Custom-Ratios Fairness: Theory and Applications

• Custom-ratios fairness is defined by using multiplicative relations (ratios) over probabilities, utilities, or error rates to assess fairness, contrasting traditional additive gap measures.
• It encompasses methodologies like differential fairness, harm ratios, and ratio metrics in preference learning and operational frameworks, highlighting tailored bias assessments.
• The approach is applied across diverse domains—including classification, resource allocation, and federated learning—with insights on balancing precision, prevalence, and computational challenges.

As an umbrella term, custom-ratios fairness denotes fairness formulations in which acceptable treatment is specified through multiplicative relations—ratios of probabilities, utilities, service levels, losses, or prevalence rates—rather than only through additive gaps. Recent work spans direct group-outcome ratio bounds, counterfactual harm ratios, normalized fill-rate guarantees, top-versus-bottom tail ratios over user error, and modeling environments that allow ratio operations, thresholded comparisons, and user-defined metric composition (Foulds et al., 2018, Ebadian et al., 2024, Gowaikar et al., 2024, d'Aloisio et al., 2024). The resulting landscape is heterogeneous: some systems treat ratio fairness as a first-class formal object, whereas others support it only indirectly through custom losses, post-processing penalties, or subgroup configuration.

1. Direct ratio-based fairness definitions

A central line of work defines fairness directly through multiplicative bounds on group-conditional outcomes. Differential fairness requires that, for all protected-group pairs and all outcomes,

$$e^{-\epsilon} \leq \frac{P_{M,\theta}(M(\mathbf{x})=y\mid \mathbf{s}_i,\theta)}{P_{M,\theta}(M(\mathbf{x})=y\mid \mathbf{s}_j,\theta)} \leq e^\epsilon.$$

The parameter $\epsilon$ is a tunable stringency parameter, so a desired allowable ratio $r$ corresponds to $\epsilon=\log r$. The framework is explicitly motivated by the legal 80% rule and extends it from one favorable outcome to all outcomes and from one protected attribute to multiple protected attributes and their intersections (Foulds et al., 2018).

A different ratio-based construction appears in collective decision making through the individual harm ratio and group harm ratio. An outcome $o$ is $\alpha$-IHR if there do not exist agents $i,j$ and an alternative outcome $o'$ such that

$\frac{1}{2}u_i(o') > \alpha u_i(o)$

while every agent $k\in N\setminus\{i,j\}$ is weakly preserved: $\epsilon$0 The groupwise extension replaces the factor $\epsilon$1 with $\epsilon$2, yielding a family of $\epsilon$3-GHR guarantees for coalitions $\epsilon$4 harmed by coalitions $\epsilon$5. In this framework, lower $\epsilon$6 means stronger fairness, and under compactness plus upper convexity every maximum Nash welfare outcome is proportionally fair and therefore has group harm ratio $\epsilon$7 and individual harm ratio $\epsilon$8 (Ebadian et al., 2024).

Preference learning adopts yet another ratio object. There, fairness is defined over the distribution of per-user model fit, and the explicitly ratio-based metric is the Kuznets Ratio,

$\epsilon$9

With $r$0, the ratio compares the aggregate error of the worst-served and best-served tails. In the reported experiments, $r$1, so the operative statistic is a top-20% versus bottom-20% user-error ratio (Gowaikar et al., 2024).

2. Intersectional and prevalence-sensitive ratios in classification

Differential fairness is explicitly intersectional. If full intersectional groups $r$2 satisfy the $r$3-ratio bound, then any nonempty proper subset of the protected attributes inherits the same $r$4-differential fairness guarantee. This theorem is one of the main reasons differential fairness is useful for custom-ratio fairness: one multiplicative tolerance over full intersections automatically protects every lower-order marginalization (Foulds et al., 2018).

A separate classification perspective emphasizes that fairness can depend strongly on group-specific label prevalence. In toxicity detection, fairness was evaluated for gender, race, and religion using

$r$5

alongside the counterfactual metric

$r$6

The paper’s most direct prevalence-ratio construct is selection bias,

$r$7

which is the absolute difference in toxic prevalence across groups (Elsafoury et al., 2023).

The observed toxic ratios in the original fairness dataset were unequal across identity groups, and the authors explicitly intervened on those ratios.

Attribute	Original positive ratios	Balanced positive ratios
Gender	male $r$8, female $r$9	male $\epsilon=\log r$0, female $\epsilon=\log r$1
Race	White $\epsilon=\log r$2, Asian $\epsilon=\log r$3, Black $\epsilon=\log r$4	White $\epsilon=\log r$5, Asian $\epsilon=\log r$6, Black $\epsilon=\log r$7
Religion	Christian $\epsilon=\log r$8, Muslim $\epsilon=\log r$9, Jewish $o$0	Christian $o$1, Muslim $o$2, Jewish $o$3

The balanced fairness set contains 55,476 samples, and the paper reports that fairness-metric agreement improves substantially after balancing group prevalence and contextual representation. In training-data interventions, re-stratification reduced selection bias from $o$4 for gender/race/religion to $o$5, but the strongest results came from perturbation-based balancing that equalized semantic contexts while also setting within-attribute positive ratios exactly to $o$6 for gender and religion and $o$7 for race. The paper’s central conclusion is that balancing positive-example ratios helps most when it is done at realistic prevalence levels rather than by forcing every group toward $o$8–$o$9 positive rate (Elsafoury et al., 2023).

