Balance and Fairness through Multicalibration in Nonlife Insurance Pricing

1. Problem Statement

The paper addresses two critical challenges in nonlife insurance pricing:

1. Financial Equilibrium (Autocalibration): Machine learning models (e.g., neural networks, boosted trees) often produce premiums that are not "autocalibrated." This means that for a group of policyholders paying the same premium $p$, the average observed claims do not equal $p$. This leads to cross-subsidization and adverse selection.
2. Fairness: Insurers face regulatory and ethical pressures to avoid discrimination based on sensitive features (e.g., gender, health status, disability). While some features are legally prohibited (e.g., gender in the EU), others are allowed but scrutinized (e.g., health status). The challenge is to ensure that the use of these sensitive features does not result in unfair treatment, defined as disparate expected outcomes for groups paying the same premium.

The authors argue that existing fairness notions (like individual fairness or demographic parity) are often incompatible with the goal of accurate risk classification in insurance. They propose a unified framework that combines financial equilibrium with a specific group-fairness criterion.

2. Methodology and Theoretical Framework

2.1 Core Concepts

• Autocalibration: A premium $\pi(X)$ is autocalibrated if $E[Y \mid \pi(X) = p] = p$. This ensures that the premium collected from a specific price group covers the expected claims of that group.
• Sufficiency (Group Fairness): The paper adopts a "weak" version of sufficiency based on conditional mean independence. A premium is fair with respect to a sensitive feature $S$ if $E[Y \mid \pi(X) = p, S = s] = E[Y \mid \pi(X) = p, S = s']$ for all $s, s'$. This implies that once the premium is fixed, the expected claim cost should not depend on the sensitive attribute.
• Multicalibration: The authors combine the above two concepts. A premium is multicalibrated with respect to $S$ if:
  $$E[Y \mid \pi(X) = p, S = s] = p$$
  for all premium levels $p$ and sensitive feature values $s$. This ensures that within every subgroup defined by the sensitive feature, the premium is self-financing.

2.2 Theoretical Properties

• Reinforcement: Multicalibration implies autocalibration. If a premium is multicalibrated, it is automatically autocalibrated at the portfolio level.
• Convex Order and Overfitting: The authors prove that multicalibrated premiums satisfy $\pi(X) \preceq_{cx} \mu(X) \preceq_{cx} Y$, where $\mu(X)$ is the true pure premium and $Y$ is the claim. This means multicalibrated premiums are less dispersed than the true risk and the raw claims, effectively preventing overfitting.
• Optimality: Multicalibrated predictors minimize Bregman loss functions (e.g., deviance) relative to non-multicalibrated candidates.

2.3 Implementation Strategies

The paper proposes practical algorithms to transform a candidate premium $\pi(X)$ into a multicalibrated version $\pi_{mbc}(X) = E[Y \mid \pi(X), S]$. Two main approaches are detailed for both categorical and continuous sensitive features:

A. Direct Multibalance Correction (Non-Iterative)

• Categorical $S$: Uses isotonic regression within each group $S=\ell$ to estimate the conditional mean function $m_\ell(p) = E[Y \mid \pi(X)=p, S=\ell]$. The new premium is the plug-in of these estimates.
• Continuous $S$: Uses bivariate local Generalized Linear Models (GLMs) to estimate the surface $m(p, s) = E[Y \mid \pi(X)=p, S=s]$. A centering step is applied to ensure marginal autocalibration is preserved.

B. Iterative Bias Correction (Regularized)

• This approach is designed to handle sparse data and stabilize estimates.
• Mechanism: It iteratively updates the premium by adding a correction term based on the residual bias within cells defined by premium bins and sensitive feature levels.
• Credibility Shrinkage: To prevent overfitting in groups with low exposure, the method employs exposure-driven shrinkage (similar to Bühlmann credibility). The correction for a specific cell is shrunk toward the pooled bin-level bias based on the exposure weight $w$ and a tuning parameter $c$:
  $$\hat{e} = z \cdot \hat{b}_{cell} + (1-z) \cdot \hat{b}_{bin}, \quad \text{where } z = \frac{w}{w+c}$$
• Convergence: The process stops when the maximum residual bias across all cells falls below a tolerance threshold.

3. Key Contributions

1. Unification of Fairness and Balance: The paper formally bridges the gap between actuarial financial equilibrium and machine learning fairness by introducing multicalibration as the standard for insurance pricing.
2. Theoretical Guarantees: It establishes that multicalibration is a stronger condition than autocalibration, guarantees convex order dominance (preventing overfitting), and improves predictive accuracy (lower Bregman loss).
3. Practical Algorithms: It provides two distinct, implementable methods (Direct Isotonic/Local GLM and Iterative Shrinkage) for both categorical and continuous sensitive features, addressing the "curse of dimensionality" and data sparsity issues common in insurance portfolios.
4. Handling Prohibited vs. Allowed Features: The framework clarifies how to handle features that are legally allowed but sensitive (e.g., health status) versus those that are proxies for prohibited features (e.g., vehicle power as a proxy for gender).

4. Numerical Results

The authors tested their methods on a French Motor Third-Party Liability (MTPL) dataset (677,991 policies).

• Setup: A baseline Poisson GAM was used to generate initial premiums. The sensitive feature was Vehicle Age (tested as both categorical bins and a continuous variable).
• Findings:
  • Bias Reduction: The baseline model showed significant under-pricing in high-premium bins and systematic bias across vehicle age groups.
  • Autocalibration: Both iterative and isotonic autocalibration removed global bias but left significant conditional bias across vehicle age groups.
  • Multicalibration: Both the iterative and direct multibalance correction methods successfully eliminated conditional bias, aligning residuals to zero across all vehicle age groups and premium bins.
  • Predictive Accuracy (Deviance): Contrary to the common belief that fairness constraints reduce accuracy, multicalibration improved predictive performance. The multicalibrated models achieved the lowest Poisson deviance on the test set, outperforming both the baseline and the simple autocalibrated models.
  • Discriminatory Power (Gini): While the Gini coefficients were slightly lower for multicalibrated models compared to simple autocalibration (due to the smoothing effect of fairness constraints), they remained significantly higher than the baseline, indicating that fairness did not destroy the model's ability to rank risks.

5. Significance and Conclusion

This paper makes a significant contribution to the actuarial and machine learning literature by demonstrating that fairness and accuracy are not mutually exclusive in insurance pricing.

• Regulatory Compliance: It offers a rigorous mathematical framework for insurers to prove that their pricing models are fair with respect to sensitive attributes, satisfying regulatory requirements (like EU directives) without sacrificing financial solvency.
• Model Improvement: By enforcing multicalibration, insurers effectively correct systematic errors in their pricing models, leading to more accurate risk assessment and better financial equilibrium.
• Practicality: The proposed iterative shrinkage methods are robust to data sparsity, making them suitable for real-world insurance portfolios where certain risk groups may have limited exposure.

In summary, the paper argues that multicalibration is the optimal standard for nonlife insurance pricing, as it simultaneously ensures that premiums are fair across sensitive groups and that the insurer's portfolio remains financially balanced and statistically accurate.