Sequential Allocation Rules are Separable: Refuting a Conjecture on Scoring-Based Allocation of Indivisible Goods

Baumeister et al. [2] introduced scoring allocation correspondences and rules, parameterized by an aggregation function ⋆ (such as + and min) and a scoring vector s. Among the properties they studied is separability, a.k.a. consistency [15], a central property important in many social decision contexts. Baumeister et al. [2] show that some common scoring allocation rules fail to be separable and conjecture that "(perhaps under mild conditions on s and ⋆), no positional scoring allocation rule is separable." We refute this conjecture by showing that (1) the family of sequential allocation rules-an elicitation-free protocol for allocating indivisible goods based on picking sequences [9]-is separable for each coherent collection of picking sequences, and (2) every sequential allocation rule can be expressed as a scoring allocation rule for a suitable choice of scoring vector and social welfare ordering.