  The Hylland-Zeckhauser (HZ) rule is a well-known rule for random assignment
of items. The complexity of the rule has received renewed interest recently
with Vazirani and Yannakakis (2020) proposing a strongly polynomial-time
algorithm for the rule under bi-valued utilities, and making several general
insights. We study the rule under the case of agents having bi-valued
utilities. We point out several characterizations of the HZ rule, drawing
clearer relations with several well-known rules in the literature. As a
consequence, we point out alternative strongly polynomial-time algorithms for
the HZ solution. We also give reductions from computing the HZ solution to
computing well-known solutions based on leximin or Nash social welfare. An
interesting contrast is that the HZ rule is group-strategyproof whereas the
unconstrained competitive equilibrium with equal incomes rule is not even
strategyproof. We clarify which results change when moving from 1-0 utilities
to the more general bi-valued utilities. Finally, we prove that the closely
related Nash bargaining solution violates envy-freeness and strategyproofness
even under 1-0 utilities.
