  We consider the compound decision problem of estimating a vector of $n$
parameters, known up to a permutation, corresponding to $n$ independent
observations, and discuss the difference between two symmetric classes of
estimators. The first and larger class is restricted to the set of all
permutation invariant estimators. The second class is restricted further to
simple symmetric procedures. That is, estimators such that each parameter is
estimated by a function of the corresponding observation alone. We show that
under mild conditions, the minimal total squared error risks over these two
classes are asymptotically equivalent up to essentially O(1) difference.
