Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic Optimization and Coupling

Frank Yang, Kai Hao Yang

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.

Stochastic Optimization and Coupling

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.
Paper Structure (46 sections, 24 theorems, 340 equations, 4 figures)

This paper contains 46 sections, 24 theorems, 340 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Abstract Results.
  3. Consistent Comparisons of Experiments.
  4. Nested Optimization and Stackelberg Principals.
  5. Related Literature.
  6. Primitives
  7. Notation.
  8. Integral Stochastic Orders.
  9. Stochastic Optimization.
  10. Nested Stochastic Optimization.
  11. Abstract Results
  12. Consequences under Min-Closure
  13. Exposed Points of Multidimensional MPS and FOSD
  14. Multidimensional MPS.
  15. Multidimensional MPC.
  16. ...and 31 more sections

Key Result

Theorem 1

The following are equivalent:

Figures (4)

  • Figure 1: Trapezoid Graph. $X=\{0,1\}$.
  • Figure 2: An exposed point $\nu$ of $\mathrm{MPS}(\mu)$
  • Figure 3: An exposed point $\nu$ of $\mathrm{LSD}(\mu)$
  • Figure 4: Comparison of envelopes

Theorems & Definitions (26)

  • Theorem 1: Equivalence Theorem
  • Remark 1
  • Remark 2
  • Corollary 1: Pointwise Optimization and Envelope
  • Corollary 2: Extreme Points of Chains
  • Corollary 3: Reduction of Nested Optimization
  • Proposition 1: Extreme and Exposed Points of Stochastic Orbits
  • Proposition 2: Exposed Points of Multidimensional MPS
  • Proposition 3: Exposed Points of Multidimensional FOSD
  • Theorem 2
  • ...and 16 more