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We propose a novel payment-based incentive scheme for peer-to-peer (P2P) live media streaming. Using this approach, peers earn points by forwarding data to others. The data streaming is divided into fixed-length periods; during each of these periods, peers compete with each other for good parents (data suppliers) for the next period in a first-price-auction-like procedure using their points. We design a distributed algorithm to regulate peer competitions and consider various individual strategies for parent selection from a game-theoretic perspective. We then discuss possible strategies that can be used to maximize a peer's expected media quality by planning different bids for its substreams. Finally, in order to encourage off-session users to remain online and continue contributing to the network, we develop an optimal data forwarding strategy that allows peers to accumulate points that can be used in future services. Simulation results show that the proposed methods effectively differentiate the media qualities received by peers making different contributions (which originate from, for example, different forwarding bandwidths or servicing times) and at the same time maintain high overall system performance.