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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.