“Dice”-sion–Making Under Uncertainty: When Can a Random Decision Reduce Risk?
研究了在随机规划和分布鲁棒优化中，随机决策何时优于确定性决策。发现无分布模糊时，若风险度量和可行域均为凸或风险度量是混合拟凹的，则确定性决策最优；否则随机决策可能更优。有分布模糊时，对任何满足连续性的模糊厌恶风险度量，都存在随机决策严格优于所有确定性决策的问题。
Stochastic programming and distributionally robust optimization seek deterministic decisions that optimize a risk measure, possibly in view of the most adverse distribution in an ambiguity 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 some 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, however, 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. This paper was accepted by Teck Ho, optimization.