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.

翻译：本文研究了在有限集上使用完全偏好关系近似不完全偏好关系$\succsim$的问题。我们的目标是使得基于这两种偏好（一种不完全、另一种完全）的选择在总体上具有尽可能小的差异。为此，我们采用偏好上的顶部差度量，并定义$\succsim$的最佳完全近似为在该度量下距离$\succsim$最近的完全偏好关系。我们证明此类近似必须是$\succsim$的极大完全化，且实际上是任何具有最大索引的$\succsim$的完全化。最后，我们利用这些结果给出偏好最佳完全近似是其典范完全化的充分条件。这为若干感兴趣的不完全偏好关系情况下的最佳近似问题提供了闭式解。