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Partially Identified Ambiguity

Cheaheon Lim

TL;DR

This paper develops a theory of learning under ambiguity arising from the DM’s beliefs about a partially identified data-generating process. It extends Bayesian concepts to set-valued priors via Aumann plausibility, showing that consistent experiments induce Aumann-plausible posteriors if and only if the prior set is partially identified, and, under maximal partial identification, that Aumann-plausible information structures coincide with consistent experiments. It then extends Blackwell-type results to the PI setting, showing that more informative experiments yield higher maxmin utility when using reduced-form posteriors, and applies these ideas to robust Bayesian analysis with a time-consistent $ ext{$ ext{Gamma}^*$}$-minimax criterion and to partially identified persuasion. The framework yields a natural benchmark for communication under ambiguity and provides tractable methods (via reduction to reduced-form parameters) for solving information-design and decision problems under partial identification, with implications for econometrics, robust Bayesian inference, and strategic communication.

Abstract

This paper develops a theory of learning under ambiguity induced by the decision maker's beliefs about the collection of data correlated with the true state of the world. Within our framework, two classical results on Bayesian learning extend to the setting with ambiguity: experiments are equivalent to distributions over posterior beliefs, and Blackwell's more informative and more valuable orders coincide. When applied to the setting of robust Bayesian analysis, our results clarify the source of time inconsistency in the Gamma-minimax problem and provide an argument in favor of the conditional Gamma-minimax criterion. We also apply our results to a persuasion game to illustrate that our model provides a natural benchmark for communication under ambiguity.

Partially Identified Ambiguity

TL;DR

This paper develops a theory of learning under ambiguity arising from the DM’s beliefs about a partially identified data-generating process. It extends Bayesian concepts to set-valued priors via Aumann plausibility, showing that consistent experiments induce Aumann-plausible posteriors if and only if the prior set is partially identified, and, under maximal partial identification, that Aumann-plausible information structures coincide with consistent experiments. It then extends Blackwell-type results to the PI setting, showing that more informative experiments yield higher maxmin utility when using reduced-form posteriors, and applies these ideas to robust Bayesian analysis with a time-consistent ext{Gamma}^*-minimax criterion and to partially identified persuasion. The framework yields a natural benchmark for communication under ambiguity and provides tractable methods (via reduction to reduced-form parameters) for solving information-design and decision problems under partial identification, with implications for econometrics, robust Bayesian inference, and strategic communication.

Abstract

This paper develops a theory of learning under ambiguity induced by the decision maker's beliefs about the collection of data correlated with the true state of the world. Within our framework, two classical results on Bayesian learning extend to the setting with ambiguity: experiments are equivalent to distributions over posterior beliefs, and Blackwell's more informative and more valuable orders coincide. When applied to the setting of robust Bayesian analysis, our results clarify the source of time inconsistency in the Gamma-minimax problem and provide an argument in favor of the conditional Gamma-minimax criterion. We also apply our results to a persuasion game to illustrate that our model provides a natural benchmark for communication under ambiguity.
Paper Structure (34 sections, 20 theorems, 67 equations, 2 figures)

This paper contains 34 sections, 20 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Partially Identified Prior Sets
  5. Consistent Experiments
  6. Updating Beliefs
  7. Main Equivalence Characterizations
  8. Aumann Plausibility
  9. Consistent Experiments to Aumann-Plausible Information Structures
  10. Aumann-Plausible Information Structures to Consistent Experiments
  11. Value of Information
  12. Preliminaries
  13. Comparing Experiments
  14. Application: Robust Bayesian Analysis
  15. Preliminaries
  16. ...and 19 more sections

Key Result

Lemma 1

Suppose $\mathcal{P} \in \mathcal{K}(\Delta \Theta)$. There exists a non-trivial experiment consistent with $\mathcal{P}$ if and only if $\dim(\mathcal{P}) < \dim(\Delta \Theta)$.

Figures (2)

  • Figure 1: Graphical illustration of the prior and posterior sets in Example \ref{['ex_diagram']}.
  • Figure 2: Plots of $\hat{v}(\tau)$ (black) and its concavification $\overline{V}(\tau)$ (red) for different $g$.

Theorems & Definitions (61)

  • Definition 1: Partially Identified
  • Example 1: Ellsberg Urn
  • Definition 2: Full Ambiguity
  • Example 2: Ellsberg Urn, v2
  • Definition 3: Consistent
  • Remark 1: Consistency and Second-Order Beliefs
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 51 more