Partial Law Invariance and Risk Measures
Yi Shen, Zachary Van Oosten, Ruodu Wang
Management Science · 2026 · Volume 72 · Issue 7
Abstract
We introduce partial law invariance, generalizing law invariance, and probabilistic sophistication widely used in decision theory, as well as statistical and financial applications. Partial law invariance may be interpreted as law invariance restricted to events for which there is no model uncertainty, reflecting practical situations in decision theory and financial risk management. We fully characterize partially law-invariant coherent risk measures via a novel representation formula. Strong partial law invariance is defined to bridge the gap between the above characterization and the classic representation formula of Kusuoka. We propose a few classes of new risk measures, including partially law-invariant versions of the expected shortfall and the entropic risk measures, and illustrate their applications in risk assessment under different types of uncertainty. We provide a tractable optimization formula for computing a class of partially law-invariant coherent risk measures and give a numerical example. This paper was accepted by Aurélien Baillon, behavioral economics and decision analysis. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141, RGPIN-2020-04356, and RGPIN-2024-03728]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.06518 .
Keywords
law invariance, probabilistic sophistication, ambiguity, coherent risk measures, source theory, expected shortfall
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