Stochastic programming and distributionally robust optimization seek
deterministic
deci-
sions that optimize a risk measure, possibly in view of the most adverse distribution in an am-
biguity set. We investigate under which circumstances such deterministic decisions are strictly
outperformed by
random
decisions which depend on a randomization device producing uniformly
distributed samples that are independent of all uncertain factors affecting the decision problem.
We find that in the absence of distributional ambiguity, deterministic decisions are optimal if
both the risk measure and the feasible region are convex, or alternatively if the risk measure
is mixture-quasiconcave. We show that several risk measures, such as mean (semi-)deviation
and mean (semi-)moment measures, fail to be mixture-quasiconcave and can therefore give rise
to problems in which the decision maker benefits from randomization. Under distributional
ambiguity, on the other hand, we show that for any ambiguity averse risk measure satisfying a
mild continuity property we can construct a decision problem in which a randomized decision
strictly outperforms all deterministic decisions.