Sensitivity-based measures of discrimination in insurance pricing

Different notions of fairness and discrimination have been extensively discussed in the machine learning, operations research, and insurance pricing literatures. As not all fairness criteria can be concurrently satisfied, metrics are needed that allow assessing the materiality of discriminatory effects and the trade-offs between various criteria. Methods from sensitivity analysis have been deployed for the measurement of demographic unfairness, that is, the statistical dependence of risk predictions on protected attributes. We produce a sensitivity-based measure for the distinct phenomenon of proxy discrimination, referring to the implicit inference of protected attributes from other covariates. For this, we first define a set of admissible prices that avoid proxy discrimination. Then, the measure is defined as the normalised L2-distance of a price from the closest element in that set. We use variance-based sensitivity analysis and Shapley values to attribute the proxy discrimination measure to individual (or subsets of) covariates and investigate how properties of the data generating process are reflected in those metrics. Furthermore, we build on the global (i.e., portfolio-wide) measures of demographic unfairness and proxy discrimination to propose local (i.e., instance- or policyholder-specific) measures, which allow a fine-grained understanding of discriminatory effects. Finally, we apply the methods developed in the paper to a real-world insurance dataset, where ethnicity is a protected variable. We observe substantial proxy-discriminatory effects for one ethnic group and identify the key variables driving this.