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The Extreme Points of Fusions

Andreas Kleiner, Benny Moldovanu, Philipp Strack, Mark Whitmeyer

TL;DR

The paper characterizes the extreme and Lipschitz-exposed points of the multidimensional set of fusions $F_\mu$, linking these points to geometric partitions of the state space via power diagrams. A central result shows that Lipschitz-exposed, finitely supported fusions are precisely those whose per-cell restrictions align with a power-diagram partition, with affinely independent cell-supports ensuring uniqueness under Lipschitz objectives. A complementary result shows a partial, hierarchical relation between general extreme fusions and convex-partitional fusions, and the Monge-Ampère framework underpins certain sufficiency arguments. The authors then connect these structural insights to moment persuasion, offering canonical solutions as partitions plus cell-wise unconstrained problems, and apply the theory to categorization under information-acquisition and memory constraints. Overall, the work provides a geometric/duality-based blueprint for designing optimal information structures and categorization schemes in multidimensional settings, with implications for Bayesian persuasion and decision-making under information-processing constraints.

Abstract

Our work explores fusions, the multidimensional counterparts of mean-preserving contractions and their extreme and exposed points. We reveal an elegant geometric/combinatorial structure for these objects. Of particular note is the connection between Lipschitz-exposed points (measures that are unique optimizers of Lipschitz-continuous objectives) and power diagrams, which are divisions of a space into convex polyhedral ``cells'' according to a weighted proximity criterion. These objects are frequently seen in nature--in cell structures in biological systems, crystal and plant growth patterns, and territorial division in animal habitats--and, as we show, provide the essential structure of Lipschitz-exposed fusions. We apply our results to several questions concerning categorization.

The Extreme Points of Fusions

TL;DR

The paper characterizes the extreme and Lipschitz-exposed points of the multidimensional set of fusions , linking these points to geometric partitions of the state space via power diagrams. A central result shows that Lipschitz-exposed, finitely supported fusions are precisely those whose per-cell restrictions align with a power-diagram partition, with affinely independent cell-supports ensuring uniqueness under Lipschitz objectives. A complementary result shows a partial, hierarchical relation between general extreme fusions and convex-partitional fusions, and the Monge-Ampère framework underpins certain sufficiency arguments. The authors then connect these structural insights to moment persuasion, offering canonical solutions as partitions plus cell-wise unconstrained problems, and apply the theory to categorization under information-acquisition and memory constraints. Overall, the work provides a geometric/duality-based blueprint for designing optimal information structures and categorization schemes in multidimensional settings, with implications for Bayesian persuasion and decision-making under information-processing constraints.

Abstract

Our work explores fusions, the multidimensional counterparts of mean-preserving contractions and their extreme and exposed points. We reveal an elegant geometric/combinatorial structure for these objects. Of particular note is the connection between Lipschitz-exposed points (measures that are unique optimizers of Lipschitz-continuous objectives) and power diagrams, which are divisions of a space into convex polyhedral ``cells'' according to a weighted proximity criterion. These objects are frequently seen in nature--in cell structures in biological systems, crystal and plant growth patterns, and territorial division in animal habitats--and, as we show, provide the essential structure of Lipschitz-exposed fusions. We apply our results to several questions concerning categorization.
Paper Structure (17 sections, 15 theorems, 63 equations, 5 figures)

This paper contains 17 sections, 15 theorems, 63 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Finitely Supported Extreme Points of TEXT
  4. Exposed Fusions
  5. Power Diagrams
  6. The Main Result
  7. The Gap Between Necessity and Sufficiency
  8. Convex partitional fusions and moment persuasion
  9. Moment Persuasion
  10. An Application to Categorization
  11. Decision Problems with Flexible Information Acquisition
  12. Decision Problems with Finite Memory
  13. Decision-Making With Complexity Constraints
  14. Omitted Proofs
  15. Proof of Proposition \ref{['prop:necessity_extreme']}
  16. ...and 2 more sections

Key Result

Proposition 1

Let $X\subseteq \mathbb{R}^{n}$ be compact and convex, and let $\mu$ be an absolutely continuous probability measure on $X$. Suppose that $\nu$ is an extreme point of $F_{\mu }$ that is finitely supported. Then there exists a partition $\mathcal{P}$ of $X$ into convex sets such that, for each $P\in

Figures (5)

  • Figure 1: A power diagram (eppstein2014)
  • Figure 2: A regular (a) and irregular (b) subdivision (lee_santos_subdivisions_2017).
  • Figure 3: The lower faces of a polyhedron in $\mathbb{R}^3$ and the corresponding polyhedral subdivision.
  • Figure 4: Example 1
  • Figure 5: Example 2

Theorems & Definitions (40)

  • Definition 1.1
  • Proposition 1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 2
  • proof : Proof of Proposition \ref{['prop:characterization_strongly_exposed']}
  • Lemma 1
  • proof
  • Corollary 1
  • ...and 30 more