In IEEE 802.11 networks, the access point (AP) selection based on the strongest signal strength often results in the extremely unfair bandwidth allocation among mobile users (MUs). In this paper, we propose a distributed AP selection algorithm to achieve a fair bandwidth allocation for MUs. The proposed algorithm gradually balances the AP loads based on max-min fairness for the available multiple bit rate choices in a distributed manner. We analyze the stability and overhead of the proposed algorithm, and show the improvement of the fairness via computer simulation.