Equal Treatment of Equals (ETE)

• ETE is a fairness principle defined by treating agents identically when they are equal in all morally and contextually relevant aspects.
• It is applied across domains such as algorithmic fairness, auction design, and school choice by ensuring identical welfare or outcome distributions for equals.
• Computational methods like explanation-based audits and reassignment procedures operationalize ETE while balancing trade-offs with efficiency, stability, and representational fairness.

Equal Treatment of Equals (ETE) is a fairness principle according to which agents who are equal in the relevant sense should receive identical treatment, identical welfare, or identical lotteries. Across political philosophy, algorithmic fairness, social choice, mechanism design, and random assignment, the central idea is stable: irrelevant differences should not affect outcomes, while the definition of “equals” depends on the domain’s admissible features, preferences, or institutional roles. In political philosophy and algorithmic fairness, ETE says that competitors who are equal in all morally relevant respects must be treated alike; in auction design it requires equal welfare for agents with identical preferences; in probabilistic assignment and school choice it requires identical assignment distributions within equality classes; and in voting it is often formalized through anonymity, neutrality, or transitive automorphism groups (Khan et al., 2022, Kazumura et al., 18 Feb 2026, Okumura, 20 Aug 2025, Bartholdi et al., 2018).

1. Core meanings and formal variants

ETE is not a single formal axiom but a family of closely related requirements. In individual-level formulations, it requires that if two individuals coincide on relevant non-protected features, then the decision rule should treat them identically. In welfare-based formulations, equals may receive different objects or payments provided they are indifferent between them. In lottery-based formulations, equals must receive the same probability distribution over admissible outcomes. In procedural formulations, equality is captured by invariance under relabeling of agents or by equal institutional roles.

Setting	“Equals”	Formalization
Algorithmic fairness	Same relevant, non-protected features	If $x_i^R=x_j^R$ then $f(x_i)=f(x_j)$
Auction design	Identical preference reports	If $R_i=R_j$, then $f_i(R) I_i f_j(R)$
Probabilistic assignment	Same admissible characteristics	$\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$
School choice	Same preferences and same priority ties	$p_{ic}=p_{jc}$ for all $c$
Voting and social choice	Identical procedural role	Anonymity, neutrality, or transitive $\mathrm{Aut}_f$

This diversity matters because ETE is weaker than some stronger symmetry requirements and stronger than others. In auction design, ETE is weaker than anonymity because anonymity requires invariance to relabelings of agents, whereas ETE only requires equal welfare for agents who are equals. In voting, anonymity is the classical strong form of ETE, but equitable voting weakens full symmetry by requiring only that the automorphism group act transitively on voters, so that every voter has the same role even when the rule is not fully anonymous. In randomized allocation, ETE depends on how equality classes are defined; the extended definition in probabilistic assignment uses admissible characteristics and a feasibility-preserving exchangeability condition so that equals are indistinguishable with respect to the constraints (Kazumura et al., 18 Feb 2026, Bartholdi et al., 2018, Okumura, 20 Aug 2025, Okumura, 7 May 2026, Xia, 2022).

A common source of confusion is the belief that ETE always means identical outcomes. Several literatures reject that interpretation. In mechanism design, ETE is stated in terms of indifference, not literal identity of bundles. In voting and judgment aggregation, ETE may concern equal procedural roles rather than identical influence in every realized profile. In probabilistic settings, the relevant object is often the distribution of outcomes rather than any single deterministic assignment (Kazumura et al., 18 Feb 2026, Botan et al., 2021, Okumura, 20 Aug 2025).

2. Algorithmic fairness and AI regulation

In political philosophy and algorithmic fairness, ETE is the core nondiscrimination principle according to which competitors who are equal in all morally relevant respects must be treated alike, while race, gender, family income, disability status, and related morally arbitrary characteristics must not affect access to desirable positions or the social goods attached to them. A standard individual-fairness formalization uses a task-specific similarity metric over relevant qualifications:

$|f(x)-f(x')| \le L\, d(x,x').$

The choice of $d$ encodes the normative judgment of who counts as equals; in practice it should be constructed on morally relevant attributes alone or in a way that neutralizes sensitive attributes. In the same literature, calibration across groups and predictive parity are presented as score-based interpretations of treating equals equally at a fair contest, while demographic parity is explicitly criticized for ignoring merit and therefore potentially treating unequals equally (Khan et al., 2022).

