Ulric B. and Evelyn L. Bray Social Sciences Seminar

Abstract: We investigate the problem of approximating an incomplete preference relation % 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 % as a complete preference relation nearest to % relative to this metric. We prove that such an approximation must be a maximal completion of %, and that it is, in fact, any one completion of % 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.