  This paper studies channel coding for the discrete memoryless multiple-access
channel with a given (possibly suboptimal) decoding rule. A multi-letter
successive decoding rule depending on an arbitrary non-negative decoding metric
is considered, and achievable rate regions and error exponents are derived both
for the standard MAC (independent codebooks), and for the cognitive MAC (one
user knows both messages) with superposition coding. In the cognitive case, the
rate region and error exponent are shown to be tight with respect to the
ensemble average. The rate regions are compared with those of the
commonly-considered decoder that chooses the message pair maximizing the
decoding metric, and numerical examples are given for which successive decoding
yields a strictly higher sum rate for a given pair of input distributions.
