标题： 最佳偏好关系的完全逼近 显示英文标题

标题： Best Complete Approximations of Preference Relations

摘要： 我们研究在有限集上通过一个完整的偏好关系来近似一个不完整的偏好关系$\succsim$的问题。 我们的目标是以这样的方式获得这种近似，使得基于两种偏好（一种不完整，另一种完整）的选择在总体上具有最小的差异。 为此，我们使用偏好上的顶差度量，并将$\succsim$的最佳完整近似定义为相对于此度量最接近$\succsim$的完整偏好关系。 我们证明，这样的近似必须是$\succsim$的极大完成，并且事实上，它是$\succsim$的具有最大索引的任一完成。 最后，我们利用这些结果为偏好的最佳完整近似是其规范完成提供了一个充分条件。 这导致了在几种感兴趣的不完整偏好关系情况下最佳近似问题的显式解。 显示英文摘要

摘要： 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.