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"Dice"-sion Making under Uncertainty: When Can a Random Decision Reduce Risk?

Erick Delage (erick.delage***at***hec.ca)
Daniel Kuhn (daniel.kuhn***at***epfl.ch)
Wolfram Wiesemann (ww***at***imperial.ac.uk)

Abstract: 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 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.

Keywords: Stochastic Programming; Risk Measures; Distributionally Robust Optimization; Ambiguity Aversion; Randomized Decisions

Category 1: Robust Optimization

Category 2: Stochastic Programming


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Entry Submitted: 08/09/2016
Entry Accepted: 08/09/2016
Entry Last Modified: 05/04/2018

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