The explanation-based reformulation of equal treatment develops this criticism further. “Beyond Demographic Parity: Redefining Equal Treatment” argues that demographic parity, formalized as $f(x_i)=f(x_j)$0, captures equal outcome rather than equal treatment. Its proposed criterion defines equal treatment through explanation distributions:

$f(x_i)=f(x_j)$1

with $f(x_i)=f(x_j)$2 an explanation vector such as Shapley values. Under the Shapley efficiency axiom,

$f(x_i)=f(x_j)$3

the paper shows that equal treatment implies demographic parity, but demographic parity does not imply equal treatment. The non-equivalence is attributed to Yule’s effect: contributions can cancel at the prediction level while still differing systematically across groups. The operational test is a classifier two-sample test on explanation vectors, using the AUC of an “Equal Treatment Inspector”; under the null $f(x_i)=f(x_j)$4, any classifier has $f(x_i)=f(x_j)$5 (Mougan et al., 2023).

This distinction also clarifies the relation between ETE and other algorithmic fairness notions. Equalized odds and equality of opportunity condition on labels, whereas the explanation-based equal-treatment criterion does not require labels. Counterfactual fairness requires invariance under interventions on the protected attribute within a structural causal model, whereas the explanation-based criterion is not reliant on causal graphs. The paper therefore presents explanation-based equal treatment as a population-level criterion that supports ETE by ensuring that the model relies on relevant features in the same way across protected groups, while also noting that individual-level ETE still requires conditioning or matching on relevant features (Mougan et al., 2023).

A broader regulatory version appears in the closed-loop view of AI. There, ETE is a one-pass property: the system provides the same information $f(x_i)=f(x_j)$6 to all users within a class defined by non-protected attributes, and there exists a constant $f(x_i)=f(x_j)$7 such that $f(x_i)=f(x_j)$8 for all users in the class and all short-run indices $f(x_i)=f(x_j)$9, independent of initial conditions. Equal impact is the long-run analogue, defined through ergodic averages and a stationary measure $R_i=R_j$0. This makes explicit a recurring tension: a policy can satisfy one-pass ETE and still generate unequal long-run impacts under feedback, or equalize long-run impacts while differentiating short-run treatment (Zhou et al., 2022).

The normative literature adds a second tension. When base rates differ across groups, calibration or predictive parity and balance conditions such as equalized odds cannot generally be satisfied simultaneously. That conflict is interpreted as the incompatibility between backward-looking and forward-looking conceptions of a fair contest under unequal life chances. On this view, pointwise ETE may be appropriate when scores validly measure relevant merit and the decision is not a gateway to future opportunity, but substantive doctrines such as luck-egalitarian equal opportunity or Rawls’s fair equality of opportunity are required when qualifications reflect accumulated social advantage or disadvantage (Khan et al., 2022).

3. Mechanism design, probabilistic assignment, and school choice

In auction design with unit-demand agents and non-quasilinear preferences, ETE is formalized by

$R_i=R_j$1

This means that agents with identical preferences obtain identical welfare at any preference profile. Within the classical domain, the main theorem states that a mechanism is strategy-proof, individually rational, and satisfies equal treatment of equals, no wastage, and no subsidy if and only if it is a minimum Walrasian equilibrium price mechanism. The result is an equity-based characterization of the MWEP mechanism and complements its efficiency-based characterization. The paper also emphasizes that ETE is weaker than anonymity and that ties are handled through indifference: equals may receive different objects or payments if they are indifferent between them (Kazumura et al., 18 Feb 2026).

