Discrimination-insensitive pricing

Paper: arXiv 2603.16720

Authors: Kathleen Miao, Silvana Pesenti

Abstract

Rendering fair prices for financial, credit, and insurance products is of ethical and regulatory interest. In many jurisdictions, discriminatory covariates, such as gender and ethnicity, are prohibited from use in pricing such instruments. In this work, we propose a discrimination-insensitive pricing framework, where we require the pricing principle to be insensitive to the (exogenously determined) protected covariates, that is the sensitivity of the pricing principle to the protected covariate is zero. We formulate and solve the optimisation problem that finds the nearest (in Kullback-Leibler (KL) divergence) “pricing” measure to the real world probability, such that under this pricing measure the principle is discrimination-insensitive. We call the solution the discrimination-insensitive measure and provide conditions for its existence and uniqueness. In situations when there are more than one protected covariates, the discrimination-insensitive pricing measure might not exist, and we propose a two-step procedure. First, for each protected covariate separately, we find the measure under which the pricing principle becomes insensitivity to that covariate. Second we reconcile these measures through a constrained barycentre model. We provide a close-form solution to this problem and give conditions for existence and uniqueness of the constrained barycentre pricing measure. As an intermediary result, we prove the representation, existence, and uniqueness of the KL barycentre of general probability measures, which may be of independent interest. Finally, in a numerical illustration, we compare our discrimination-insensitive premia and the constrained barycentre pricing measure with recently proposed fair premia from the actuarial literature.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for fair pricing using Kullback-Leibler divergence minimization and constrained Wasserstein/KL barycentres. While mathematically rich and highly novel in its theoretical formulation, the empirical component is limited to numerical illustrations rather than extensive empirical backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Fair & Discrimination-Insensitive Pricing] --> B[Inputs: Real-World Probability & Protected Covariates]
    B --> C{Single or Multiple Protected Covariates?}
    
    C -- Single Covariate --> D[Minimise KL Divergence for Insensitive Measure]
    C -- Multiple Covariates --> E[Step 1: Compute Individual Insensitive Measures<br/>Step 2: Reconcile via Constrained KL Barycentre]
    
    D --> F[Analytical Proofs: Existence, Uniqueness & General KL Barycentre Representation]
    E --> F
    
    F --> G[Outcome: Fair Premia Validated Against Actuarial Benchmarks]

Cited References (42)

1. Fairness in machine learning (2020)
2. Assessing algorithmic fairness with unobserved protected class using data combination (2022)
3. The fairness of credit scoring models (2026)
4. A Sequentially Fair Mechanism for Multiple Sensitive Attributes (2024)
5. A Survey on Bias and Fairness in Machine Learning (2021)
6. Sensitivity analysis: A review of recent advances (2016)
7. Reverse sensitivity testing: What does it take to break the model? (2019)
8. Marginal Fairness: Fair Decision-Making under Risk Measures — → 2505.18895

3 of 42 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.