  An experimenter seeks to learn a subject's preference relation. The
experimenter produces pairs of alternatives. For each pair, the subject is
asked to choose. We argue that, in general, large but finite data do not give
close approximations of the subject's preference, even when the limiting
(countably infinite) data are enough to infer the preference perfectly. We
provide sufficient conditions on the set of alternatives, preferences, and
sequences of pairs so that the observation of finitely many choices allows the
experimenter to learn the subject's preference with arbitrary precision. While
preferences can be identified under our sufficient conditions, we show that it
is harder to identify utility functions. We illustrate our results with several
examples, including consumer choice, expected utility, and preferences in the
Anscombe-Aumann model.
