Concepts and Applications of Stochastically Weighted Stochastic Dominance

Stochastic dominance theory provides tools to compare random entities. When comparing random vectors (say X and Y ), the problem can be viewed as one of multi-criterion decision making under uncertainty. One approach is to compare weighted sums of the components of these random vectors using univariate dominance. In this paper we propose new concepts of stochastically weighted dominance. The main idea is to treat the vector of weights as a random vector V . We show that such an approach is much less restrictive than the deterministic weighted approach. We further show that the proposed new concepts of stochastic dominance are representable by a finite number of (mixed-integer) linear inequalities when the distributions of X, Y and V have finite support. We discuss two applications to illustrate the usefulness of the stochastically weighted dominance concept. The first application discusses the effect of this notion on the feasibility regions of optimization problems. The second application presents a multi-criterion staffing problem where the goal is to decide the allocation of servers between two M/M/c queues based on waiting times. The latter example illustrates the use of stochastically weighted dominance concept for a ranking of the alternatives.