Strategy‐proofness, tops‐only, and the uniform rule

In the division problem with single‐peaked preferences, an allocation rule is strategy‐proof for same tops if no one can gain by reporting a false preference relation having the true peak. This new condition is so weak that it is implied by strategy‐proofness and tops‐only. We show that the uniform rule is the only rule satisfying this mild property under efficiency and envy‐freeness. We then analyze how largely the preference domain can be extended with admitting a rule satisfying the three axioms, and show that the single‐plateaued domain is the unique such maximal domain.