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The problem of reliable communication over the memoryless state-dependent multiple-access channel (MAC) is considered, where the encoders and the decoder are provided with various degrees of asymmetric noisy channel state information (CSI). For the case where the encoders observe causal, asymmetric noisy CSI and the decoder observes complete CSI, inner and outer bounds to the capacity region, which are tight for the sum-rate capacity, are provided. Next, single-letter characterizations for the channel capacity regions under each of the following system settings are established: 1) the CSI at the encoders are asymmetric deterministic functions of the CSI at the decoder and the encoders have noncausal noisy CSI; 2) the encoders observe asymmetric noisy CSI with asymmetric delays and the decoder observes complete CSI; 3) a degraded message set scenario with asymmetric noisy CSI at the encoders and complete and/or noisy CSI at the decoder. The main component in these results is a generalization of a recently introduced converse coding approach for the MAC with asymmetric quantized CSI at the encoders and herein considerably extended and adapted for the noisy CSI setup.