Coarse-graining and the Blackwell order

Abstract

Suppose we have a pair of information channels, κ1, κ2, with a common input. The Blackwell order is a partial order over channels that compares κ1 and k2 by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, κ1 is said to be Blackwell-inferior to κ2 if and only if κ1 can be constructed by garbling the output of κ2. A related partial order stipulates that κ2 is more capable than κ1 if the mutual information between the input and output is larger for κ2 than for κ1 for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where κ1 is less capable than κ2 but not Blackwell-inferior. We show that this may even happen when κ1 is constructed by coarse-graining the inputs of κ2. 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.