Impossibility results in the probabilistic assignment problem with stochastic same-sidedness and minimal invariance

Bogomolnaia and Moulin (2001) demonstrate the impossibility of designing a rule that simultaneously satisfies stochastic dominance efficiency, equal treatment of equals, and stochastic dominance strategy-proofness in the context of the probabilistic assignment problem with indivisible objects. Despite attempts to relax these conditions by introducing concepts like upper contour strategy-proofness or robust ex-post efficiency, the impossibility results remain. Recently, Bandhu et al. (2024) introduced the concept of stochastic same-sidedness in the random voting model. This condition stipulates that if an agent modifies their preference by swapping two consecutively ranked objects, then (1) the sum of probabilities assigned to objects strictly higher than the swapped pair should remain unchanged, and (2) the sum of probabilities assigned to the swapped pair should also remain constant. We first show that the impossibility persists even when stochastic dominance strategy-proofness is weakened to stochastic same-sidedness. We then decompose stochastic same-sidedness into three minimal invariance axioms and use these to establish further impossibility results.