Stochastic Optimization and Coupling

2026-03-11

AI Summary

This study addresses the maximization of linear functionals over probability measures subject to integral stochastic order constraints in arbitrary dimensions. By establishing equivalences among test function cones, value function properties, solution correspondences, and order-preserving couplings, the authors develop a unified theoretical framework. This framework reveals the equivalence of four fundamental properties, extends Blackwell’s theorem, and offers novel insights into information design, mechanism design, and decision theory. Leveraging tools from integral stochastic orders, convex analysis, measure couplings, and extreme point decompositions, the paper fully characterizes the structure of extreme and exposed points under multivariate mean-preserving spreads and stochastic dominance orders, thereby providing two equivalent characterizations of experimental comparison.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Web measurements

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.

Problem

• stochastic optimization
• integral stochastic order
• probability measures
• linear functional
• order-preserving coupling

Innovation

• stochastic optimization
• integral stochastic orders
• order-preserving coupling
• Blackwell's theorem
• information design