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How to efficiently and fairly allocate data rate among different users is a key problem in the field of multiuser multimedia communication. However, most of the existing optimization-based methods, such as minimizing the weighted sum of the distortions or maximizing the weighted sum of the peak signal-to-noise ratios (PSNRs), have their weights heuristically determined. Moreover, those approaches mainly focus on the efficiency issue while there is no notion of fairness. In this paper, we address this problem by proposing a game-theoretic framework, in which the utility/payoff function of each user/player is jointly determined by the characteristics of the transmitted video sequence and the allocated bit-rate. We show that a unique Nash equilibrium (NE), which is proportionally fair in terms of both utility and PSNR, can be obtained, according to which the controller can efficiently and fairly allocate the available network bandwidth to the users. Moreover, we propose a distributed cheat-proof rate allocation scheme for the users to converge to the optimal NE using alternative ascending clock auction. We also show that the traditional optimization-based approach that maximizes the weighted sum of the PSNRs is a special case of the game-theoretic framework with the utility function defined as an exponential function of PSNR. Finally, we show several experimental results on real video data to demonstrate the efficiency and effectiveness of the proposed method.