  In this paper, we focus on how to dynamically allocate a divisible resource
fairly among n players who arrive and depart over time. The players may have
general heterogeneous valuations over the resource. It is known that the exact
envy-free and proportional allocations may not exist in the dynamic setting
[Walsh, 2011]. Thus, we will study to what extent we can guarantee the fairness
in the dynamic setting. We first design two algorithms which are O(log
n)-proportional and O(n)-envy-free for the setting with general valuations, and
by constructing the adversary instances such that all dynamic algorithms must
be at least Omega(1)-proportional and Omega(n/log n)-envy-free, we show that
the bounds are tight up to a logarithmic factor. Moreover, we introduce the
setting where the players' valuations are uniform on the resource but with
different demands, which generalize the setting of [Friedman et al., 2015]. We
prove an O(log n) upper bound and a tight lower bound for this case.
