  In a compound decision problem, consisting of $n$ statistically independent
copies of the same problem to be solved under the sum of the individual losses,
any reasonable compound decision rule $\delta$ satisfies a natural symmetry
property, entailing that $\delta(\sigma(\boldsymbol{y})) =
\sigma(\delta(\boldsymbol{y}))$ for any permutation $\sigma$. We derive the
greatest lower bound on the risk of any such decision rule. The classical
problem of estimating the mean of a homoscedastic normal vector is used to
demonstrate the theory, but important extensions are presented as well in the
context of Robbins's original ideas.
