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 fundamental fairness notion known as equal treatment of equals (ETE) and ensuring various notions of efficiency. We extend the definition of ETE so that it accommodates a variety of constraints and applications. We analyze the ETE reassignment procedure, which transforms any assignment into one satisfying ETE, and examine its compatibility with three efficiency concepts: ex-post efficiency, ordinal efficiency, and rank-minimizing efficiency. 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, the existence of assignments satisfying both ETE and ordinal efficiency can be established. 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.

💡 Research Summary

This paper tackles the problem of achieving both fairness and efficiency in multi‑unit probabilistic assignment settings where objects are indivisible and various feasibility constraints may apply. The authors begin by extending the classic notion of Equal Treatment of Equals (ETE). While traditional ETE only requires agents to have identical preference rankings, the extended definition also demands that agents share the same “available” characteristics (e.g., age, regional caps) and face identical constraints. This refinement allows the concept to be compatible with affirmative‑action policies and real‑world feasibility restrictions.

The core technical tool introduced is the “ETE reassignment” procedure. Starting from any (pure) assignment—where each agent receives an integer number of copies of each object—the procedure groups agents that are equal under the extended definition and uniformly randomizes the objects among members of each group. The resulting probabilistic assignment guarantees ETE by construction.

The authors then examine how this reassignment interacts with three well‑studied efficiency notions:

1. Ex‑post Efficiency (EE) – every pure assignment that receives positive probability must be Pareto‑efficient. The paper proves that if the original assignment is EE, the ETE reassignment remains EE, because the set of realized pure assignments is unchanged.

2. Ordinal Efficiency (OE) – no other assignment first‑order stochastically dominates the given one. The authors provide a counter‑example showing that an OE assignment can lose its ordinal efficiency after ETE reassignment; the uniform reshuffling may reduce the probability that some agents receive their top‑ranked objects, creating a stochastic dominance by another feasible assignment.

3. Rank‑Minimizing Efficiency (RE) – the assignment minimizes the sum of ranks of allocated objects. RE is the strongest of the three notions; it always exists and implies OE. Crucially, the paper demonstrates that RE is preserved under ETE reassignment. Since RE optimizes the rank profile globally, redistributing objects within an equal group does not alter the total rank sum.

Because RE assignments exist, the preservation result guarantees the existence of assignments that satisfy both ETE and OE (by first finding an RE assignment, which is automatically OE, and then applying ETE reassignment). However, computing an RE assignment in the general multi‑unit, upper‑bound constrained setting is computationally hard.

To address this, the authors propose a polynomial‑time algorithm that combines a serial dictatorship (SD) rule with specially crafted priority lists that satisfy the “consecutive equals” property: agents belonging to the same equal group appear consecutively in the priority order. Under this property, the SD outcome is ordinally efficient, and its subsequent ETE reassignment also remains ordinally efficient. The algorithm therefore yields a feasible probabilistic assignment that simultaneously satisfies ETE and OE for any collection of upper‑bound constraints (e.g., per‑object capacity caps).

The paper also discusses strategic considerations. While the classic random serial dictatorship is strategy‑proof, the added ETE reassignment step introduces opportunities for manipulation: an agent can misreport preferences to obtain a more favorable stochastic outcome after redistribution. Thus, policymakers must weigh the trade‑off between fairness/efficiency gains and potential loss of incentive compatibility.

In summary, the study makes three substantive contributions: (i) an extended, constraint‑aware definition of ETE; (ii) a thorough compatibility analysis of ETE reassignment with EE, OE, and RE, establishing the preservation of EE and RE but not OE in general; and (iii) a constructive, computationally efficient method for producing assignments that meet both ETE and OE under general upper‑bound constraints, together with a discussion of its strategic vulnerabilities. These results broaden the theoretical toolkit for designers of school choice, course allocation, daycare matching, and other markets where indivisible goods must be allocated fairly and efficiently under complex feasibility rules.