The probabilistic-assignment literature generalizes ETE to multi-unit environments with rich constraints. Agents are partitioned into classes $R_i=R_j$2 by admissible characteristics $R_i=R_j$3, with Assumption 1 requiring that preferences belong to $R_i=R_j$4 and Assumption 2 requiring that equals be exchangeable with respect to feasibility constraints. A lottery $R_i=R_j$5 satisfies ETE if equals receive identical individual assignment distributions:

$R_i=R_j$6

The ETE reassignment procedure enforces this by uniformly mixing over permutations of equals within each feasible pure assignment. For any class $R_i=R_j$7,

$R_i=R_j$8

The paper proves that the ETE reassignment of an ex-post efficient assignment remains ex-post efficient, may fail to preserve ordinal efficiency in general settings, and preserves rank-minimizing efficiency. Since rank-minimizing efficiency implies ordinal efficiency, the existence of assignments satisfying both ETE and ordinal efficiency follows. Under general upper bound constraints, a computationally efficient method combines serial dictatorship with a priority list satisfying consecutive equals and then applies the ETE reassignment (Okumura, 20 Aug 2025).

School choice adopts the same reassignment idea in a more specialized setting. Equals are students who have identical strict preference rankings and are tied at every school. A lottery satisfies ETE if, within each equals group, all students have identical probability vectors over schools:

$R_i=R_j$9

Starting from a constrained efficient stable matching $f_i(R) I_i f_j(R)$0, the proposed method applies the ETE reassignment to the degenerate lottery concentrated on $f_i(R) I_i f_j(R)$1. The resulting lottery is ex ante stable, satisfies ETE, and is not ordinally dominated by any other ex ante stable lottery. The matrix implementation is especially simple: within each equals group $f_i(R) I_i f_j(R)$2, replace each student’s probability vector by the group average,

$f_i(R) I_i f_j(R)$3

The note presents this as a polynomial-time construction of an ex ante stable school-choice lottery satisfying ETE (Okumura, 7 May 2026).

Across these mechanism-design settings, ETE functions as a symmetry principle over admissible equals rather than a blanket prohibition on all differentiated treatment. The crucial choice is the equality class: identical preferences in auctions, admissible characteristics in probabilistic assignment, and identical preferences plus tied priorities in school choice (Kazumura et al., 18 Feb 2026, Okumura, 20 Aug 2025, Okumura, 7 May 2026).

4. Voting, judgment aggregation, and participatory budgeting

In social choice, the classical form of ETE is anonymity: relabeling voters should not change the outcome. “Equitable voting rules” shows that May’s symmetry assumption is stricter than necessary to capture equal treatment of voters. The paper defines a rule as equitable if, for all voters $f_i(R) I_i f_j(R)$4, there exists $f_i(R) I_i f_j(R)$5 such that $f_i(R) I_i f_j(R)$6, equivalently, $f_i(R) I_i f_j(R)$7 acts transitively on $f_i(R) I_i f_j(R)$8. This preserves identical roles across voters while admitting a far richer set of rules than full anonymity. The paper proves that every winning coalition under an equitable rule has size at least $f_i(R) I_i f_j(R)$9 and constructs neutral, positively responsive equitable rules with winning coalitions of size at most $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$0. It also shows that stronger symmetry quickly restores majority rule: for almost all $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$1, any $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$2-equitable, neutral, positively responsive rule must be majority, and for all $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$3, any $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$4-equitable rule has winning coalitions of size at least $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$5 (Bartholdi et al., 2018).

A different response to ETE appears in “Most Equitable Voting Rules.” There, ETE is captured jointly by anonymity and neutrality, and the paper studies the ANR impossibility: there is no voting rule that satisfies anonymity, neutrality, and resolvability in the minimal case of two agents and two alternatives. The proposed solution is the notion of “most equitable refinements,” together with Most-Favorable-Permutation tie-breaking. For an anonymous and neutral irresolute rule $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$6, the key object is the fixed-point set

$\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$7

and a profile is problematic exactly when $\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$8. The MFP construction yields a resolute refinement that preserves anonymity and neutrality whenever possible and is therefore a most equitable refinement of the base rule (Xia, 2022).

