  We investigate the problem of approximating an incomplete preference relation
$\succsim$ on a finite set by a complete preference relation. We aim to obtain
this approximation in such a way that the choices on the basis of two
preferences, one incomplete, the other complete, have the smallest possible
discrepancy in the aggregate. To this end, we use the top-difference metric on
preferences, and define a best complete approximation of $\succsim$ as a
complete preference relation nearest to $\succsim$ relative to this metric. We
prove that such an approximation must be a maximal completion of $\succsim$,
and that it is, in fact, any one completion of $\succsim$ with the largest
index. Finally, we use these results to provide a sufficient condition for the
best complete approximation of a preference to be its canonical completion.
This leads to closed-form solutions to the best approximation problem in the
case of several incomplete preference relations of interest.
