Equal Treatment of Equals and Efficiency in Probabilistic Assignments

This paper studies general multi-unit probabilistic assignment problems involving indivisible objects, with a particular focus on achieving the fairness notion of equal treatment of equals (ETE) and satisfying various efficiency criteria. We extend the definition of ETE so that it accommodates a wide range of constraints and applications. We introduce the ETE reassignment procedure, which transforms any assignment into one that satisfies ETE, and examine whether the efficiency properties satisfied by the original assignment -- namely, ex-post efficiency, ordinal efficiency, and rank-minimizing efficiency -- are preserved under the ETE reassignment. We show that, while the ETE reassignment of an ex-post efficient assignment remains ex-post efficient, it may fail to preserve ordinal efficiency in general settings. However, since the ETE reassignment of a rank-minimizing assignment preserves rank-minimizing efficiency, there must exist an assignment satisfying both ETE and ordinal efficiency. Furthermore, we propose a computationally efficient method for constructing assignments that satisfy both ETE and ordinal efficiency under general upper bound constraints by combining the serial dictatorship rule with appropriately specified priority lists and the ETE reassignment procedure.