Judgment aggregation broadens the notion further. It distinguishes equal treatment of agents through anonymity, equal treatment of issues through neutrality, and two egalitarian outcome principles. The maximin principle minimizes the maximum Hamming distance from the collective judgment,

$\Pr(x; x(\sigma)_i)=\Pr(x; x(\sigma)_j)$9

whereas the equity principle minimizes the maximum gap in dissatisfaction,

$p_{ic}=p_{jc}$0

The paper proves that maximin and equity are mutually incompatible, that no rule satisfying either can be majoritarian, and that both conflict with full strategyproofness. MaxHam satisfies participation and antipodal strategyproofness, whereas no rule satisfying the equity property can satisfy participation or antipodal strategyproofness (Botan et al., 2021).

Participatory budgeting provides an operational version of ETE at the level of per-voter budget trajectories and payments. For the Method of Equal Shares, Bounded Overspending Equal Shares, BOS+, and the fractional variant FrES, ETE requires that voters with identical utilities and identical initial endowments be treated identically throughout the execution. The reason is algebraic: the update rules are symmetric. In MES,

$p_{ic}=p_{jc}$1

in BOS,

$p_{ic}=p_{jc}$2

and in FrES,

$p_{ic}=p_{jc}$3

When two voters have the same $p_{ic}=p_{jc}$4 and the same utility function, the decrements are identical in every round, so their residual funds, contributions, and realized utilities remain identical. The paper therefore treats ETE as preserved by MES, BOS, BOS+, and FrES, even though these rules differ sharply in proportionality and efficiency guarantees (Papasotiropoulos et al., 2024).

5. Trade-offs with efficiency, stability, representation, and attractiveness

ETE rarely operates in isolation. In several literatures it is paired with efficiency, but the pairing can be either complementary or antagonistic. In the auction framework, ETE plus no wastage, together with strategy-proofness, individual rationality, and no subsidy, characterizes the same mechanism as the efficiency-based approach. The paper states this as an equivalence: for strategy-proof, individually rational, no-subsidy mechanisms on the classical domain, Pareto efficiency is equivalent to ETE plus no wastage (Kazumura et al., 18 Feb 2026).

In probabilistic assignment, the relation is subtler. ETE reassignment preserves ex-post efficiency and rank-minimizing efficiency, but may destroy ordinal efficiency in general settings. The school-choice note resolves part of that tension by restricting attention to ex ante stable lotteries and starting from a constrained efficient stable matching. The resulting ETE lottery is not ordinally dominated by any other ex ante stable lottery, but the note also emphasizes that it may still be ordinally dominated by an ex post stable lottery. A plausible implication is that the relevant efficiency benchmark depends on the stability concept one chooses to preserve (Okumura, 20 Aug 2025, Okumura, 7 May 2026).

Algorithmic fairness exhibits the most widely discussed trade-offs. The explanation-based account states that equal treatment can conflict with demographic parity because the former inspects how features influence predictions, not only the produced outcomes. The equal-opportunity literature interprets impossibility results as evidence that backward-looking and forward-looking conceptions of a fair contest become incompatible when base rates differ across groups. The closed-loop regulation paper adds a temporal dimension: one-pass ETE and long-run equal impact can diverge because repeated interactions and retraining alter the stationary distribution of outcomes (Mougan et al., 2023, Khan et al., 2022, Zhou et al., 2022).

Indirect representation provides another instructive example. In weighted assemblies, ETE is interpreted as equal a priori influence per citizen. If delegate $p_{ic}=p_{jc}$5 is pivotal with probability $p_{ic}=p_{jc}$6 and constituency $p_{ic}=p_{jc}$7 has size $p_{ic}=p_{jc}$8, then equal per-citizen influence requires

$p_{ic}=p_{jc}$9

Combining this with the asymptotic relation $c$0 yields the egalitarian weight rule

$c$1

Under i.i.d. individual ideal points, this reduces to the square-root rule $c$2; under sufficiently strong affiliation within constituencies, ETE instead requires a Shapley-value rule linear in constituency size (Kurz et al., 2012).

