  We initiate the work on maximin share (MMS) fair allocation of m indivisible
chores to n agents using only their ordinal preferences, from both algorithmic
and mechanism design perspectives. The previous best-known approximation is
2-1/n by Aziz et al. [IJCAI 2017]. We improve this result by giving a simple
deterministic 5/3-approximation algorithm that determines an allocation
sequence of agents, according to which items are allocated one by one. By a
tighter analysis, we show that for n=2,3, our algorithm achieves better
approximation ratios, and is actually optimal. We also consider the setting
with strategic agents, where agents may misreport their preferences to
manipulate the outcome. We first provide a O(\log (m/n))-approximation
consecutive picking algorithm, and then improve the approximation ratio to
O(\sqrt{\log n}) by a randomized algorithm. Our results uncover some
interesting contrasts between the approximation ratios achieved for chores
versus goods.
