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Ranking Statistical Experiments via the Linear Convex Order and the Lorenz Zonoid: Economic Applications

Kailin Chen

TL;DR

The paper introduces the Linear-Blackwell (LB) order as a tractable ranking of statistical experiments, defined by $\boldsymbol{p}_F(\boldsymbol{q}) \succeq_{lcx} \boldsymbol{p}_G(\boldsymbol{q})$ for all priors $\boldsymbol{q}$ and equivalently by Lorenz zonoid inclusion $\mathcal{Z}(F) \supseteq \mathcal{Z}(G)$, plus a posterior-mean variability perspective. It provides three complementary characterizations—posterior dispersion, Lorenz zonoids, and mean-variance type comparisons of posteriors—alongside a geometric view that simplifies verification relative to the Blackwell order. The framework is applied across decision problems under uncertainty, including binary-action and quasi-concave payoffs, as well as moral hazard with mixed actions and limited liability, showing that LB dominance implies higher ex ante value in binary and QCC contexts and lower minimal disutility for implementing target actions. It extends to screening with ex post signals, establishing that LB-dominant experiments expand the set of implementable mechanisms and reduce the principal’s expected loss under broad regularity assumptions. Overall, the LB order delivers a broadly applicable, tractable informativeness criterion that subsumes and extends classical Blackwell comparisons, with concrete implications for incentive design, mechanism selection, and information structure choice in economics.

Abstract

This paper introduces a novel ranking of statistical experiments, the Linear-Blackwell (LB) order, equivalently characterized by (i) more dispersed posteriors and likelihood ratios in the sense of the linear convex order, (ii) a larger Lorenz zonoid--the set of statewise profiles spanned by signals, and (iii) greater variability of the posterior mean. We apply the LB order to compare experiments in binary-action decision problems and in problems with quasiconcave payoffs, as analyzed by Kolotilin, Corrao, and Wolitzky (2025). Furthermore, the LB order enables the comparison of experiments in moral hazard problems, complementing the findings in Holmström (1979) and Kim (1995). Finally, the LB order applies to the comparison of experiments generating ex post signals in screening problems.

Ranking Statistical Experiments via the Linear Convex Order and the Lorenz Zonoid: Economic Applications

TL;DR

The paper introduces the Linear-Blackwell (LB) order as a tractable ranking of statistical experiments, defined by for all priors and equivalently by Lorenz zonoid inclusion , plus a posterior-mean variability perspective. It provides three complementary characterizations—posterior dispersion, Lorenz zonoids, and mean-variance type comparisons of posteriors—alongside a geometric view that simplifies verification relative to the Blackwell order. The framework is applied across decision problems under uncertainty, including binary-action and quasi-concave payoffs, as well as moral hazard with mixed actions and limited liability, showing that LB dominance implies higher ex ante value in binary and QCC contexts and lower minimal disutility for implementing target actions. It extends to screening with ex post signals, establishing that LB-dominant experiments expand the set of implementable mechanisms and reduce the principal’s expected loss under broad regularity assumptions. Overall, the LB order delivers a broadly applicable, tractable informativeness criterion that subsumes and extends classical Blackwell comparisons, with concrete implications for incentive design, mechanism selection, and information structure choice in economics.

Abstract

This paper introduces a novel ranking of statistical experiments, the Linear-Blackwell (LB) order, equivalently characterized by (i) more dispersed posteriors and likelihood ratios in the sense of the linear convex order, (ii) a larger Lorenz zonoid--the set of statewise profiles spanned by signals, and (iii) greater variability of the posterior mean. We apply the LB order to compare experiments in binary-action decision problems and in problems with quasiconcave payoffs, as analyzed by Kolotilin, Corrao, and Wolitzky (2025). Furthermore, the LB order enables the comparison of experiments in moral hazard problems, complementing the findings in Holmström (1979) and Kim (1995). Finally, the LB order applies to the comparison of experiments generating ex post signals in screening problems.

Paper Structure

This paper contains 32 sections, 9 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Preliminaries
  2. Statistical Experiments
  3. Linear-Blackwell Order
  4. Definition
  5. Likelihood Ratio and Posterior Expectation
  6. A Geometric Characterization
  7. Relations to Blackwell Order
  8. Decision under Uncertainty
  9. Decision Problem
  10. Binary Actions
  11. Quasi-concavity
  12. Moral Hazard
  13. Model Setting
  14. Informativeness Criteria
  15. Robustness
  16. ...and 17 more sections

Key Result

Lemma 1

$F\succeq_{LB}G\iff \forall \boldsymbol{b}=(b_0,\ldots,b_n)^{\top}\in\mathbb{R}^{n+1},$

Figures (4)

  • Figure 1: A geometric characterization of the LB order. The left panel illustrates the Lorenz zonoid $\mathcal{Z}(F):=\{\left(\mathbb{E}[h(x)\mid \theta_0],\mathbb{E}[h(x)\mid \theta_1]\right)\mid h:X \to [0,1]\}$ for $\Theta=\{\theta_0,\theta_1\}$. It contains the diagonal line segment (the blue dotted line) and is a subset of the unit cube $[0,1]^{2}$. The right panel depicts the Lorenz zonoids of $F$ and $G$ where $F$ dominates $G$ in the LB order, illustrating that $\mathcal{Z}(F) \supseteq \mathcal{Z}(G)$.
  • Figure 2: A geometric characterization of the LB order. The left panel illustrates $\mathcal{Z}(F)$ for $\Theta=\{\theta_0,\theta_1,\theta_2\}$. It contains the diagonal line segment (the blue dotted line) and is a subset of the unit cube $[0,1]^{3}$. The right panel depicts the Lorenz zonoids of $F$ and $G$ where $F\succeq_{LB}G$, illustrating that $\mathcal{Z}(F) \supseteq \mathcal{Z}(G)$.
  • Figure 3: QCC and non-QCC problems. Let $\Theta=\{\theta_0,\theta_1,\theta_2\}$. A decision problem with $A=\{a_0,a_1,a_2\}$ is characterized by a partition of the belief simplex $\widehat{\Delta}_2$ into three regions according to preferred actions. The dark grey, light grey, and white areas contain beliefs under which the DM prefers $a_0$, $a_1$, and $a_2$, respectively. The left panel illustrates a QCC problem, while the right panel illustrates a non-QCC problem.
  • Figure 4: The set of achievable utility–disutility profiles $\Gamma$ and the effective disutility function $\gamma$. The curve $\gamma$ is the lower boundary of $\operatorname{co}(\Gamma)$.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Lemma 2
  • proof
  • ...and 10 more