Multiplicative Metric Fairness Under Composition

Author Milan Mossé

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Milan Mossé
  • Department of Philosophy, University of California at Berkeley, CA, USA


Many thanks to Omer Reingold and Li-Yang Tan for their generous guidance and support with this project. Thanks to James Evershed, Wes Holliday, Niko Kolodny, Gabrielle Candès, and three anonymous reviewers for helpful comments.

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Milan Mossé. Multiplicative Metric Fairness Under Composition. In 4th Symposium on Foundations of Responsible Computing (FORC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 256, pp. 4:1-4:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Dwork, Hardt, Pitassi, Reingold, & Zemel [Dwork et al., 2012] introduced two notions of fairness, each of which is meant to formalize the notion of similar treatment for similarly qualified individuals. The first of these notions, which we call additive metric fairness, has received much attention in subsequent work studying the fairness of a system composed of classifiers which are fair when considered in isolation [Chawla and Jagadeesan, 2020; Chawla et al., 2022; Dwork and Ilvento, 2018; Dwork et al., 2020; Ilvento et al., 2020] and in work studying the relationship between fair treatment of individuals and fair treatment of groups [Dwork et al., 2012; Dwork and Ilvento, 2018; Kim et al., 2018]. Here, we extend these lines of research to the second, less-studied notion, which we call multiplicative metric fairness. In particular, we exactly characterize the fairness of conjunctions and disjunctions of multiplicative metric fair classifiers, and the extent to which a classifier which satisfies multiplicative metric fairness also treats groups fairly. This characterization reveals that whereas additive metric fairness becomes easier to satisfy when probabilities of acceptance are small, leading to unfairness under functional and group compositions, multiplicative metric fairness is better-behaved, due to its scale-invariance.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • algorithmic fairness
  • metric fairness
  • fairness under composition


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