When Is a Cheap Fairness Edit Safe? Screening Post-Hoc Edits in Two-Sided Matching Markets

Moirangthem Tiken Singha; Rabinder Kumar Prasada

cs.LGOct 27, 2026

When Is a Cheap Fairness Edit Safe? Screening Post-Hoc Edits in Two-Sided Matching Markets

Authors: Moirangthem Tiken Singha, Rabinder Kumar Prasada

Organizations: Department of Computer Science and Engineering, Dibrugarh University Institute of Engineering and Technology, Dibrugarh University, Dibrugarh, Assam, India

Abstract

Two-sided matching systems, such as admissions pipelines, allocate opportunities jointly. Reducing a protected at tribute's influence usually requires encoder retraining. We study a closed-form linear projector that does this at inference time. For markets that stay vertically ordered before and after the edit, we bound the mean welfare change by the edit-induced appeal shift and an explicit remainder. A probabilistic refinement replaces worst-case constants with a data-dependent deviation term. A pre-deployment screen combines a verticality test with a welfare sweep and returns PASS, KILL, or OUT-OF-SCOPE. An exact tier requires every opportunity to rank applicants by appeal, and a graded tier admits near-vertical markets. On 30 held-out synthetic markets, the screen agrees with every pre-specified regime verdict, with no observed false PASS. On 60 exactly vertical markets, no bound is violated, and the probabilistic bound is informative in 69% to 87% of configurations. On 2700 configurations from three realistic markets, which do not reach the exact tier, no plug-in bound expression is violated. A closed-form factorization stays within graded scope in 90% to 100% of instances, whereas two-tower and graph neural network encoders fall outside it in every instance. Architecture choice thus appears to affect whether a post-hoc edit is admissible.

Figures & tables

Method	Post-hoc	Retargetable	Closed-form	Welfare Bound
Concept erasure	Yes	Yes	Partial	No
Post-processing fairness	Yes	Yes	Partial	No
In-processing fair ranking/recommendation	No	No	No	No
Fair matching mechanisms	N/A	N/A	N/A	No (mechanism only)
This work	Yes	Yes	Yes	Yes (conditional)

Table 1: Properties of representative approaches. The welfare guarantee of this work is conditional on vertical-market assumptions.

Regime (reference)	Screening-rule verdict
Regime (reference)	PASS	KILL	OUT
LM-HB (PASS)	10	0	0
HM-LB (KILL)	0	10	0
NV (OUT)	0	0	10

Table 2: Screening-rule verdicts against the reference verdict of each synthetic regime, with

n=200

agents per side and 10 held-out seeds per regime (30 markets).

Regime	Bound	Mean RHS	Tightness	Informative	Violations
LM-HB	Theorem 1	$2.6\times 10^{5}$	$2.6\times 10^{-6}$	$0.00$	0/30
LM-HB	Theorem 2	$0.87$	$1.0\times 10^{-1}$	$0.69$	0/30
LM-HB	Proposition 1	$1.4\times 10^{5}$	$4.4\times 10^{-6}$	$0.00$	0/30
HM-LB	Theorem 1	$2.6\times 10^{5}$	$3.6\times 10^{-7}$	$0.00$	0/30
HM-LB	Theorem 2	$0.38$	$2.0\times 10^{-2}$	$0.87$	0/30
HM-LB	Proposition 1	$3.1\times 10^{5}$	$3.1\times 10^{-7}$	$0.00$	0/30

Table 3: Welfare change against the three bounds on exactly vertical markets (

n=200

, 30 markets per regime, 10 values of

\gamma

). Informative is the share of configurations whose right-hand side is below the full feasible welfare range in Equation ( 14 ). Violations are counted per market.

	Worst-Case		Probabilistic
Regime	Mean RHS	Tightness	Mean RHS	Tightness
LM-HB	$1.10\times 10^{5}$	$2.7\times 10^{-6}$	1.79	$7.10\times 10^{-2}$
HM-LB	$2.87\times 10^{6}$	$2.6\times 10^{-6}$	1.86	$9.36\times 10^{-2}$
NV	$7.82\times 10^{10}$	$3.6\times 10^{-11}$	$7.74\times 10^{5}$	$1.7\times 10^{-6}$

Table 4: Mean right-hand side and mean tightness ratio of the worst-case and probabilistic bounds in the three synthetic regimes (

n=200

agents per side, 30 seeds, 11 values of

\gamma

). Worst-case tightness uses all 330 configurations. Probabilistic tightness uses the 300 configurations with

\gamma>0

, because the probabilistic right-hand side is zero at

\gamma=0

. NV denotes the non-vertical regime.

