  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.
