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Discrimination-free Insurance Pricing with Privatized Sensitive Attributes

This paper proposes statistical methods for estimating discrimination-free insurance premiums using only privatized sensitive attributes held by a trusted third party, providing theoretical guarantees and empirical validation for fair pricing under both known and unknown privacy mechanisms.

Original authors: Tianhe Zhang, Suhan Liu, Peng Shi

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tianhe Zhang, Suhan Liu, Peng Shi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to set the price for a car insurance policy. To do this fairly and accurately, you need to know how much risk a driver brings. Usually, you look at things like their age, driving history, and the type of car they drive.

But there's a problem: In many places, the law says you cannot use certain personal details to set prices, like a person's gender or race. This is to prevent discrimination. However, there's a catch: even if you don't ask for gender, your computer model might still figure it out by looking at other clues (like zip codes or shopping habits) that are strongly linked to gender. This is called "indirect discrimination."

Recent experts have come up with a mathematical recipe called a "Discrimination-Free Premium." This recipe tries to calculate a price that ignores gender completely, ensuring that two people with the same driving habits pay the same, regardless of their gender.

The Big Problem:
To use this "Discrimination-Free" recipe, you usually need to see the actual gender data to do the math. But, because of privacy laws, insurance companies often aren't allowed to see the real gender data at all. It's like trying to bake a cake using a recipe that requires eggs, but the store only sells you "egg-flavored powder" that looks like eggs but isn't quite the same.

The Solution in This Paper:
The authors of this paper (Tianhe Zhang, Suhan Liu, and Peng Shi) figured out how to bake that cake using only the "powder." They created a new method that works even when the sensitive information (like gender) has been "scrambled" or "noised up" to protect privacy.

Here is how they did it, using some simple analogies:

1. The "Trusted Neighbor" Setup

Imagine a three-person team:

The Insurance Company: They have the customer's driving data (age, car type) and the claim history (how much money was paid out). They don't have the gender data.
The Trusted Third Party (TTP): A neutral, secure vault that holds the scrambled gender data.
The Customer: They provide their data.

The Insurance Company sends their driving data to the Trusted Neighbor. The Neighbor mixes it with the scrambled gender data to do the math, then sends back the final "fair price" without ever revealing who is actually male or female.

2. The "Scrambled Egg" (Privatized Data)

The paper deals with two scenarios regarding how "scrambled" the data is:

Scenario A: We know the scramble recipe.
Imagine the Trusted Neighbor tells the Insurance Company, "I added exactly 10% noise to the gender data." The authors show that if you know exactly how the data was scrambled, you can mathematically "unscramble" the bias and calculate the fair price perfectly. It's like knowing exactly how much salt was added to a soup, so you can calculate how much water to add to fix the taste.
Scenario B: We don't know the scramble recipe.
This is harder. Imagine the Neighbor says, "I scrambled it, but I forgot exactly how much noise I added." The authors developed a clever trick to guess the noise level using a special "anchor point" in the data (a specific type of driver where we are 100% sure of their gender). Once they estimate the noise level, they can still calculate a fair price, though it's slightly less precise than if they knew the exact recipe.

3. The "Magic Filter" (Transformed Data)

The paper also introduces a cool trick. Before sending data to the Trusted Neighbor, the Insurance Company runs the driving data through a "magic filter" (a neural network). This filter learns to highlight the parts of the data that actually predict risk (like "smoking status" or "age") while smoothing out the noise.

The Result: Even if the gender data is very noisy (very scrambled), using this "magic filter" on the driving data helps the model stay accurate. It's like wearing noise-canceling headphones; even if the room is loud, you can still hear the music clearly.

What the Experiments Showed

The authors ran computer simulations and tested their method on real-world health and car insurance data. They found:

Fairness Works: Their method successfully removed both direct and indirect discrimination.
Privacy is Safe: The insurance company never needed to see the real gender data.
Noise Matters: If the privacy "noise" is too high (too much scrambling), the prices get less accurate. However, their method handles this better than older methods.
Underestimating is Risky: If you guess the noise level is lower than it actually is, your prices get messed up more than if you guess it's higher. It's better to be slightly "conservative" in your guess.

The Bottom Line

This paper provides a practical toolkit for insurance companies. It allows them to use advanced AI to set fair prices that don't discriminate based on gender or race, even when strict privacy laws prevent them from ever seeing the real gender data. They can do this by working with a trusted third party and using statistical tricks to correct for the "noise" added to protect privacy.

