ETE Reassignment Procedure

• ETE Reassignment Procedure is a symmetrization method that enforces equal treatment of equals by averaging assignments within specified equality classes.
• It transforms any lottery over pure assignments into one with uniform marginal distributions within equals while preserving feasibility and certain efficiency properties.
• In school choice, it converts deterministic matchings into ex ante stable lotteries, remedying arbitrary asymmetries among students with identical priorities.

The ETE reassignment procedure is a symmetrization operator on probabilistic assignments that enforces equal treatment of equals by averaging assignments within exogenously specified equality classes. In the general multi-unit assignment framework, it transforms any feasible lottery over pure assignments into another feasible lottery in which all agents in the same class receive the same distribution over bundles; in the school-choice specialization, it is applied to a constrained efficient stable matching to obtain an ex ante stable lottery satisfying equal treatment of equals (Okumura, 20 Aug 2025, Okumura, 7 May 2026).

1. Definition and conceptual scope

In the general formulation, the procedure is defined on a finite set of agents $A$, a finite set of object types $O$, and a finite set $Y$ of feasible pure assignments $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$. A probabilistic assignment is a lottery $\sigma:Y\to[0,1]$ with $\sum_{y\in Y}\sigma(y)=1$. Equality is represented by a partition $\mathcal{A}=\{A_1,\dots,A_N\}$ of the agent set, where each $A_n$ is an equivalence class of agents regarded as equals (Okumura, 20 Aug 2025).

The fairness requirement is equal treatment of equals: for any two agents in the same class, the induced lotteries over individual bundles must coincide. The reassignment procedure does not alter the aggregate feasibility structure by introducing new bundles or new pure assignments outside the feasible set. Instead, it permutes assignments only within equality classes and then averages over those permutations. This makes the procedure a fairness-preserving symmetrization rather than a new allocation rule in its own right (Okumura, 20 Aug 2025).

The same basic idea appears in school choice, but there the equals classes are defined more specifically. Students are equal only when they have identical preferences and are tied in priority at every school. The procedure is then used to randomize within those classes after a deterministic stable matching has been selected (Okumura, 7 May 2026).

2. Formal construction

The general construction starts from a feasible probabilistic assignment $\sigma$. For a fixed pure assignment $y\in Y$, consider every bijection $O$0 that permutes agents only within each equality class. Applying such a permutation to the rows of $O$1 produces a derived assignment $O$2, and the collection of all such derived assignments is denoted $O$3 (Okumura, 20 Aug 2025).

For each pure assignment $O$4, the procedure defines a lottery $O$5 that is uniform over $O$6: $O$7 The ETE reassignment of $O$8 is then

$O$9

Operationally, this means: first sample a pure assignment $Y$0 from $Y$1; then replace it by a uniformly chosen within-class permutation of $Y$2 (Okumura, 20 Aug 2025).

In the school-choice specialization, the same construction is written with deterministic matchings $Y$3 rather than pure object-allocation matrices. If $Y$4 permutes students only within equals groups, the derived matching is

$Y$5

For a fixed matching $Y$6, the induced lottery $Y$7 is uniform over all derived matchings, and the ETE reassignment of a lottery $Y$8 is

$Y$9

When the input lottery is degenerate on a single constrained efficient stable matching $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$0, the output is simply the symmetrized lottery over all within-class permutations of $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$1 (Okumura, 7 May 2026).

3. Marginal equalization and fairness guarantee

The central characterization of the procedure is its effect on marginal bundle distributions. If $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$2 is the ETE reassignment of $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$3, then for every class $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$4, every agent $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$5, and every bundle $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$6,

$$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$7

Thus, each agent in a class receives the class-average marginal distribution from the original assignment (Okumura, 20 Aug 2025).

This immediately implies ETE, because the right-hand side depends only on the class $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$8, not on the individual $$y=(y_{ao})_{a\in A,o\in O}\in \mathbb{Z}_+^{|A|\times |O|}$$9. The procedure therefore converts any feasible probabilistic assignment into one satisfying equal treatment of equals, provided that feasibility is invariant under swapping equal agents. In the general framework, that invariance is encoded as the assumption that if two equal agents exchange their assignments in a feasible pure assignment, the resulting assignment remains feasible (Okumura, 20 Aug 2025).

In school choice, the same averaging identity appears at the level of school probabilities. For each equals group $\sigma:Y\to[0,1]$0, each student $\sigma:Y\to[0,1]$1, and each school $\sigma:Y\to[0,1]$2,

$\sigma:Y\to[0,1]$3

This shows that the reassignment procedure preserves the multiset of schools allocated to each equality class in each deterministic realization, while washing out within-class asymmetries in the ex ante distribution (Okumura, 7 May 2026).

4. Efficiency properties and their limits

The procedure has a sharply differentiated interaction with efficiency notions. In the general assignment framework, the ETE reassignment of an ex-post efficient assignment remains ex-post efficient. It may, however, fail to preserve ordinal efficiency in general settings. By contrast, if the input assignment is rank-minimizing efficient, then its ETE reassignment remains rank-minimizing efficient (Okumura, 20 Aug 2025).

The rank-minimizing result is structurally important. Let $\sigma:Y\to[0,1]$4 denote the rank of bundle $\sigma:Y\to[0,1]$5 in agent $\sigma:Y\to[0,1]$6's preference ordering, and define total expected rank by

$\sigma:Y\to[0,1]$7

An assignment is rank-minimizing efficient if it minimizes $\sigma:Y\to[0,1]$8 over all feasible lotteries. Because within-class permutations do not change the multiset of bundles held by each equality class, and equals have identical preferences, symmetrization leaves the total rank objective unchanged. This is the basis for the preservation theorem (Okumura, 20 Aug 2025).

