Coarse-Graining and the Blackwell Order

Abstract

Suppose we have a pair of information channels, κ 1 , κ 2 , with a common input. The Blackwell order is a partial order over channels that compares κ 1 and κ 2 by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, κ 1 is said to be Blackwell-inferior to κ 2 if and only if κ 1 can be constructed by garbling the output of κ 2 . A related partial order stipulates that κ 2 is more capable than κ 1 if the mutual information between the input and output is larger for κ 2 than for κ 1 for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where κ 1 is less capable than κ 2 but not Blackwell-inferior. We show that this may even happen when κ 1 is constructed by coarse-graining the inputs of κ 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.

Keywords

• Channel preorders
• Blackwell order
• degradation order
• garbling
• more capable
• coarse-graining