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A Unified Approach to Quantifying Algorithmic Unfairness: Measuring Individual & Group Unfairness via Inequality Indices
Till Speicher, Hoda Heidari, Nina Grgic-Hlaca, Krishna P. Gummadi, Adish Singla, Adrian Weller, Muhammad Bilal Zafar
TL;DR
The paper asks how to compare the degrees of unfairness produced by algorithms, since prior work mainly specifies fairness conditions without satisfactory unfairness measures. It uses economics-based inequality indices over algorithmic benefits to quantify individual- and group-level unfairness. The resulting decomposition shows that reducing between-group unfairness can increase within-group and overall unfairness, while the paper also characterizes tradeoffs with prediction accuracy.
Problem
Prior fairness research largely defines conditions for fairness but lacks satisfactory measures for comparing how unfair different algorithms are.
Method
The paper applies axiomatized inequality indices to benefit functions derived from algorithmic outcomes, using subgroup decomposability to measure individual- and group-level unfairness.
Results
Reducing between-group unfairness can increase within-group unfairness and consequently increase an algorithm’s overall individual unfairness.
Takeaways & Limitations
The framework provides a general basis for comparing unfair algorithms and for analyzing tradeoffs among fairness notions and prediction accuracy.
Takeaways & Limitations
The current work focuses on supervised learning, leaving careful exploration of unsupervised settings such as clustering and ranking for future work.
Abstract
from arXiv · showhide
Discrimination via algorithmic decision making has received considerable attention. Prior work largely focuses on defining conditions for fairness, but does not define satisfactory measures of algorithmic unfairness. In this paper, we focus on the following question: Given two unfair algorithms, how should we determine which of the two is more unfair? Our core idea is to use existing inequality indices from economics to measure how unequally the outcomes of an algorithm benefit different individuals or groups in a population. Our work offers a justified and general framework to compare and contrast the (un)fairness of algorithmic predictors. This unifying approach enables us to quantify unfairness both at the individual and the group level. Further, our work reveals overlooked tradeoffs between different fairness notions: using our proposed measures, the overall individual-level unfairness of an algorithm can be decomposed into a between-group and a within-group component. Earlier methods are typically designed to tackle only between-group unfairness, which may be justified for legal or other reasons. However, we demonstrate that minimizing exclusively the between-group component may, in fact, increase the within-group, and hence the overall unfairness. We characterize and illustrate the tradeoffs between our measures of (un)fairness and the prediction accuracy.
1 INTRODUCTION
The paper addresses the missing principled way to compare degrees of unfairness among algorithms by using inequality indices to quantify unequal benefits. This framework unifies individual- and group-level unfairness and exposes tradeoffs between between-group, within-group, overall unfairness, and prediction accuracy.
- Motivation: Prior fairness work defines conditions for fairness but lacks a principled measure for comparing how unfair two algorithms are.This gap matters when designers must choose among feasible but unfair algorithms.
- Approach: The framework uses economics-based inequality indices to measure how unequally algorithmic outcomes benefit individuals or groups.A benefit function maps each individual’s algorithmic outcome to a non-negative value and can be adapted to different fairness conditions.
- Approach: Generalized entropy indices provide subgroup decomposability, separating total unfairness into between-group and within-group components.This supports analysis across sensitive-attribute groups and intersecting subgroups such as young white women or old black men.
- Findings: Existing fair-learning methods may eliminate between-group unfairness while targeting only a small fraction of total unfairness and increasing within-group unfairness.The resulting overall unfairness can therefore become worse despite improvements on the between-group component.
- Findings: The proposed measures reveal tradeoffs between individual fairness and prediction accuracy while enabling comparisons across different algorithmic predictors.Their axiomatic basis includes anonymity, population invariance, progressive transfer preference, and subgroup decomposability.
2 MEASURING ALGORITHMIC UNFAIRNESS VIA INEQUALITY INDICES
The paper defines algorithmic unfairness by measuring inequality in the benefits that predictions provide to individuals or groups. It constructs benefit functions for fairness notions, applies inequality indices, and decomposes individual unfairness into between-group and within-group components.
- Benefit functions: The framework supports group-level and individual-level unfairness measurement across fairness notions by assigning benefits to true positives, true negatives, false positives, and false negatives.Group benefits are defined from the mean benefits of individuals in a group, while fairness notions determine which outcomes are considered and how they are valued.
- Unfairness as benefit inequality: The approach maps each algorithmic outcome to a non-negative benefit and measures inequality in the resulting benefit distribution using inequality indices.Benefit assignments depend on the fairness notion, such as whether accurate outcomes or positive labels are considered more desirable.
- Scope: The framework is developed for supervised learning, while extension beyond supervised tasks requires specifying an appropriate notion of individual benefit.The paper specifically leaves careful exploration of unsupervised settings such as clustering or ranking for future work.
- Inequality indices: The paper focuses on generalized entropy indices, which satisfy anonymity, population-invariance, the Pigou-Dalton transfer principle, zero-normalization, subgroup decomposability, and scale invariance.These properties provide the axiomatic basis for comparing unequal benefit distributions.
- Decomposition: Individual unfairness can be decomposed into between-group unfairness and within-group unfairness, allowing the framework to quantify inequality across sensitive and intersecting subgroups.Existing group fairness notions resemble the between-group component but may ignore group proportions, whereas this measure incorporates group size.
3 THEORETICAL CHARACTERIZATION
The paper characterizes how inequality-based individual unfairness relates to accuracy and group fairness, including tradeoffs between within-group and between-group inequality. It shows that group granularity and the choice of fairness objective determine how well reducing group unfairness addresses overall unfairness.