3. Modeling and operationalizing custom ratio metrics

MODNESS is the clearest explicit environment for user-authored custom ratio metrics. It is a model-driven engineering framework implemented with the EMF ecosystem and organized around a Bias and Fairness Metamodel with packages for bias definition, fairness analysis specification, and metric definition. A metric contains an EqualityOperator, a toleranceValue, and a Function; the supported function classes include Operation, Logarithm, Summation, ExpectedValue, GroupSize, and Probability. The paper is explicit that fairness metrics can either be established in existing literature or custom-defined by the user, and that MODNESS can generate operator specifications that define and compose different metrics (d'Aloisio et al., 2024).

The TPL use case gives a direct ratio example. The custom metric coverage is generated as $\frac{1}{2}u_i(o') > \alpha u_i(o)$7 with threshold = 1.0 and tolerance_value = 0.2. The numerator counts low-frequency items with high ranking; the denominator counts all high-ranked items. This is not merely a ratio in principle but an implemented, generated ratio metric. The same framework also supports SingleOperator and RangeOperator, so thresholded ratio criteria and bounded intervals are part of the modeling vocabulary (d'Aloisio et al., 2024).

Other toolchains are more limited. MMM-fair is strongest on multi-attribute fairness, multi-objective optimization, Pareto-front exploration, and fairness-aware boosting, but the paper does not document first-class support for arbitrary user-authored ratio formulas such as lower and upper bounds on ratios of group metrics. Its API exposes named constraints such as "EO", "DP", and "EP", protected-attribute structure via saIndex and saValue, and trade-off hyperparameters such as alpha and gamma; its documented strength is configurable group-fairness optimization over standard notions rather than arbitrary algebraic ratio constraints (Swati et al., 9 Sep 2025).

A more generic post-processing route is the fairness-adjuster framework with objective

$\alpha$0

Because $\alpha$1 is user-chosen, ratio-style fairness can often be represented indirectly, especially when group-level rates are written as smooth functions of adjusted scores. The paper’s experiments center on disparate impact, defined as the proportion of favorable outcomes predicted for the protected group divided by the proportion of favorable outcomes predicted for the non-protected group, and show tuning toward fairness values near $\alpha$2 on Adult, German, and COMPAS. At the same time, the framework is not presented as an exact hard-constrained solver for arbitrary ratio formulas; the paper explicitly notes that some fairness definitions may require specialized loss surrogates (Eberhard et al., 22 Apr 2025).

Custom-loss work on COMPAS is adjacent rather than fully ratio-centric. The main contribution there is Group Accuracy Parity (GAP), implemented through

$\alpha$3

with weighted binary cross-entropy used as the cross-entropy term. The same paper catalogs ratio-based fairness notions such as disparate impact, error ratio, and false-positive-rate ratio, but its own proposed objective is group-comparative parity rather than a ratio objective (Lee et al., 3 Jan 2025).

4. Resource allocation and service-ratio fairness

In online allocation and matching, custom-ratios fairness often appears as a normalized service-rate guarantee. In rideshare assignment during high-demand hours, fairness is defined as

$\alpha$4

the minimum ratio of expected successful matches to expected arrivals across request types. The platform designer chooses parameters $\alpha$5 and $\alpha$6, with $\alpha$7, and the algorithm achieves competitive ratios at least $\alpha$8 simultaneously on profit and fairness. Here the ratio is not only an evaluation metric but also the object of an explicit guarantee (Nanda et al., 2019).

The stationary-arrival online bipartite matching literature uses closely related group-level service-rate fairness. Long-run fairness is

$\alpha$9

and short-run fairness is defined by conditioning on realized arrivals and then averaging the worst realized group fill rate. The objective is still egalitarian—maximize the minimum service rate across groups—but the normalization by expected arrivals makes the criterion inherently ratio-based (Ma et al., 2020).

FORA-IU makes the custom-ratio structure explicit through priority coefficients $i,j$0. Its fairness criterion, FE-FR-$i,j$1, requires that for every group $i,j$2,

$i,j$3

Equivalently,

$i,j$4

The $i,j$5 values are customized entitlement weights: higher $i,j$6 means a higher target fill rate per unit of expected demand. The optimal universal guarantee under arbitrary time-varying arrivals is $i,j$7, where

$i,j$8

and under stationary arrivals the exact finite-horizon guarantee becomes

$i,j$9

This is one of the cleanest formalizations of custom service-ratio fairness in online allocation (Averbakh et al., 5 May 2026).

