Strategy-proof and fair reallocation with single-peaked preferences

We consider the strategy-proof rules for reallocating individual endowments of an infinitely divisible good when agents’ preferences are single-peaked. In social endowment setting, the seminal work established by Sprumont (Econometrica 59:509–519, 1991) proves that the uniform rule is the unique one which satisfies strategy-proofness, efficiency, and envy-freeness. However, the uniform rule is not so appealing in our model since it disregards the differences in individual endowments. In other words, the uniform rule is not individually rational. In this paper, we propose a new rule named the uniform proportion rule. First, we prove that it is the unique rule which satisfies strategy-proofness, efficiency, and envy-freeness on proportion and we show that it is individually rational. Then, we show that our rule is indeed a member of the class of sequential allotment rules characterized by Barberà et al. (Games Econ Behav 18:1–21, 1997).