  We consider a simple model of imprecise comparisons: there exists some
$\delta>0$ such that when a subject is given two elements to compare, if the
values of those elements (as perceived by the subject) differ by at least
$\delta$, then the comparison will be made correctly; when the two elements
have values that are within $\delta$, the outcome of the comparison is
unpredictable. This model is inspired by both imprecision in human judgment of
values and also by bounded but potentially adversarial errors in the outcomes
of sporting tournaments.
  Our model is closely related to a number of models commonly considered in the
psychophysics literature where $\delta$ corresponds to the {\em just noticeable
difference unit (JND)} or {\em difference threshold}. In experimental
psychology, the method of paired comparisons was proposed as a means for
ranking preferences amongst $n$ elements of a human subject. The method
requires performing all $\binom{n}{2}$ comparisons, then sorting elements
according to the number of wins. The large number of comparisons is performed
to counter the potentially faulty decision-making of the human subject, who
acts as an imprecise comparator.
  We show that in our model the method of paired comparisons has optimal
accuracy, minimizing the errors introduced by the imprecise comparisons.
However, it is also wasteful, as it requires all $\binom{n}{2}$. We show that
the same optimal guarantees can be achieved using $4 n^{3/2}$ comparisons, and
we prove the optimality of our method. We then explore the general tradeoff
between the guarantees on the error that can be made and number of comparisons
for the problems of sorting, max-finding, and selection. Our results provide
strong lower bounds and close-to-optimal solutions for each of these problems.
