Bogomolnaia and Moulin (J Econ Theory 100:295–328, 2001) show that there is no rule satisfying stochastic dominance efficiency,equal treatment of equals and stochastic dominance strategy-proofness for a probabilistic assignment problem of indivisible objects. Recently, Mennle and Seuken (Partial strategyproofness: relaxing strategy-proofness for the random assignment problem. Mimeo, 2017) show that stochastic dominance strategy-proofness is equivalent to the combination of three axioms, swap monotonicity,upper invariance, and lower invariance. In this paper, we introduce a weakening of stochastic dominance strategy-proofness, called upper-contour strategy-proofness, which requires that if the upper-contour sets of some objects are the same in two preference relations, then the sum of probabilities assigned to the objects in the two upper-contour sets should be the same. First, we show that upper-contour strategy-proofness is equivalent to the combination of two axioms, upper invariance and lower invariance. Next, we show that the impossibility result still holds even though stochastic dominance strategy-proofness is weakened to upper-contour strategy-proofness.
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