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Characterizing the set of achievable distortions in lossy transmission of dependent sources over multi-user channels is an open problem. Even for a channel such as the multiple access channel, where the capacity is well-understood, the complete characterization of achievable distortions is still unknown. Since separation of source coding and channel coding is sub-optimal for this case, bounds on achievable distortions require tools beyond prior channel capacities and rate-distortion bounds. In this paper we propose a systematic approach for finding lower bounds on the set of distortions achievable through joint source-channel coding.