  This paper studies the second-order asymptotics of the discrete memoryless
multiple-access channel with degraded message sets. For a fixed average error
probability $\epsilon\in(0,1)$ and an arbitrary point on the boundary of the
capacity region, we characterize the speed of convergence of rate pairs that
converge to that point for codes that have asymptotic error probability no
larger than $\epsilon$, thus complementing an analogous result given previously
for the Gaussian setting.
