  The reliability function of a channel is the maximum achievable exponential
rate of decay of the error probability as a function of the transmission rate.
In this work, we derive bounds on the reliability function of discrete
memoryless multiple-access channels (MAC) with noiseless feedback. We show that
our bounds are tight for a variety of MACs, such as $m$-ary additive and two
independent point-to-point channels. The bounds are expressed in terms of a new
information measure called ``variable-length directed information". The upper
bound is proved by analyzing stochastic processes defined based on the entropy
of the message, given the past channel's outputs. Our method relies on tools
from the theory of martingales, variable-length information measures, and a new
technique called time pruning. We further propose a variable-length achievable
scheme consisting of three phases: (i) data transmission, (ii) hybrid
data-confirmation, and (iii) full confirmation. We show that two-phase-type
schemes are strictly suboptimal in achieving the MAC's reliability function.
Moreover, we study the shape of the lower-bound and show that it increases
linearly with respect to a specific Euclidean distance measure defined between
the transmission rate pair and the capacity boundary. As side results, we
derive an upper bound on the capacity of MAC with noiseless feedback and study
a new problem involving a hybrid of hypothesis testing and data transmission.
