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 $δ$ satisfies a natural symmetry property, entailing that $δ(σ(\boldsymbol{y})) = σ(δ(\boldsymbol{y}))$ for any permutation $σ$. 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.