  In this note, we prove two new impossibility results for random assignment
mechanisms: Bogomolnaia and Moulin (2001) showed that no assignment mechanism
can satisfy strategyproofness, ordinal efficiency, and symmetry at the same
time, and Mennle and Seuken (2017) gave a decomposition of strategyproofness
into the axioms swap monotonicity, upper invariance, and lower invariance. For
our first impossibility result, we show that upper invariance, lower
invariance, ordinal efficiency, and symmetry are incompatible. This refines the
prior impossibility result because it relaxes swap monotonicity. For our second
impossibility result, we show that no assignment mechanism satisfies swap
monotonicity, lower invariance, ordinal efficiency, anonymity, neutrality, and
non-bossiness. By contrasts, the Probabilistic Serial (PS) mechanism that
Bogomolnaia and Moulin (2001) introduced, satisfies these axioms when lower
invariance is replaced by upper invariance. It follows that there cannot exists
a lower invariant counterpart to PS.
