  We offer a simple analysis of the problem of choosing a statistical
experiment to optimize the induced distribution of posterior medians, or more
generally $q$-quantiles for any $q \in (0,1)$. We show that all implementable
distributions of the posterior $q$-quantile are implemented by a single
experiment, the $q$-quantile matching experiment, which pools pairs of states
across the $q$-quantile of the prior in a positively assortative manner, with
weight $q$ on the lower state in each pair. A dense subset of implementable
distributions of posterior $q$-quantiles can be uniquely implemented by
perturbing the $q$-quantile matching experiment. A linear functional is
optimized over distributions of posterior $q$-quantiles by taking the optimal
selection from each set of $q$-quantiles induced by the $q$-quantile matching
experiment. The $q$-quantile matching experiment is the only experiment that
simultaneously implements all implementable distributions of the posterior
$q$-quantile.
