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{\alpha}-Fair Insurance Pricing: A Fairness Continuum

This paper introduces the $\alpha$ -FISP framework, a constrained optimization model that resolves the tension between actuarial and solidarity fairness in insurance pricing by generating a continuum of solvable premium solutions parameterized by $\alpha$ , thereby allowing regulators and insurers to balance risk differentiation with cross-subsidization while ensuring solvency.

Original authors: Tianhe Zhang, Xiguang Liu, Peng Shi

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tianhe Zhang, Xiguang 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 running a giant potluck dinner where everyone brings a dish, but some people bring expensive ingredients while others bring simple snacks. The goal is to figure out how much each person should pay for their meal.

This paper tackles a very old, tricky argument about how to charge people for insurance: Should everyone pay exactly what their personal risk costs (Actuarial Fairness), or should everyone pay the same price to help those who are riskier (Solidarity Fairness)?

Here is the paper's solution, broken down into simple concepts.

1. The Two Opposing Views

The authors describe a tug-of-war between two ways of thinking about "fairness":

The "Actuarial" View (The Strict Accountant): This view says, "If you are a young, non-smoking driver, you should pay less. If you are an older smoker, you should pay more." It's based on the idea that your price should match your specific risk perfectly. If you are low-risk, you shouldn't have to subsidize high-risk people.
The "Solidarity" View (The Community Helper): This view says, "We are all in this together." It argues that insurance is a social safety net. If we charge high-risk people too much, they can't afford insurance. So, low-risk people should pay a little extra to help cover the costs for the high-risk group.

The Problem: In the past, these two ideas were seen as enemies. You couldn't have both. If you used too much data (like gender, race, or zip code) to calculate risk, you might end up charging vulnerable groups so much they can't afford coverage. But if you ignore risk entirely, the insurance company might go broke because the "accountants" aren't collecting enough money.

2. The Solution: The "Fairness Slider" ( $\alpha$ )

The authors propose a new framework called $\alpha$ -Fairness. Think of this as a slider switch or a dimmer knob that lets you choose exactly how much "solidarity" you want in the pricing.

The Knob is labeled $\alpha$ (alpha) and goes from 0 to 1.
At 0 (The "Actuarial" End): The knob is all the way down. Everyone pays exactly what their specific risk costs. No one helps anyone else.
At 1 (The "Solidarity" End): The knob is all the way up. Everyone in the same basic group (e.g., same age and smoking status) pays the exact same price, regardless of other details. The low-risk people fully subsidize the high-risk people.
In the Middle (e.g., 0.5): This is the sweet spot. The low-risk people pay a little extra to help the high-risk people, but not so much that they are being punished. The high-risk people get a discount, but not a free ride.

3. The "Safety Net" Rule (Solvency)

The paper adds one crucial rule that many other fairness ideas forget: The insurance company must not go broke.

They call this Individual Solvency. Imagine a specific group of people (say, 30-year-old smokers). The total money collected from this group must be enough to pay for their expected claims. You can't take money from 30-year-old smokers to pay for 60-year-old smokers. The "pot" for each group must stay in the black.

This ensures that while we are being fair to different groups, we aren't creating a financial disaster for the insurance company.

4. How It Works in Practice

The authors turned this idea into a math problem (an optimization task).

Input: They start with the "perfect" price based on risk (what the accountant would charge).
Adjustment: They then apply the $\alpha$ -slider. If you set $\alpha$ to 0.8, the math forces the prices for people in the same group to be very close to each other (within 80% of each other).
Constraint: The math also checks to make sure the group still has enough money to pay its bills.

5. What They Found

The authors tested this with computer simulations and real health insurance data.

It Works: They showed that you can smoothly slide from "strict risk-based pricing" to "community-based pricing" without breaking the math.
It's Safe: Unlike some other "fair" pricing methods they tested, their method never resulted in the insurance group losing money (negative solvency).
It's Flexible: Because regulators in different places have different rules (some states want more fairness, some want less), this framework allows a company to just turn the $\alpha$ -knob to match the local laws.

The Big Picture Analogy

Imagine a school cafeteria.

Actuarial Fairness: You pay exactly for the ingredients in your lunch. If you order steak, you pay \ $20. If you order a sandwich, you pay \$ 5.
Solidarity Fairness: Everyone pays $10. The steak-eaters subsidize the sandwich-eaters so everyone can eat.
The Paper's Solution: The principal (the regulator) says, "We need a middle ground." They set a rule: "The price difference between the steak and sandwich must be no more than 50%."
- The steak might drop to $15.
- The sandwich might rise to $10.
- The cafeteria still makes enough money to buy food (Solvency), but the gap between the rich and poor eaters isn't as huge.

The paper gives us the mathematical tool to find that perfect middle ground, ensuring insurance remains both affordable for people and financially healthy for the company.

Technical Summary: $\alpha$ -Fair Insurance Pricing

Problem Statement
Insurance pricing faces a fundamental tension between two competing notions of fairness: actuarial fairness and solidarity fairness. Actuarial fairness dictates that premiums should reflect individual expected losses, often achieved through granular risk classification. Conversely, solidarity fairness emphasizes social pooling, where risks are shared across groups to ensure affordability and access, often requiring cross-subsidization.

