Beyond Incompatibility: Trade-offs between Mutually Exclusive Fairness Criteria in Machine Learning and Law

TL;DR

This work tackles the incompatibility of three key ML fairness notions—calibration within groups, balance for the negative class, and balance for the positive class—by introducing FAIM, a post-processing algorithm that continuously interpolates between them using optimal transport. FAIM constructs a per-group mapping via three criterion-specific distributions, computes their Wasserstein-2 barycenter with weights $\theta^A,\theta^B,\theta^C$, and applies an OT map to obtain fair scores, thereby producing a tunable compromise rather than an absolute trade-off. The authors validate FAIM on synthetic data, the COMPAS recidivism dataset, and a Zalando e-commerce ranking dataset, illustrating how different weightings yield calibrated, balanced, or intermediate fairness outcomes while preserving as much predictive performance as possible. They also discuss the normative and legal implications, showing how FAIM can help align ML systems with EU regulatory regimes such as the AI Act and the Digital Markets Act, albeit with careful attention to ground-truth reliability and the limits of post-hoc fairness adjustments. Overall, FAIM provides a rigorous, flexible framework for operationalizing fairness trade-offs in high-stakes domains and for translating technical fairness objectives into legally interpretable and auditable policies.

Abstract

Fair and trustworthy AI is becoming ever more important in both machine learning and legal domains. One important consequence is that decision makers must seek to guarantee a 'fair', i.e., non-discriminatory, algorithmic decision procedure. However, there are several competing notions of algorithmic fairness that have been shown to be mutually incompatible under realistic factual assumptions. This concerns, for example, the widely used fairness measures of 'calibration within groups' and 'balance for the positive/negative class'. In this paper, we present a novel algorithm (FAir Interpolation Method: FAIM) for continuously interpolating between these three fairness criteria. Thus, an initially unfair prediction can be remedied to, at least partially, meet a desired, weighted combination of the respective fairness conditions. We demonstrate the effectiveness of our algorithm when applied to synthetic data, the COMPAS data set, and a new, real-world data set from the e-commerce sector. Finally, we discuss to what extent FAIM can be harnessed to comply with conflicting legal obligations. The analysis suggests that it may operationalize duties in traditional legal fields, such as credit scoring and criminal justice proceedings, but also for the latest AI regulations put forth in the EU, like the Digital Markets Act and the recently enacted AI Act.

Figures (4)

• Figure 1: (Best seen in color) Synthetic true (solid lines) and predicted (dashed lines) score distributions for demographic two groups (orange and blue). The decision boundary at $\text{score}=0$ determines ground truth positive and negative labels based on the true scores, and predicted positive and negative labels based on the predicted scores. This simulates a situation in which the scoring algorithm overestimates the qualification of the advantaged group, and further disadvantages the disadvantaged group.
• Figure 2: (Best seen in color). Prediction scores before and after applying FAIM to shift scores to achieve (a-c) each of the three fairness criteria, respectively, and (d) an equal mix of all three fairness criteria. The prediction scores before FAIM are the same as in Fig. \ref{['fig:experiments:data sets:synthetic']} but scaled to values between 0 and 1. The score distributions (top row) have been smoothed using kernel density estimation to make comparison between distributions easier to interpret. The discrete score transport plots (bottom row) shows how mixing of fairness criteria (d) yields a score transport somewhere in between that corresponding to each respective fairness critera.
• Figure 3: (Best seen in color) Accuracy rates disaggregated by COMPAS score. Across all demographics, accuracy rates show that the mid-range scores 4--7 do not provide very meaningful insight on recidivism risk.
• Figure 4: (Best seen in color). Product rankings with low and high visibility brands under different settings. Color indicates the average overall brand visibility of a product (group membership: low and high), whereas fill pattern indicates customer relevance (ground truth: relevant and not relevant). Fig. \ref{['fig:experiments:results:zalando']}a exemplifies the outcome of an ideal ranker that ranks products by relevance irrespective of brand (and brand visibility). Fig. \ref{['fig:experiments:results:zalando']}b shows the distribution of relevant vs not relevent products from low and high visibility brands as shown by Zalando's ranking algorithm by the time of data collection. We see that high visibility brands do indeed have an advantage over low visibility brands, as there are almost no products from the low brand visibility group in the first few bins. Fig. \ref{['fig:experiments:results:zalando']}c--f show how the ranking changes after FAIM is applied with different values for $\theta^A$, $\theta^B$, and $\theta^C$. Depending on our preference for calibration, balance for the positive class, or balance for the negative class, we see varying degrees of ranking improvement for relevant products from the low brand visibility group.

Theorems & Definitions (6)

• Theorem 1
• Remark 2
• Definition 3: Optimal transport map
• Definition 4: Displacement interpolation, cf. Remark 2.13 in ambrosio2013user
• Theorem 5: Barycenter in Wasserstein space agueh2011bary
• Remark 6