Regime	$L_{\mathrm{mix}}$	$\kappa$	$\nu$	$c$	$\mathrm{ent}$	$\delta_{\gamma}$
LM-HB	4.90	0.21	6.11	1.35	0.02	4.71
HM-LB	5.95	2.12	1.59	0.69	0.63	2.92
NV	$4.7\times 10^{7}$	0.51	0.46	0.17	0.17	0.62

Table 5: Constants of Proposition 1 in the three synthetic regimes, as means over 30 seeds.

L_{\mathrm{mix}}

is the exchange-cost constant of Assumption 6 ,

\kappa

and

\nu

are the merit-regularity constants of Assumption 8 ,

c

is the appeal alignment of the erased direction, and

\delta_{\gamma}

is the shift scale of Equation ( 6 ) averaged over the sweep. All are plug-in estimates, with merit and quality measured in the coordinates stated in Section 4.1 . NV denotes the non-vertical regime.

Statistic	Value
Spearman $\rho$	0.779
$p$ -value	$4.5\times 10^{-51}$
Grid samples ( $n$ )	245
Positive-loss samples	102
Log–log slope	0.79
95% CI (slope)	$[0.64,\;1.07]$

Table 6: Relationship between entanglement and welfare loss over the

7\times 7

(\omega,\beta)

grid.

Statistic	Value
Exact partner-match rate	$26.8\%$ CI $[21.4,31.4]\%$
Kendall’s $\tau$ (partner-quality rank)	$0.76$ CI $[0.62,0.87]$
$\tau$ vs. $1-$ discordance, Spearman $\rho$	$0.92$ ( $p=3.2\times 10^{-17}$ )
$\tau$ vs. rank-one share, Spearman $\rho$	$0.46$ ( $p=0.003$ )

Table 7: Assortative-prediction agreement with exact deferred acceptance, across mixed nominally vertical and non-vertical regimes.

Comparison	Value	Test statistic
Random / structured cost ratio	$0.19$ CI $[0.17,0.21]$	$p=0.002$ , rank-biserial $-1.0$
Residual gap: one-sided $A$	$0.116$
Residual gap: one-sided $B$	$0.104$	$p=0.002$ , effect $1.0$ ( $A$ vs. both)
Residual gap: two-sided	$0.070$
Gap AUC: one-sided $A$	$0.098$
Gap AUC: one-sided $B$	$0.086$	$p=0.002$ , effect $1.0$

Table 8: Structural ablations: coherent vs. incoherent erasure, and one- vs. two-sided erasure at

\gamma=0.5

(

n=10

seeds, Wilcoxon signed-rank, exact, paired). One-sided

A

and one-sided

B

edit only side

\mathcal{A}

or only side

\mathcal{B}

; two-sided erasure is Algorithm 2 .

Figure 1: Encoder-dependence of the screening rule’s entanglement diagnostic, over

30

instances per cell. Bar heights are means, and whiskers give the observed minimum and maximum across instances. The reference line is

\tau_{\mathrm{ent}}=0.50

. A0’s entanglement on the score-priority datasets is near

1

, while the trained encoders spread across nearly the full range on identical data.

Figure 2: Encoder-dependence of the discordance statistic P2, over

30

instances per cell. Bar heights are means, and whiskers give the observed minimum and maximum across instances. Lower values indicate closer agreement among side-

\mathcal{B}

rankings. The dashed line is the calibrated screen threshold

\tau_{\mathrm{disc}}=0.55

, and the shaded band above it is the excluded region. No cell attains the exact tier, which requires a discordance of zero.

Figure 3: Encoder-dependence of the mean right-hand side of the probabilistic bound, log scale, over 30 instances per cell. Bar heights are means over instances. Solid red hatching marks cells that are out of scope in most instances, and dotted red hatching marks cells that are out of scope in some instances.

Dataset	Arch.	Ent.	P2, discordance (range)	Rank-1 share	PASS	OUT	Mean RHS	Prop. 1 RHS	Viol.
Law School	A0	0.98	0.42 [0.29, 0.59]	0.99	0.00	0.10	1.05	$2.6\times 10^{7}$	0/300
Law School	A1	0.08	0.99 [0.94, 1.00]	0.99	0.00	1.00	2.04	$2.3\times 10^{5}$	0/300
Law School	A2	0.63	1.00 [1.00, 1.00]	0.87	0.00	1.00	1.71	$8.0\times 10^{5}$	0/300
ENEM	A0	0.96	0.27 [0.16, 0.41]	0.99	0.70	0.00	0.95	$2.5\times 10^{5}$	0/300
ENEM	A1	0.07	1.00 [0.93, 1.00]	0.99	0.00	1.00	1.06	$1.1\times 10^{5}$	0/300
ENEM	A2	0.43	1.00 [1.00, 1.00]	0.82	0.00	1.00	0.81	$1.1\times 10^{5}$	0/300

