  We introduce the concept of partial law invariance, generalizing the concepts
of law invariance and probabilistic sophistication widely used in decision
theory, as well as statistical and financial applications. This new concept is
motivated by practical considerations of decision making under uncertainty,
thus connecting the literature on decision theory and that on 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.