In modern data-driven environments, the availability of granular data allows for increasingly precise risk differentiation, potentially exacerbating premium disparities and raising concerns about affordability and social exclusion. Simultaneously, regulators (e.g., the EU Gender Directive, Colorado SB 21-169) are increasingly restricting the use of sensitive attributes (such as gender, race, or socioeconomic proxies) in pricing. Existing literature often treats these concepts as binary or focuses on predictive fairness in machine learning (e.g., demographic parity), failing to address the specific economic allocation of financial burdens in insurance or the critical operational requirement of solvency (ensuring premiums cover expected losses).

Methodology
The authors propose the $\alpha$ -Fair Individual Solvent Premium ( $\alpha$ -FISP) framework, which models insurance pricing as a constrained optimization problem.

Definitions and Metrics:
- Actuarial Fair Premium ( $\mu(X, D)$ ): The expected loss $E[Y|X, D]$ , fully differentiating by non-sensitive ( $X$ ) and sensitive ( $D$ ) attributes.
- Solidarity Fair Premium ( $\mu(X)$ ): The expected loss $E[Y|X]$ , pooling risks across sensitive attributes within a non-sensitive risk class.
- $\alpha$ -Fairness: A pricing function $f$ $f$ is $\alpha$ $α$ -fair if, for any two individuals with identical non-sensitive attributes ( $x = x'$ $x = x^{'}$ ), the ratio of their premiums is at least $\alpha$ $α$ (where $\alpha \in [0, 1]$ $α \in [0, 1]$ ). Specifically, $\min(f(x, d)/f(x, d'), f(x, d')/f(x, d)) \geq \alpha$ $min (f (x, d) / f (x, d^{'}), f (x, d^{'}) / f (x, d)) \geq α$ .
  - $\alpha = 0$ : Pure actuarial fairness (full differentiation).
  - $\alpha = 1$ : Pure solidarity fairness (uniform pricing within the risk class).
- Individual Solvency: A constraint requiring that, for any fixed non-sensitive attribute $x$ , the weighted average of premiums across sensitive groups must cover the weighted average of their expected losses. This prevents cross-subsidization between heterogeneous risk classes ( $x \neq x'$ ).
Optimization Formulation:
The problem is formulated as minimizing the Mean Squared Error (MSE) between the predicted premium and realized loss, subject to:
- $\alpha$ -Fairness Constraints: Linear inequalities ensuring the ratio of premiums within a risk class stays within $[\alpha, 1/\alpha]$ .
- Solvency Constraints: Ensuring the group-level premium revenue covers the group-level expected loss.
The solution, $\alpha$ -FISP, is derived as the unique minimizer of this constrained risk minimization problem.

Key Theoretical Properties
The paper establishes several theoretical guarantees for the $\alpha$ -FISP framework:

Existence and Uniqueness: Under standard regularity conditions (positive expected losses, bounded premiums), a unique optimal solution exists.
Boundedness: The resulting premiums lie within the convex hull of the actuarially fair premiums for a given risk class. No premium is pushed below the minimum expected loss or inflated beyond the maximum actuarially fair benchmark within that class.
Order Preservation: The relative ordering of premiums across sensitive attributes is preserved. If one group is actuarially riskier than another, the $\alpha$ -fair premium for the riskier group will not be lower than that of the less risky group.
Statistical Guarantees: The framework provides finite-sample excess risk bounds. The cost of enforcing fairness is shown to be an additive bias term that depends on the heterogeneity of risks and the chosen $\alpha$ . The convergence rate remains minimax optimal ( $O(n^{-1/2})$ ).
Solvency: The framework explicitly guarantees individual-level solvency, a property often overlooked in fairness-focused pricing studies.

Experimental Results
Numerical experiments were conducted on synthetic datasets and a U.S. Health Insurance dataset:

Continuum Behavior: As $\alpha$ varies from 0 to 1, premiums smoothly transition from actuarially fair levels to solidarity-fair levels, validating the continuum concept.
Order Preservation: Simulations confirmed that premiums maintain monotonicity with respect to risk levels, even as $\alpha$ increases, provided certain conditions on low-risk group mass are met.
Comparison with DFIP: When compared to the Discrimination-Free Insurance Pricing (DFIP) method (Lindholm et al., 2022), $\alpha$ -FISP consistently maintained a positive solvency margin. In contrast, DFIP produced insolvent premiums (negative solvency margin) across various fairness budgets.
Computational Tractability: The framework is computationally efficient, solvable via standard constrained optimization techniques (e.g., Lagrangian methods with neural networks).
Regulatory Alignment: The framework aligns with heterogeneous state-level regulatory requirements in the U.S., where different jurisdictions may demand different levels of cross-subsidization (captured by varying $\alpha$ ).

Significance and Claims
The paper claims to advance the literature on fairness in both machine learning and actuarial science by:

Formalizing a Fairness Continuum: Introducing $\alpha$ -fairness as a tunable parameter that explicitly captures the trade-off between actuarial and solidarity fairness, moving beyond binary definitions.
Integrating Solvency: Being the first framework to explicitly guarantee individual-level solvency while enforcing fairness constraints, addressing a critical operational gap in existing fair-pricing studies.
Structural Allocation vs. Prediction: Reframing fairness not merely as a predictive constraint (e.g., minimizing error disparity) but as an allocation mechanism that governs how financial burdens are distributed. This bridges the gap between traditional actuarial principles and modern algorithmic fairness.
Practical Implementability: Providing a computationally tractable, interpretable, and regulatorily adaptable pipeline that allows decision-makers to select an operating point ( $\alpha$ ) aligned with specific strategic or regulatory priorities without sacrificing model stability or solvency.