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

Benno Kuckuck, Jörg Rothe

Abstract: Baumeister et al. (2017) 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 (Thomson, 2011), a central property important in many social decision contexts. Baumeister et al. (2017) 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 (Kohler and Chandrasekaran, 1971)---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.