# The Fairness of Risk Scores Beyond Classification: Bipartite Ranking and the xAUC Metric

arXiv: https://arxiv.org/abs/1902.05826 · HTML: https://mlsanity.com/paper/1902.05826/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2019.

## Built on by (most cited first, 30 of 41)

- [AUC Maximization in the Era of Big Data and AI: A Survey](https://mlsanity.com/paper/2203.15046/) (2022)
- [Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning](https://mlsanity.com/paper/2108.08435/) (2021)
- [Fairness in Ranking under Uncertainty](https://mlsanity.com/paper/2107.06720/) (2021)
- [A Sociotechnical View of Algorithmic Fairness](https://mlsanity.com/paper/2110.09253/) (2021)
- [The KL-Divergence between a Graph Model and its Fair I-Projection as a Fairness Regularizer](https://mlsanity.com/paper/2103.01846/) (2021)
- [Deep ROC Analysis and AUC as Balanced Average Accuracy to Improve Model Selection, Understanding and Interpretation](https://mlsanity.com/paper/2103.11357/) (2021)
- [Minimax AUC Fairness: Efficient Algorithm with Provable Convergence](https://mlsanity.com/paper/2208.10451/) (2022)
- [The Data Addition Dilemma](https://mlsanity.com/paper/2408.04154/) (2024)
- [Slicing Through Bias: Explaining Performance Gaps in Medical Image Analysis using Slice Discovery Methods](https://mlsanity.com/paper/2406.12142/) (2024)
- [A comparison of approaches to improve worst-case predictive model performance over patient subpopulations](https://mlsanity.com/paper/2108.12250/) (2021)
- [On (assessing) the fairness of risk score models](https://mlsanity.com/paper/2302.08851/) (2023)
- [Fairness for AUC via Feature Augmentation](https://mlsanity.com/paper/2111.12823/) (2021)
- [Stochastic Methods for AUC Optimization subject to AUC-based Fairness Constraints](https://mlsanity.com/paper/2212.12603/) (2022)
- [Fairness Through Regularization for Learning to Rank](https://mlsanity.com/paper/2102.05996/) (2021)
- [Repairing Regressors for Fair Binary Classification at Any Decision Threshold](https://mlsanity.com/paper/2203.07490/) (2022)
- [Disparate Censorship & Undertesting: A Source of Label Bias in Clinical Machine Learning](https://mlsanity.com/paper/2208.01127/) (2022)
- [Toward Operationalizing Pipeline-aware ML Fairness: A Research Agenda for Developing Practical Guidelines and Tools](https://mlsanity.com/paper/2309.17337/) (2023)
- [Pairwise Fairness for Ordinal Regression](https://mlsanity.com/paper/2105.03153/) (2021)
- [Deep AUC Maximization for Medical Image Classification: Challenges and Opportunities](https://mlsanity.com/paper/2111.02400/) (2021)
- [Fairness for Robust Learning to Rank](https://mlsanity.com/paper/2112.06288/) (2021)
- [Preserving AUC Fairness in Learning with Noisy Protected Groups](https://mlsanity.com/paper/2505.18532/) (2025)
- [Bipartite Ranking Fairness through a Model Agnostic Ordering Adjustment](https://mlsanity.com/paper/2307.14668/) (2023)
- [Retiring DeltaDP: New Distribution-Level Metrics for Demographic Parity](https://mlsanity.com/paper/2301.13443/) (2023)
- [Threshold-Independent Fair Matching through Score Calibration](https://mlsanity.com/paper/2405.20051/) (2024)
- [Standardized Interpretable Fairness Measures for Continuous Risk Scores](https://mlsanity.com/paper/2308.11375/) (2023)
- [Aligning Evaluation with Clinical Priorities: Calibration, Label Shift, and Error Costs](https://mlsanity.com/paper/2506.14540/) (2025)
- [Consider the Alternatives: Navigating Fairness-Accuracy Tradeoffs via Disqualification](https://mlsanity.com/paper/2110.00813/) (2021)
- [Beyond Submodularity: A Unified Framework of Randomized Set Selection with Group Fairness Constraints](https://mlsanity.com/paper/2304.06596/) (2023)
- [Algorithmic Fairness: A Tolerance Perspective](https://mlsanity.com/paper/2405.09543/) (2024)
- [Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints](https://mlsanity.com/paper/2505.12530/) (2025)

## Abstract (arXiv)

Where machine-learned predictive risk scores inform high-stakes decisions, such as bail and sentencing in criminal justice, fairness has been a serious concern. Recent work has characterized the disparate impact that such risk scores can have when used for a binary classification task. This may not account, however, for the more diverse downstream uses of risk scores and their non-binary nature. To better account for this, in this paper, we investigate the fairness of predictive risk scores from the point of view of a bipartite ranking task, where one seeks to rank positive examples higher than negative ones. We introduce the xAUC disparity as a metric to assess the disparate impact of risk scores and define it as the difference in the probabilities of ranking a random positive example from one protected group above a negative one from another group and vice versa. We provide a decomposition of bipartite ranking loss into components that involve the discrepancy and components that involve pure predictive ability within each group. We use xAUC analysis to audit predictive risk scores for recidivism prediction, income prediction, and cardiac arrest prediction, where it describes disparities that are not evident from simply comparing within-group predictive performance.