5. Ratio fairness over users, clients, and structural advantage

Preference learning reformulates fairness as the equality of representational quality across users. The distributed quantity is per-user error $o'$0, and the framework measures its dispersion using $o'$1, variance, the Gini coefficient, generalized entropy, the Atkinson index, and the Kuznets Ratio. The empirical pattern is that high average performance can coexist with substantial inequality across users: on Jester, the best-MSE preprocessing methods had $o'$2, whereas Mehestan scaling reduced the Kuznets ratio to $o'$3 at a substantial performance cost. In this setting, ratio fairness is neither demographic parity nor error-rate parity; it is the ratio geometry of the user-error distribution itself (Gowaikar et al., 2024).

Graph federated learning introduces a different ratio object: overlap ratios among clients. FairGFL defines node overlap

$o'$4

and link overlap

$o'$5

combines them through

$o'$6

and then aggregates client updates using inverse-overlap weights

$o'$7

Its composite objective is

$o'$8

The fairness notion here is cross-client performance consistency under imbalanced structural advantage, not a user-specified target ratio, but the method is still ratio-aware in a mathematically direct sense (Zhou et al., 29 Dec 2025).

Biometric quality assessment provides a nearby but distinct case. The paper does not define fairness as a ratio, but it studies thresholded discard disparities derived from quality scores, most directly through the Mean-Discard-Gap. For thresholds

$o'$9

it compares the maximum and minimum discard percentages across groups and defines

$\frac{1}{2}u_i(o') > \alpha u_i(o)$0

The construction is threshold-dependent and operationally close to discard-rate ratios, even though the paper itself stays with absolute gaps rather than multiplicative criteria (Dörsch et al., 2024).

6. Limits, boundary cases, and methodological tensions

Not every fairness toolkit that advertises customization supports arbitrary ratios as a first-class object. MMM-fair is a clear case: it provides multi-attribute fairness, multiple fairness definitions, Pareto exploration, a chat-based interface, custom fairness constraint definition, and deployment-ready models, but the paper does not provide an explicit general mathematical language for formulas such as

$\frac{1}{2}u_i(o') > \alpha u_i(o)$1

Its documented mechanism is closer to weighted objective optimization plus post hoc Pareto selection than to a solver for arbitrary ratio inequalities (Swati et al., 9 Sep 2025).

Post-processing frameworks face a related issue. A generic fairness penalty $\frac{1}{2}u_i(o') > \alpha u_i(o)$2 can often encode ratio-style objectives, but hard ratio constraints on thresholded rates are typically nonconvex, nondifferentiable, or numerically unstable when denominators are small. This is why the post-processing framework emphasizes smooth or surrogate losses and why its strongest empirical evidence is on disparate impact rather than on exact arbitrary ratio intervals (Eberhard et al., 22 Apr 2025).

The toxicity-detection literature adds a distinct warning: equalizing group-conditioned positive-label ratios is not sufficient if the intervention also shifts the global positive rate far from the task’s natural distribution. Re-stratification toward roughly $\frac{1}{2}u_i(o') > \alpha u_i(o)$3 toxic prevalence improved some fairness quantities but reduced AUC and often worsened threshold-based gaps, whereas perturbation that kept group ratios equal at realistic low prevalence produced the most consistent fairness gains (Elsafoury et al., 2023).

Several papers also show that ratio-like disparities can emerge unintentionally. In content-provider fairness, the two-flow metric

$\frac{1}{2}u_i(o') > \alpha u_i(o)$4

records de facto asymmetric throughput shares under heterogeneous congestion control. The study finds that some employed congestion control algorithms lead to significantly asymmetric bandwidth shares, while FQ_CoDel yields fairness close to equilibrium. This is not custom-ratio fairness in the normative sense; it is uncontrolled ratio emergence caused by stack tuning, pacing, queue size, and queueing discipline (Rüth et al., 2019).

A final recurring tension is that ratio fairness and exact fairness are not synonymous. Differential fairness protects tiny intersectional groups because it does not weight by group prevalence, but accurate estimation becomes harder under sparsity (Foulds et al., 2018). Harm-ratio guarantees can be exact and strong under compactness plus upper convexity, yet $\frac{1}{2}u_i(o') > \alpha u_i(o)$5-IHR need not exist for indivisible goods, and checking $\frac{1}{2}u_i(o') > \alpha u_i(o)$6-IHR is coNP-complete in additive indivisible-goods allocation (Ebadian et al., 2024). Custom-ratios fairness therefore remains both a formal design choice and a domain-dependent engineering problem: the ratio itself may be transparent, but its estimation, optimization, and interpretation are often not.