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The Geometry of Heterogeneous Extremes: Optimal Transport and Entropic Design

I. Sebastian Buhai

Abstract

Extreme economic outcomes are not shaped by tails alone. They are also shaped by unequal access to opportunities. This paper develops a theory of heterogeneous extremes by taking the distribution of opportunity access as the object of study. In a mixed Poisson search setting, normalized maxima admit a Laplace mixture representation that yields order comparisons and a clean benchmark against the homogeneous economy. The main contribution is geometric: a canonical coupling turns differences in heterogeneity into optimal transport bounds for the whole induced law of extremes, the full schedule of top quantiles, and structured counterfactual paths between economies. The paper also derives a second order expansion that separates classical extreme value approximation error from heterogeneity effects. As a complementary normative exercise, it studies an entropy regularized design problem for reallocating opportunities under a mean constraint. A stylized labor market network application interprets heterogeneity as unequal access to job opportunities and shows how the framework can be used for tail counterfactuals and robustness analysis of top wage distributions.

The Geometry of Heterogeneous Extremes: Optimal Transport and Entropic Design

Abstract

Extreme economic outcomes are not shaped by tails alone. They are also shaped by unequal access to opportunities. This paper develops a theory of heterogeneous extremes by taking the distribution of opportunity access as the object of study. In a mixed Poisson search setting, normalized maxima admit a Laplace mixture representation that yields order comparisons and a clean benchmark against the homogeneous economy. The main contribution is geometric: a canonical coupling turns differences in heterogeneity into optimal transport bounds for the whole induced law of extremes, the full schedule of top quantiles, and structured counterfactual paths between economies. The paper also derives a second order expansion that separates classical extreme value approximation error from heterogeneity effects. As a complementary normative exercise, it studies an entropy regularized design problem for reallocating opportunities under a mean constraint. A stylized labor market network application interprets heterogeneity as unequal access to job opportunities and shows how the framework can be used for tail counterfactuals and robustness analysis of top wage distributions.
Paper Structure (86 sections, 37 theorems, 229 equations, 1 table)

This paper contains 86 sections, 37 theorems, 229 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and overall contribution
  3. Main results
  4. A structural decomposition of heterogeneous extremes.
  5. Wasserstein geometry and quantitative robustness.
  6. A complementary entropy-regularized design problem on the same type space.
  7. Application to labor market networks.
  8. Empirical interface and identification of heterogeneity
  9. Related literature
  10. Heterogeneous extreme value theory.
  11. Optimal transport, coupling, and entropy projection.
  12. Labor market networks.
  13. Roadmap and regime map
  14. Environment: heterogeneous extreme value primitives
  15. Baseline extreme value theory
  16. ...and 71 more sections

Key Result

Lemma 1

Under Assumption ass:mixedPoisson, for any $y\in[0,1]$ and any $\theta>0$,

Theorems & Definitions (94)

  • Remark 1: Why we exclude an atom at zero
  • Lemma 1: Probability generating function of mixed Poisson
  • Remark 2: Search technology versus type heterogeneity
  • Proposition 1: Heterogeneous extreme value law
  • Remark 3: Formal scope of the limit law
  • Remark 4: Connection to random-sample-size maxima
  • Definition 1: Convex order
  • Proposition 2: Convex order implies Laplace order
  • Corollary 1: Heterogeneity lowers extremes relative to the homogeneous benchmark
  • Remark 5: Tail index versus level effects
  • ...and 84 more