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Two New Impossibility Results for the Random Assignment Problem

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.