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On the Monotonicity of Information Costs

Xiaoyu Cheng, Yonggyun Kim

Abstract

We study the monotonicity of information costs: more informative experiments must be more costly. As criteria for informativeness, we consider the standard information orders introduced by Blackwell (1951, 1953) and Lehmann (1988). We provide simple necessary and sufficient conditions for a cost function to be monotone with respect to each order, grounded in their garbling characterizations. Finally, we examine several well-known cost functions from the literature through the lens of these conditions.

On the Monotonicity of Information Costs

Abstract

We study the monotonicity of information costs: more informative experiments must be more costly. As criteria for informativeness, we consider the standard information orders introduced by Blackwell (1951, 1953) and Lehmann (1988). We provide simple necessary and sufficient conditions for a cost function to be monotone with respect to each order, grounded in their garbling characterizations. Finally, we examine several well-known cost functions from the literature through the lens of these conditions.
Paper Structure (57 sections, 22 theorems, 100 equations, 5 figures)

This paper contains 57 sections, 22 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Binary Experiments
  4. Blackwell Monotonicity
  5. Preliminaries
  6. Blackwell Information Order
  7. Information Costs
  8. Blackwell Monotonicity
  9. Parallelogram Hull
  10. The Characterization
  11. Permutation Invariance
  12. Decreasing in Signal Replacement
  13. Proof Sketch for Sufficiency
  14. Further Characterizations with Binary States
  15. Lehmann Monotonicity
  16. ...and 42 more sections

Key Result

Lemma 1

For any $f,g \in \mathcal{E}_2$, $f \succeq_{B} g$ if and only if $g$ lies in the parallelogram hull of $f$ and $\mathbf{1}-f$, defined as In other words, $S_{B}(f) = \text{PARL}(f, \mathbf{1}-f)$.This geometric characterization of the Blackwell order coincides with the zonotope order in bertschinger2014blackwell, which is further applied in deOliveira2024robust

Figures (5)

  • Figure 1: A Graphical Illustration with Binary States
  • Figure 2: Geometry of MLRP experiments and the Lehmann order
  • Figure 3: Shrinking Operations
  • Figure 4: The graphs of $\tilde{F}^{-1} ( \tilde{G} (y|\omega_i) |\omega_i )$
  • Figure 5: Operations of shrinking $\mathbf{S}_i(f)$

Theorems & Definitions (53)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Definition 2
  • Theorem 2
  • Remark 1
  • Definition 3
  • ...and 43 more