This paper considers the problem of allocating multiple divisible commodities among a group of agents with single-peaked preferences. We show that the uniform rule is the only allocation rule that satisfies strategy-proofness, envy-freeness, and same-sidedness: the third axiom is a weak efficiency condition that is equivalent to Pareto efficiency in the single-commodity case. Therefore, this result generalizes the characterization of the uniform rule by Sprumont (1991. Econometrica 59 (2), 509-519) in the single-commodity case: it is the only rule satisfying strategy-proofness, envy-freeness, and Pareto efficiency.
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