Table 9: Summary over three realistic markets and three architectures, 30 instances per cell. Each instance draws its own subsample, interaction noise, and encoder initialization. Ent. and Mean RHS are means over instances. P2 is the discordance of Definition 2 , with its observed minimum and maximum across instances, and the next column is the leading-singular-value share of Remark 2 . Lower discordance indicates closer agreement among side-

\mathcal{B}

rankings, and the graded screen admits an instance at

\text{P2}\leq 0.55

. PASS and OUT are the fractions of instances receiving those verdicts; the remainder are KILL. Mean RHS is the probabilistic right-hand side, and the next column is the right-hand side of Proposition 1 on the same instances. Viol. counts violations of the probabilistic bound over the ten non-zero values of

\gamma

in each instance.

Method	Gap (%)	Welfare (%)	Verdict	Retrain
Ours (with verdict)	61.1	102.4	Yes	No
INLP [ 18 ]	69.9	102.5	No	No
RLACE-style [ 19 ]	36.5	101.7	No	No
Group score standardization [ 10 ]	97.0	101.2	No	No
Reject-option-style [ 14 ]	5.5	98.2	No	No

Table 10: Comparison with post-hoc baselines on the LM-HB market (

\omega=0.05

\beta=0.65

n=400

, 30 seeds), relative to the uncorrected system. Values are means over seeds. Welfare retention above

100\%

means that welfare improved relative to the uncorrected matching. RLACE-style, group score standardization, and reject-option-style are simplified reimplementations.

Method	Gap (%)	Welfare (%)	Verdict	Retrain
Ours (with verdict)	85.6	95.0	Yes	No
INLP [ 18 ]	6.3	100.1	No	No
RLACE-style [ 19 ]	47.1	87.8	No	No
Group score standardization [ 10 ]	98.0	96.9	No	No
Reject-option-style [ 14 ]	25.3	99.0	No	No

Table 11: Comparison with post-hoc baselines on the HM-LB market (

\omega=0.65

\beta=0.05

n=400

, 30 seeds), relative to the uncorrected system. The row for the proposed method averages the

8

seeds returning KILL; baseline rows average all

30

seeds. RLACE-style, group score standardization, and reject-option-style are simplified reimplementations.

Claim	Test / Metric	Key Evidence
Screening accuracy on labeled markets	Wilson 95% CI	0/20 false PASS among markets with a non-PASS reference verdict; 30/30 verdicts agree with the reference regime verdict; upper 95% bound = 0.161
Scope test detects sign failures	Discordance vs. rank-one share	Rank-one share 0.99 with discordance 0.99 (Law School and ENEM under A1); share 0.10 with discordance 0.47 (Ratings under A0)
Entanglement is associated with welfare loss	Spearman, bootstrap	$\rho=0.779$ , $p=4.5\times 10^{-51}$ ; log–log slope $0.79$ (95% CI $[0.64,1.07]$ )
Residual-bound expression (Prop. 1 )	Violation count; dominance check	No observed violation in 990 synthetic configurations across three regimes and 2700 real-data configurations; $\delta_{\gamma}$ dominated the realized shift in every instance
No bound violation within the exact scope	Violation count	0/60 exactly vertical markets violate any evaluated bound; probabilistic bound informative in 68.7% (LM-HB) and 87.3% (HM-LB) of configurations
Assortativity tracks discordance	Kendall $\tau$ , Spearman	$\tau=0.76$ (95% CI $[0.62,0.87]$ ); $\rho=0.92$ with $1-$ discordance, $p=3.2\times 10^{-17}$

Table 12: Principal empirical claims and their quantitative evidence.

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University of Toronto, Toronto, Ontario, Canada · Georgia Institute of Technology, Atlanta, Georgia, USA

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Public institutions increasingly procure AI systems whose design they cannot inspect or change. In higher education, proprietary Early Warning Systems (EWS) leave colleges with few options beyond adjusting model outputs to address inequity. This raises the question of how fairness work is coordinated among vendors, institutions, advisors, and students with unequal power to change these systems? Using student records from a public college in Ontario, Canada, we evaluate six post-hoc fairness interventions on a research EWS under simulated procurement constraints. We compare fairness, accuracy, and demographic disparities, introducing error-type profiling to trace how interventions redistribute false positives and false negatives. Interventions redistributed disparities without consistently reducing them. Two implementations favored already-advantaged groups because they used group size to define disadvantage; small, marginalized groups remained poorly served. These findings show how procurement constraints and implementation choices shape the possibilities for fairness work. We call the resulting condition fairness theatre; dashboard metrics converge while groups' error burdens persist or worsen.

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