The group-draw literature makes the trade-off especially concrete. There, ETE requires that equal teams have identical ex ante placement and opponent probabilities. The paper studies 32 sets of geographical restrictions and an HHI-based inequality index

$c$3

with $c$4 indicating perfect ETE and $c$5 maximal inequality. Under the 2025 World Men’s Handball Championship data, the cloud of 64 points spans $c$6 and $c$7. Constraints A–D reduce unattractive matches at modest fairness cost, whereas Constraint E is identified as the main driver of non-uniformity-induced ETE violations under the Skip mechanism. The recommended remedy is uniform randomization over the feasible assignment set, which yields exact ETE among equals as far as the constraints allow (Csató et al., 25 Feb 2025).

These examples show that ETE is not opposed to efficiency by definition. Rather, it restructures the optimization problem by fixing which distinctions are admissible, after which efficiency, stability, representational accuracy, or attractiveness must be pursued within that restricted space (Kazumura et al., 18 Feb 2026, Kurz et al., 2012, Csató et al., 25 Feb 2025).

6. Computational methods and contemporary implementations

Recent work treats ETE as operational rather than purely axiomatic. In explanation-based auditing, the workflow is explicit: train a predictive model, compute Shapley-based attributions on validation and test data, construct explanation vectors, train an Equal Treatment Inspector to predict the protected attribute from those vectors, evaluate its AUC on held-out data, and test $c$8 using a Brunner–Munzel test or confidence intervals via DeLong or related methods. The accompanying explanationspace Python package provides wrappers over SHAP, the equal-treatment inspector, significance tests, and tutorials for ACS datasets and synthetic cases (Mougan et al., 2023).

Mechanism-design implementations are likewise concrete. The MWEP mechanism is computed by an ascending-price process in the spirit of Demange–Gale–Sotomayor: initialize prices at zero, identify overdemanded sets, raise their prices uniformly until a demand change occurs, and stop when there are no overdemanded or weakly underdemanded sets. In participatory budgeting, BOS runs in polynomial time by checking at most $c$9 critical fractions per project plus $\mathrm{Aut}_f$0, then selecting the candidate that minimizes $\mathrm{Aut}_f$1. In school choice, once a constrained efficient stable matching has been found, the ETE reassignment is only a group-average transformation of the assignment matrix and therefore runs in $\mathrm{Aut}_f$2 after the stable matching stage (Kazumura et al., 18 Feb 2026, Papasotiropoulos et al., 2024, Okumura, 7 May 2026).

Social-choice computation has produced its own symmetry-preserving procedures. For full rankings, MFP tie-breaking is polynomial-time computable. In judgment aggregation, MaxEq outcome determination is complete for the search analogue of $\mathrm{Aut}_f$3, and the paper gives Answer Set Programming encodings that compute Hamming distances, max and min distances, and the inequity objective. These encodings make the egalitarian rules applicable to moderately sized instances despite the second-level polynomial-hierarchy hardness results (Xia, 2022, Botan et al., 2021).

Federated learning introduces a different implementation path. E2FL operationalizes user-level equality by the standard deviation across client performances and group-level equity by the standard deviation across group mean performances. Instead of adding fairness regularizers, it uses rank-based training and equal-weight voting at two levels:

$\mathrm{Aut}_f$4

Clients with similar data distributions are grouped together, each client contributes one vote within the group, and each group contributes one vote to the global ranking. On FairMNISTRotate, the reported downlink is 5.99 MB for E2FL versus 62.0 MB for IFCA, while E2FL reduces both client variance and group variance relative to FedAvg. This suggests an implementation pattern for ETE in distributed learning: equalize influence structurally through the aggregation rule rather than only through post hoc disparity penalties (Mozaffari et al., 2022).

Taken together, these methods show that ETE has become a design principle with explicit tests, reassignment operators, tie-breaking rules, rank aggregators, and optimization routines. The unifying theme is not any single formula, but the insistence that symmetry among admissible equals be built into the mechanism, the audit, or the aggregation step itself (Mougan et al., 2023, Okumura, 7 May 2026, Mozaffari et al., 2022).