A maximal domain of preferences for strategy-proof, efficient, and simple rules in the division problem (Q1762856)

The paper is concerned with the division of a quantity \(M\) of a perfectly divisible good among \(n\) individuals. The perspective is that of finding a maximal set of preferences over allocations which admit allocation rules satisfying certain properties. The main properties under consideration are strategy-proofness and (Pareto) efficiency, but the focus of the paper is on \textit{simple} rules, where simple means depending not on the complete preferences' specifications but only on their preferred shares (e.g.\ if they are single-peaked, only on the preferred element). On the other hand the rules considered are not imposed to be symmetric, which might be a price to pay to reach results on classes restricted to be simple, but is nonetheless peculiar (symmetry means, in the case of 2 persons for example, that if preferences are swapped allocations are too). The result from which the analysis of the paper takes departure is by \textit{S. Ching} and \textit{S. Serizawa} [J. Econ. Theory 78, No. 1, 157--166 (1998; Zbl 0914.90010)], and states that the maximal domain for strategy-proof efficient symmetric rules is the set of single-plateaued preferences (which have a unique plateau instead of peak). For fixed \(M\), the paper finds that substituting simpleness for symmetry enlarges the maximal set to a family admitting some indifference outside the plateau. On the other hand, if also \(M\) is allowed to vary, the set coincides with the one found by Ching-Serizawa.