- 3.2 Individual vs. Group Fairness: Additively decomposable inequality indices express overall individual unfairness as within-group plus between-group components.The decomposition formally connects individual-level unfairness to group-level unfairness for partitions of the population.
- 3.1 Accuracy vs. Individual Fairness: Perfect individual fairness is achievable exactly when perfect prediction accuracy is achievable, under zero-normalized inequality indices and classifiers closed under complements.This equivalence does not imply that minimizing prediction error generally eliminates unfairness.
- 3.1 Accuracy vs. Individual Fairness: Fairness-optimal and accuracy-optimal classifiers may differ, and the fairness-optimal classifier can have arbitrarily worse accuracy.The divergence arises under broad conditions rather than only in isolated examples.
- 3.2 Individual vs. Group Fairness: Reducing only between-group unfairness can increase within-group unfairness and thereby increase overall individual unfairness.This risk is greatest when groups are coarse and benefits within groups are highly unequal; finer partitions make between-group unfairness more central but harder to control.
- 3.2 Individual vs. Group Fairness: Between-group unfairness increases as groups become more granular, from broad sensitive-feature partitions toward intersections of features.With each individual placed in its own group, between-group unfairness equals overall individual unfairness.
4 EMPIRICAL ANALYSIS
The empirical analysis evaluates fairness–accuracy tradeoffs and decomposes individual unfairness into between-group and within-group components across Adult and COMPAS datasets. Results show that accuracy-optimal thresholds need not minimize unfairness, and reducing between-group unfairness can increase within-group or overall unfairness.
- Empirical setup: The study evaluates fairness–accuracy tradeoffs using Adult and COMPAS datasets, with gender and race as sensitive features and repeated 70%-30% train-test splits.Logistic regression, RBF-kernel SVM, random forest, and an oracle are evaluated; hyperparameters use an additional train-validation split.
- 4.1 Fairness vs. Accuracy Tradeoffs: For the oracle, perfect accuracy coincides with zero unfairness near τ = 0.75 on Adult and τ = 0.45 on COMPAS.These thresholds equal the respective fractions of instances in the negative class; beyond the optimum, accuracy and unfairness reverse trend.
- 4.1 Fairness vs. Accuracy Tradeoffs: For non-oracle classifiers, the threshold optimizing imperfect accuracy is far from the threshold optimizing unfairness, while unfairness can increase as accuracy first rises and then falls.Overall unfairness is not necessarily monotone in the decision ranking threshold.
- 4.2 Fairness Decomposability: Between-group unfairness follows multimodal threshold trends and is often a small fraction of overall unfairness, although its contribution increases with more sensitive-feature groups.The Adult trend includes local minima and maxima across τ; COMPAS shows a similar pattern to a lesser extent.
- 4.2 Fairness Decomposability: Logistic regression, SVM, and random forests achieve accuracies of 66%, 67%, and 65%, with overall unfairness values of 0.145, 0.151, and 0.134 on COMPAS.Despite similar accuracy and overall unfairness, classifiers can have different between-group unfairness across sensitive-feature sets.
- 4.2 Fairness Decomposability: Reducing between-group unfairness can increase within-group unfairness for Whites and, for some covariance thresholds, increase overall unfairness relative to an unconstrained classifier.This pattern is observed when tightening false-negative-rate constraints from Zafar et al.
5 CONCLUSION
The conclusion presents inequality indices as a principled framework for measuring total algorithmic unfairness. It emphasizes that focusing only on between-group unfairness can increase overall unfairness.
- 5 CONCLUSION: Inequality indices provide a principled scalar measure of total unfairness for algorithmic decision systems.The approach is based on axiomatic principles and can capture prior fairness conditions through different benefit functions.
- 5 CONCLUSION: The framework unifies group and individual fairness and supports quantitative analysis of earlier unfairness-mitigation methods.It also admits generalizations beyond previous fairness conditions.
- 5 CONCLUSION: Minimizing only between-group unfairness may increase overall unfairness.The conclusion notes that this focus may nevertheless be justified for legal reasons or to redress particular social prejudices.
A APPENDIX: TECHNICAL MATERIAL
The appendix supplies technical proofs and examples for the paper’s fairness propositions. It shows that fairness-optimal decisions can be much less accurate, and that added features can worsen the fairness of accuracy-optimal classifiers.
- Technical propositions: A zero-error classifier assigns equal benefit to everyone, yielding zero inequality under the index.This supports the proposition that perfect accuracy minimizes the inequality measure when such a classifier exists.
- Technical propositions: For a probabilistic classifier, the benefit distribution assigns probabilities p(1−q), pq + (1−p)(1−q), and (1−p)q to benefits 0, 1, and 2.The appendix compares this distribution with the accuracy-optimal classifier’s distribution.
- Proof techniques: The appendix uses progressive transfers and inequality-index properties such as scale invariance and the Dalton principle to establish unfairness comparisons.These proof steps show that the constructed benefit distributions have lower or equal inequality under the relevant index.
- Technical examples: The fairness-optimal classifier can assign label 1 to every instance, achieving accuracy p while an accuracy-optimal classifier achieves accuracy (1−p).The ratio (1−p)/p can become arbitrarily large as p becomes sufficiently small.
- Technical examples: Adding a new feature can worsen the fairness of the accuracy-optimal classifier.Example A.2 constructs two subpopulations whose accuracy-optimal predictions produce a higher generalized-entropy value after feature addition.