In a bipartite graph $G=(U\cup V,E)$ where $E\subseteq U\times V$, a semi-matching is defined as a set of edges $M\subseteq E$, such that each vertex in U is incident with exactly one edge in M. Many previous works focus on the problem of finding the fairest semi-matchings: ones that assign U-vertices with V-vertices as fairly as possible. In these works, fairness is usually measured according to a specific index. In fact, there exist various fairness measures, and they often disagree on the fairness comparison of some semi-matching pairs. In this paper, we prove that there always exists one (or a set of equally fair) semi-matching(s), universally agreed by all the existing fairness measures, to be the fairest among all the semi-matchings of a given bipartite graph. In other words, given that fairness measures disagree on many comparisons between semi-matchings, they nonetheless are all in agreement on the (set of) fairest semi-matching(s) for a given bipartite graph. To prove this, we propose a partially ordered relationship (named Transfer-based Comparison) among the semi-matchings, showing that the greatest elements always exist in such a partially ordered set. We then show that such greatest elements can guarantee to be the fairest ones under the fairness measure of Majorization [1]. This further indicates that such fairest semi-matchings are agreed by all the fairness measures which are compatible with Majorization. To the best knowledge of us, this is true for all existing fairness measures.

在二部图中 $G=（ü\cup V，Ë）$ 哪里 $Ë\subseteq ü\times V$，半匹配定义为一组边$\mathrm{中号}\subseteq Ë$，使得U中的每个顶点都恰好与M中的一个边入射。先前的许多工作着重于寻找最公平的半匹配的问题：将U顶点分配给V的半匹配-vertices尽可能公平。在这些作品中，公平通常是根据特定指标来衡量的。实际上，存在各种公平性度量，并且它们在某些半匹配对的公平性比较上经常存在分歧。在本文中，我们证明了在所有二分图的所有半匹配中，总是存在一个（或一组同样公平的）半匹配，并且所有现有的公平性度量都普遍认为这是最公平的。 。换句话说，鉴于公平性度量在半匹配之间的许多比较中均存在分歧，因此，对于给定的二分图，它们都在（一组）最公平的半匹配上是一致的。为了证明这一点，我们提出了半匹配之间的部分有序关系（称为基于传输的比较），表明最大的元素始终存在于这样的部分有序集合中。然后，我们证明在专业化[1]的公平性度量下，这样的最大要素可以保证是最公平的要素。这进一步表明，最公平的半匹配被所有与Majorization兼容的公平性措施所同意。就我们所知，所有现有的公平措施都是如此。