OptiGrad: A Fair and more Efficient Price Elasticity Optimization via a Gradient Based Learning
Vincent Grari, Marcin Detyniecki
TL;DR
OptiGrad tackles price elasticity optimization in non-life insurance by replacing discrete, reverse-engineered ratebook approaches with a differentiable, continuous framework. It combines a differentiable pure premium model $h_{w_h}$ and a differentiable conversion model $f_{w_f}$ with a learnable pricing coefficient $c_{w_c}$ to maximize Global Written Margin while enforcing a minimum conversion; a sigmoid reparameterization and Lagrangian terms enable gradient-based optimization within bounds $[a,b]$. The framework is extended to fairness by incorporating an HGR-based adversarial component that minimizes dependence between the final commercial premium and sensitive attributes, operationalized via a min-max objective with hyperparameters $\lambda_f$ and $\lambda_S$. Empirical results on Atoti/Kaggle insurance data show OptiGrad outperforms traditional methods in margin-efficiency and, when fairness is enforced, significantly reduces demographic disparities (as measured by HGR) without sacrificing primary profitability. The work demonstrates a practical pathway to jointly optimize profitability, conversion, and fairness in continuous pricing for insurance.
Abstract
This paper presents a novel approach to optimizing profit margins in non-life insurance markets through a gradient descent-based method, targeting three key objectives: 1) maximizing profit margins, 2) ensuring conversion rates, and 3) enforcing fairness criteria such as demographic parity (DP). Traditional pricing optimization, which heavily lean on linear and semi definite programming, encounter challenges in balancing profitability and fairness. These challenges become especially pronounced in situations that necessitate continuous rate adjustments and the incorporation of fairness criteria. Specifically, indirect Ratebook optimization, a widely-used method for new business price setting, relies on predictor models such as XGBoost or GLMs/GAMs to estimate on downstream individually optimized prices. However, this strategy is prone to sequential errors and struggles to effectively manage optimizations for continuous rate scenarios. In practice, to save time actuaries frequently opt for optimization within discrete intervals (e.g., range of [-20\%, +20\%] with fix increments) leading to approximate estimations. Moreover, to circumvent infeasible solutions they often use relaxed constraints leading to suboptimal pricing strategies. The reverse-engineered nature of traditional models complicates the enforcement of fairness and can lead to biased outcomes. Our method addresses these challenges by employing a direct optimization strategy in the continuous space of rates and by embedding fairness through an adversarial predictor model. This innovation not only reduces sequential errors and simplifies the complexities found in traditional models but also directly integrates fairness measures into the commercial premium calculation. We demonstrate improved margin performance and stronger enforcement of fairness highlighting the critical need to evolve existing pricing strategies.