Stochastic Optimization and Coupling

TL;DR

This paper explores the properties of optimization problems involving probability measures under integral stochastic orders, establishing equivalences and deriving implications for decision theory and information design.

Contribution

It introduces new equivalences for optimization under stochastic orders and generalizes Blackwell's theorem, linking order properties with applications in decision and information theory.

Findings

Equivalence of four key properties for stochastic order optimization

Characterization of extreme points and exposed points in stochastic dominance

Generalization of Blackwell's theorem for experiment comparison

Abstract

We study optimization problems in which a linear functional is maximized over probability measures that are dominated by a given measure according to an integral stochastic order in an arbitrary dimension. We show that the following four properties are equivalent for any such order: (i) the test function cone is closed under pointwise minimum, (ii) the value function is affine, (iii) the solution correspondence has a convex graph with decomposable extreme points, and (iv) every ordered pair of measures admits an order-preserving coupling. As corollaries, we derive the extreme and exposed point properties involving integral stochastic orders such as multidimensional mean-preserving spreads and stochastic dominance. Applying these results, we generalize Blackwell's theorem by completely characterizing the comparisons of experiments that admit two equivalent descriptions -- through instrumental values and through information technologies. We also show that these results immediately yield new insights into information design, mechanism design, and decision theory.