# Coarse-graining and the Blackwell order

## TL;DR

This paper investigates the relationship between coarse-graining of information channels and the Blackwell order, revealing that coarse-graining can sometimes improve expected utility, contrary to traditional intuition that it only reduces information.

## Contribution

It demonstrates that coarse-graining of channels can lead to higher expected utility, challenging the assumption that it always diminishes information and utility.

## Key findings

- Coarse-graining can increase expected utility of channels.
- Blackwell-inferiority does not always imply less capability.
- Counterexamples show coarse-graining can improve utility.

## Abstract

Suppose we have a pair of information channels, $\kappa_{1},\kappa_{2}$, with a common input. The Blackwell order is a partial order over channels that compares $\kappa_{1}$ and $\kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, $\kappa_{1}$ is said to be Blackwell-inferior to $\kappa_{2}$ if and only if $\kappa_{1}$ can be constructed by garbling the output of $\kappa_{2}$. A related partial order stipulates that $\kappa_{2}$ is more capable than $\kappa_{1}$ if the mutual information between the input and output is larger for $\kappa_{2}$ than for $\kappa_{1}$ for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where $\kappa_{1}$ is less capable than $\kappa_{2}$ but not Blackwell-inferior. We show that this may even happen when $\kappa_{1}$ is constructed by coarse-graining the inputs of $\kappa_{2}$. Such a coarse-graining is a special kind of "pre-garbling" of the channel inputs. This example directly establishes that the expected value of the shared utility function for the coarse-grained channel is larger than it is for the non-coarse-grained channel. This contradicts the intuition that coarse-graining can only destroy information and lead to inferior channels. We also discuss our results in the context of information decompositions.