  There is an innate human tendency, one might call it the "league table
mentality," to construct rankings. Schools, hospitals, sports teams, movies,
and myriad other objects are ranked even though their inherent
multi-dimensionality would suggest that -- at best -- only partial orderings
were possible. We consider a large class of elementary ranking problems in
which we observe noisy, scalar measurements of merit for $n$ objects of
potentially heterogeneous precision and are asked to select a group of the
objects that are "most meritorious." The problem is naturally formulated in the
compound decision framework of Robbins's (1956) empirical Bayes theory, but it
also exhibits close connections to the recent literature on multiple testing.
The nonparametric maximum likelihood estimator for mixture models (Kiefer and
Wolfowitz (1956)) is employed to construct optimal ranking and selection rules.
Performance of the rules is evaluated in simulations and an application to
ranking U.S kidney dialysis centers.