Technical Summary: Discrimination-Free Insurance Pricing with Privatized Sensitive Attributes

Problem Statement
The paper addresses the challenge of achieving fairness in insurance pricing when sensitive attributes (e.g., gender, race) are not directly accessible to the insurer due to regulatory restrictions (e.g., the EU Gender Directive, Colorado SB 21-169) or privacy constraints. While recent actuarial research has established the concept of the "discrimination-free premium"—which removes both direct and indirect causal effects of sensitive attributes while maintaining actuarial consistency—implementing this typically requires access to the true sensitive attributes. In practice, insurers often only possess non-sensitive attributes and outcomes, while sensitive data may be held by a trusted third party (TTP) in a privatized, noise-perturbed form. The core problem is how to estimate discrimination-free premiums when the sensitive attribute $D$ is unobserved and only a privatized version $S$ is available.

Methodology
The authors propose a multi-party learning framework involving an insurer and a trusted third party (TTP).

Data Setting: The insurer observes non-sensitive attributes $X$ and outcomes $Y$ . The TTP holds the privatized sensitive attributes $S$ , generated via a privacy mechanism (e.g., Local Differential Privacy via Randomized Response) that introduces noise to the true sensitive attribute $D$ . The insurer transforms $X$ into $\tilde{X}$ (potentially via a neural network) and sends $\{\tilde{X}, Y\}$ to the TTP.
Pricing Rule: The goal is to estimate the discrimination-free price $h^*(X) = \int \mu(X, d) dP^*(d)$ , where $\mu(X, D)$ is the conditional expected loss and $P^*$ is a reference distribution.
Estimation Strategies:
- Known Noise Rates: When the privacy mechanism parameters (noise rates $\pi, \bar{\pi}$ ) are known, the TTP constructs an equivalent risk function (Risk-LDP) that minimizes the expected loss with respect to the true attributes $D$ using only the privatized data $S$ . This involves inverting the transition matrix of the privacy mechanism to correct for the noise. The TTP estimates group-specific score functions $f_k(\tilde{X})$ for each sensitive group $k$ and aggregates them using the reference distribution.
- Unknown Noise Rates: When noise rates are unknown, the paper introduces a procedure to estimate them from the data using "anchor points" (covariate values where the sensitive attribute is known with certainty). The TTP partitions the data into groups, estimates the noise parameter $\pi$ for each group, and aggregates these estimates to construct the inverse matrix required for the risk minimization.
Theoretical Guarantees: The paper establishes statistical bounds on the estimation error for both scenarios. For known noise rates, the bound depends on the Rademacher complexity of the loss class and the noise parameters. For unknown noise rates, the bound accounts for the estimation uncertainty of the noise parameters, showing that the error grows if the noise rate is underestimated.

Key Contributions

Statistical Framework: Development of a method to estimate discrimination-free premiums using only privatized sensitive attributes, bridging actuarial fairness concepts with statistical learning under privacy constraints.
Theoretical Guarantees: Establishment of convergence rates and error bounds for the proposed estimators under two practical settings: known noise mechanisms and unknown noise rates (estimated via anchor points).
Empirical Validation: Demonstration of the method's performance through numerical experiments on synthetic data and real-world datasets (U.S. Health Insurance and Automobile Insurance). The experiments analyze the impact of noise levels, sample sizes, and the accuracy of noise rate estimation on pricing performance.

Results

Noise Impact: Estimation accuracy deteriorates as the noise level increases (i.e., as the privacy parameter $\pi$ decreases). The paper notes a non-linear degradation, where performance drops sharply as $\pi$ approaches the non-informative limit ( $1/|D|$ ).
Asymmetry in Estimation Error: When noise rates are unknown and must be estimated, underestimating the noise rate ( $\pi$ ) has a significantly more detrimental effect on performance than overestimating it. This is due to the mathematical structure of the inverse transition matrix, where terms involving $1/(\pi - 1/|D|)$ explode as $\pi$ approaches the lower bound.
Feature Transformation: Using transformed attributes $\tilde{X}$ (learned via supervised methods) rather than raw attributes $X$ improves model convergence and robustness to noise. This is particularly beneficial when noise rates are high or sample sizes are limited.
Group Size in Estimation: For unknown noise rates, the number of groups ( $n_1$ ) used to estimate $\pi$ affects performance. A larger $n_1$ improves accuracy when the effective sample size is large, but can degrade performance when the noise rate is high (small effective sample size) due to instability in small group estimates.

Significance
The paper claims to provide a practical solution for implementing fair insurance pricing in environments where direct access to sensitive attributes is restricted. By integrating the discrimination-free premium framework with privacy-preserving statistical learning, the authors demonstrate that insurers can construct fair pricing models without violating privacy regulations or requiring direct access to protected data. The work highlights the trade-offs between privacy protection (noise level) and pricing accuracy, offering theoretical guidance on managing these trade-offs, particularly regarding the risks of underestimating noise levels in privacy mechanisms.