A Theoretical Foundation of Ambiguity Measurement: A Reply

Yehuda Izhakian

Abstract

Fu, Melenberg, and Schweizer (FMS, 2023) revisits the ambiguity measure, mho-square, introduced in Izhakian (2020), and its underlying decision-theoretic models. This reply highlights four incompatibilities of FMS's setting with the models it aims to revisit, which invalidate its conclusions.

(i) FMS replaces acts with lotteries, which is invalid when the preference relation is not probabilistically sophisticated; the preference relation in Izhakian (2020) is not probabilistically sophisticated.

(ii) FMS violates the independence between outcome ranking and event ranking, which is crucial in the axiomatic foundation of the underlying decision-theoretic models.

(iii) FMS treats nonadditive probabilities (capacities) as if they were additive, which is a fundamental inconsistency with the functional representation of preferences in the underlying decision-theoretic models.

(iv) FMS relies upon an equivalence between first-order stochastic dominance (FOSD) and higher expected utility to prove its main claim. There is no equivalence between conventional FOSD and higher expected utility when probabilities are nonadditive.

The alternative measure FMS proposes does not reflect the ordering of any known preference relation and has a major flaw---sensitivity to the ordering of events by their associated outcomes.