achieve competitive performance in terms of resource allocation efficiency compared with Vickrey-Clarke-Groves (VCG) approaches, while having the capability to strike a proper balance between efficiency and fairness by adjusting the payment function. Specifically, with a smaller payment parameter value, the RA algorithms can also achieve much better connection blocking probability performance than VCG algorithms.