  For better or for worse, rankings of institutions, such as universities,
schools and hospitals, play an important role today in conveying information
about relative performance. They inform policy decisions and budgets, and are
often reported in the media. While overall rankings can vary markedly over
relatively short time periods, it is not unusual to find that the ranks of a
small number of "highly performing" institutions remain fixed, even when the
data on which the rankings are based are extensively revised, and even when a
large number of new institutions are added to the competition. In the present
paper, we endeavor to model this phenomenon. In particular, we interpret as a
random variable the value of the attribute on which the ranking should ideally
be based. More precisely, if $p$ items are to be ranked then the true, but
unobserved, attributes are taken to be values of $p$ independent and
identically distributed variates. However, each attribute value is observed
only with noise, and via a sample of size roughly equal to $n$, say. These
noisy approximations to the true attributes are the quantities that are
actually ranked. We show that, if the distribution of the true attributes is
light-tailed (e.g., normal or exponential) then the number of institutions
whose ranking is correct, even after recalculation using new data and even
after many new institutions are added, is essentially fixed. Formally, $p$ is
taken to be of order $n^C$ for any fixed $C>0$, and the number of institutions
whose ranking is reliable depends very little on $p$.
