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Weighted Garbling

Daehyun Kim, Ichiro Obara

TL;DR

An information order on experiments based on weighted garbling is introduced, a generalization of the standard notion of garbling that is equivalent to ordinary garbling conditional on a payoff-irrelevant event and characterized in terms of induced posterior belief distributions, showing that it depends only on their support.

Abstract

We introduce an information order on experiments based on weighted garbling, a generalization of the standard notion of garbling. In this order, an experiment is more informative than another if the latter is a weighted garbling of the former. We show that this is equivalent to ordinary garbling conditional on a payoff-irrelevant event. We also characterize the order in terms of induced posterior belief distributions, showing that it depends only on their support. Our main results provide two decision-theoretic characterizations of this order. First, in static decision problems, one experiment dominates another if and only if its value of information is at least a fixed fraction of the other's across all problems. Second, in a class of stopping time problems with a hidden Markov process and repeated experimentation, one experiment dominates another if and only if it yields weakly higher expected payoffs for every problem with a regular prior.

Weighted Garbling

TL;DR

An information order on experiments based on weighted garbling is introduced, a generalization of the standard notion of garbling that is equivalent to ordinary garbling conditional on a payoff-irrelevant event and characterized in terms of induced posterior belief distributions, showing that it depends only on their support.

Abstract

We introduce an information order on experiments based on weighted garbling, a generalization of the standard notion of garbling. In this order, an experiment is more informative than another if the latter is a weighted garbling of the former. We show that this is equivalent to ordinary garbling conditional on a payoff-irrelevant event. We also characterize the order in terms of induced posterior belief distributions, showing that it depends only on their support. Our main results provide two decision-theoretic characterizations of this order. First, in static decision problems, one experiment dominates another if and only if its value of information is at least a fixed fraction of the other's across all problems. Second, in a class of stopping time problems with a hidden Markov process and repeated experimentation, one experiment dominates another if and only if it yields weakly higher expected payoffs for every problem with a regular prior.

Paper Structure

This paper contains 19 sections, 18 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weighted Garbling
  3. Decision Problems and Experiments
  4. Weighted Garbling
  5. Transitivity of Weighted Garbling Order
  6. Belief-Based Characterization of Weighted-Garbling Order
  7. Weighted-Garbling Order and Payoff Guarantee
  8. Conditional Informativeness
  9. Weighted Garbling as Payoff Guarantee
  10. Weighted-Garbling Order and Optimal Stopping Problems with Hidden Markov Processes
  11. Discussion: Weighted-Garbling Order and Other Extensions of Blackwell Order
  12. Conclusion
  13. Appendix
  14. Proof of \ref{['prop:n2']}
  15. Proof of \ref{['thm:1']}
  16. ...and 4 more sections

Key Result

Proposition 1

If $\Pi$ is a weighted garbling of $\Pi'$ with weight $\gamma_1$ and $\Pi'$ is a weighted garbling of $\Pi"$ with weight $\gamma_2$, then $\Pi$ is a weighted garbling of $\Pi"$ with some weight $\gamma$ such that $\overline{\gamma} \leq \overline{\gamma}_1 \overline{\gamma}_2$.

Figures (2)

  • Figure 1: Experiments in Example 1. $q' > q$
  • Figure 3: Posterior beliefs from one-shot and twice-repeated experiments. The blue points are the (extreme) posterior beliefs associated with $\Pi'$, whereas the red points are those associated with $\Pi^\epsilon$ with $\epsilon = \frac{1}{100}$. In particular, $q_\epsilon(s_2) = \left( \frac{49}{198}, \frac{149}{396}, \frac{149}{396} \right).$

Theorems & Definitions (39)

  • Definition 1: Weighted Garbling and Size
  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Definition 2: Conditional Informativeness
  • ...and 29 more