Probabilistic assignment of indivisible objects when agents have the same preferences except the ordinal ranking of one object

Bogomolnaia and Moulin (2001) show that there is no rule satisfying equal treatment of equals, stochastic dominance efficiency, and stochastic dominance strategyproofness for a probabilistic assignment problem of indivisible objects. Kasajima (2013) shows that the incompatibility result still holds when agents are restricted to have single-peaked preferences. In this paper, we further restrict the domain and investigate the existence of rules satisfying the three properties. As it turns out, the three properties are still incompatible even if all agents have the same preferences except the ordinal ranking of one object.