Marginal Fairness: Fair Decision-Making under Risk Measures

TL;DR

This work addresses how to achieve fair decision-making in risk-sensitive domains by separating predictive modeling from risk-based decisions. It introduces marginal fairness, a derivative-based, individual fairness criterion that enforces insensitivity of the decision rule to small perturbations in protected attributes while allowing these attributes to inform the predictive model. The authors develop a general theory for constructing marginally fair decision rules across continuous, bounded, discrete, and multivariate protected attributes, and extend the framework with cascade sensitivity to account for dependencies among covariates via copulas. They demonstrate the approach through numerical simulations and an empirical auto insurance study, showing that marginal fairness can be attained with limited sacrifice to predictive performance and risk segmentation, aligning with regulatory concerns such as EU gender-neutral pricing. The framework provides a practical, regulatorily aligned method for fairness that directly targets the decision rule under generalized distortion risk measures, including Expected Shortfall.

Abstract

This paper introduces marginal fairness, a new individual fairness notion for equitable decision-making in the presence of protected attributes such as gender, race, and religion. This criterion ensures that decisions based on generalized distortion risk measures are insensitive to distributional perturbations in protected attributes, regardless of whether these attributes are continuous, discrete, categorical, univariate, or multivariate. To operationalize this notion and reflect real-world regulatory environments (such as the EU gender-neutral pricing regulation), we model business decision-making in highly regulated industries (such as insurance and finance) as a two-step process: (i) a predictive modeling stage, in which a prediction function for the target variable (e.g., insurance losses) is estimated based on both protected and non-protected covariates; and (ii) a decision-making stage, in which a generalized distortion risk measure is applied to the target variable, conditional only on non-protected covariates, to determine the decision. In this second step, we modify the risk measure such that the decision becomes insensitive to the protected attribute, thus enforcing fairness to ensure equitable outcomes under risk-sensitive, regulatory constraints. Furthermore, by utilizing the concept of cascade sensitivity, we extend the marginal fairness framework to capture how dependencies between covariates propagate the influence of protected attributes through the modeling pipeline. A numerical study and an empirical implementation using an auto insurance dataset demonstrate how the framework can be applied in practice.

Figures (13)

• Figure 1: A graphical representation of the decision process (arrows indicate statistical or functional dependence, not causality). The predicted outcome $Y$ is modeled as a function of both protected attributes ${\mathbf{D}}$ and non-protected covariates ${\mathbf{X}}$. The decision $\rho_\gamma(Y \mid {\mathbf{X}})$ is a function of the conditional distribution of $Y$ given ${\mathbf{X}}$, which is modeled as a function of only ${\mathbf{X}}$. The dashed arrow between ${\mathbf{D}}$ and ${\mathbf{X}}$ indicates that dependence between ${\mathbf{D}}$ and ${\mathbf{X}}$ may or may not exist.
• Figure 1: Sensitivity to $D$ under the assumption that $D\sim Ber(p)$ and independent of $X$ from \ref{['ex:discrete-expected-value']}. The sensitivity is given in \ref{['eq:sens-bernoulli']} with $\beta_2 = 1$. The $x$-axis is the success rate, i.e. $p \in (0,1)$.
• Figure 1: Cascading perturbation of $X_\delta$ due to a perturbation of $D \sim Bern(p)$ from \ref{['mortgage-cascade']}. Blue lines correspond to $p = 0.8$ and red lines to $p =0.2$. Solid lines are $\delta = 0$ and dashed lines are the perturbation $\delta = 0.2$.
• Figure 1: Comparison of fair decision strategies. Unaware decision with expected value (black), discrimination-free with expected value (grey), and marginally fair with expected value (blue) and ES (red).
• Figure 1: Comparison of fair decision strategies for 50 randomly selected policyholders under the expected value risk measure. Blue dots correspond to marginally fair decision, orange crosses to unaware decision, and the greed squares to the discrimination-free decision.

Theorems & Definitions (46)

• Definition 2.1: Generalized distortion risk measure
• Example 2.2: Expected Shortfall
• Example 2.3: Discrimination in insurance
• Definition 2.4: Notions of fairness
• Definition 3.1: Marginal fairness
• Example 3.2
• Definition 3.3: Multi-marginal fairness
• Proposition 3.4: Marginal sensitivity
• Example 3.5
• Remark 3.6