The failure of ordinal-efficiency preservation is not a minor edge case but a genuine structural limitation. The general paper gives counterexamples under broad feasibility conditions, including settings with general upper bounds, showing that one cannot in general take an arbitrary ordinally efficient assignment, apply ETE reassignment, and conclude that ordinal efficiency is preserved (Okumura, 20 Aug 2025).

In school choice, the corresponding guarantee is narrower but stronger in its own domain. Starting from a constrained efficient stable matching $\sigma:Y\to[0,1]$9, the degenerate lottery concentrated on $\sum_{y\in Y}\sigma(y)=1$0 is ex ante stable and not ordinally dominated by any ex ante stable lottery. The paper proves that ETE reassignment preserves ex ante stability in this setting, and the resulting lottery is still not ordinally dominated by any other ex ante stable lottery (Okumura, 7 May 2026).

5. Constructive methods and computational use

The general paper distinguishes between existence and construction. Because rank-minimizing efficient assignments exist and ETE reassignment preserves rank-minimizing efficiency, assignments satisfying both ETE and ordinal efficiency exist in very general settings. For direct construction under general upper bound constraints, the paper proposes a computationally efficient method: run serial dictatorship with a priority list satisfying consecutive equals, then apply ETE reassignment to the resulting pure assignment (Okumura, 20 Aug 2025).

Under this construction, the priority list is chosen so that each equality class appears as a contiguous block. Serial dictatorship then produces a pure assignment that is ordinally efficient under general upper bounds, and the subsequent ETE reassignment preserves ordinal efficiency in that setting. The resulting lottery satisfies both ETE and ordinal efficiency, although the paper notes that it need not be rank-minimizing efficient (Okumura, 20 Aug 2025).

The school-choice note adopts an even simpler pipeline. First compute a constrained efficient stable matching using known polynomial-time methods cited there. Then form the degenerate lottery on that matching, and finally apply ETE reassignment. The result is a polynomial-time construction of an ex ante stable school-choice lottery satisfying equal treatment of equals (Okumura, 7 May 2026).

A practical implication is that ETE reassignment is not a standalone matching rule competing with deferred acceptance or serial dictatorship. It is a post-processing operator: it takes a deterministic or probabilistic assignment already selected for other reasons and enforces symmetry across equals without altering class-level assignment totals.

6. School-choice specialization

In the school-choice model, a deterministic matching $\sum_{y\in Y}\sigma(y)=1$1 assigns each student to a school subject to quotas. Stability is defined by absence of justified envy: there do not exist students $\sum_{y\in Y}\sigma(y)=1$2 such that $\sum_{y\in Y}\sigma(y)=1$3 prefers $\sum_{y\in Y}\sigma(y)=1$4's school to her own and has higher priority than $\sum_{y\in Y}\sigma(y)=1$5 at that school. A lottery is ex ante stable if it has no ex ante justified envy across its support (Okumura, 7 May 2026).

The ETE reassignment procedure is applied after a constrained efficient stable matching has already been chosen. Its effect is to randomize only within groups of students who are identical from the mechanism’s perspective: same preference ordering and tied priorities at every school. If, in the deterministic matching, one member of such a group receives school $\sum_{y\in Y}\sigma(y)=1$6 and another receives school $\sum_{y\in Y}\sigma(y)=1$7, then the reassigned lottery gives each of them the average of those assignments rather than preserving the arbitrary deterministic asymmetry (Okumura, 7 May 2026).

This construction yields three properties simultaneously in the school-choice domain: ex ante stability, ETE, and absence of ordinal domination by any other ex ante stable lottery. The paper also identifies a boundary of the result: the ETE-reassigned lottery need not be undominated relative to the larger class of ex post stable lotteries. The undominatedness claim is specifically tied to the class of ex ante stable lotteries (Okumura, 7 May 2026).

The school-choice use case therefore illustrates the procedure’s general role with unusual clarity. A deterministic matching may be stable and constrained efficient yet still violate equal treatment of equals. ETE reassignment repairs that fairness defect by symmetrizing within equality classes, while preserving the relevant stability notion for the lottery domain.

7. Interpretation and significance

The ETE reassignment procedure is best understood as a canonical fairness operator for settings in which equality is defined exogenously and feasibility is permutation-invariant within equality classes. Its mathematical action is simple: preserve the aggregate assignment profile of each class and average away arbitrary within-class asymmetries. Its normative role is equally specific: once a planner has decided that certain agents are indistinguishable for allocation purposes, the procedure ensures that this indistinguishability is reflected in the induced lottery (Okumura, 20 Aug 2025).

Its significance lies in what it preserves and what it does not. It preserves feasibility by construction, ex-post efficiency in the general model, rank-minimizing efficiency in the general model, and ex ante stability in the school-choice specialization. It does not preserve ordinal efficiency in full generality, and in school choice its optimality guarantee is relative to ex ante stable lotteries rather than all ex post stable lotteries (Okumura, 20 Aug 2025, Okumura, 7 May 2026).

A concise way to summarize the procedure is: $\sum_{y\in Y}\sigma(y)=1$8 In the general assignment setting, this symmetrization underwrites existence results for assignments satisfying both ETE and ordinal efficiency. In school choice, it yields a simple polynomial-time route from a constrained efficient stable matching to an ex ante stable lottery satisfying equal treatment of equals (Okumura, 20 Aug 2025, Okumura, 7 May 2026).