Jensen gaps and the decomposition principle

Abstract

We provide axiomatic characterizations of Jensen-gap functionals, which have the form $ρ(X)=\mathbb{E}[f(X)]-f(\mathbb{E}[X])$ for some real-valued function $f$ called the generator. This class includes important examples such as the excess growth rate, the utility premium, the Kullback-Leibler (KL) divergence, and the variance, and is also closely related to measures of variability. We analyze various forms of convexity for Jensen gaps. To characterize Jensen gaps, we formulate conditional versions of law-invariant functionals, which are needed to state the decomposition principle. On an atomless probability space, we show that law invariance together with a binary decomposition principle is sufficient to characterize Jensen gaps on simple random variables. Under suitable continuity conditions, the characterization extends to broader classes, including compactly supported random variables and integrable random variables. We then identify the generator $f$ by imposing either of two natural invariance conditions: scaling invariance or translation invariance. The former identifies logarithmic, $x\log x$, and power generators, whereas the latter identifies exponential and quadratic generators.