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Best Complete Approximations of Preference Relations

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  • Hiroki Nishimura
  • Efe A. Ok

Abstract

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.

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  • Hiroki Nishimura & Efe A. Ok, 2023. "Best Complete Approximations of Preference Relations," Papers 2311.06641, arXiv.org.
    • Handle: RePEc:arx:papers:2311.06641
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