  We revisit the problem of fairly allocating a sequence of time slots when
agents may have different levels of patience (Mackenzie and Komornik, 2023).
For each number of agents, we provide a lower threshold and an upper threshold
on the level of patience such that (i) if each agent is at least as patient as
the lower threshold, then there is a proportional allocation, and (ii) if each
agent is at least as patient as the upper threshold and moreover has weak
preference for earlier time slots, then there is an envy-free allocation. In
both cases, the proof is constructive.
