We consider the problem of ranking a set of objects based on their performance when the measurement of said performance is subject to noise. In this scenario, the performance is measured repeatedly, resulting in a range of measurements for each object. If the ranges of two objects do not overlap, then we consider one object as 'better' than the other, and we expect it to receive a higher rank; if, however, the ranges overlap, then the objects are incomparable, and we wish them to be assigned the same rank. Unfortunately, the incomparability relation of ranges is in general not transitive; as a consequence, in general the two requirements cannot be satisfied simultaneously, i.e., it is not possible to guarantee both distinct ranks for objects with separated ranges, and same rank for objects with overlapping ranges. This conflict leads to more than one reasonable way to rank a set of objects. In this paper, we explore the ambiguities that arise when ranking with ties, and define a set of reasonable rankings, which we call partial rankings. We develop and analyse three different methodologies to compute a partial ranking. Finally, we show how performance differences among objects can be investigated with the help of partial ranking.

翻译：我们研究在性能测量受噪声影响时对一组对象进行排序的问题。在此场景下，对每个对象的性能进行重复测量，从而获得一系列测量值。若两个对象的测量值范围互不重叠，则认为其中一个对象"优于"另一个，并预期其获得更高排名；然而，若测量范围存在重叠，则这些对象不可比较，我们希望为其分配相同排名。遗憾的是，测量范围的不可比性关系通常不具备传递性；因此，这两个要求通常无法同时满足，即既不能保证测量范围分离的对象获得不同排名，又不能保证测量范围重叠的对象获得相同排名。这种冲突导致存在多种合理的对象排序方式。本文探讨了并列排序中出现的歧义性，定义了一组称为偏序排名的合理排序方式。我们开发并分析了三种计算偏序排名的不同方法。最后，我们展示了如何借助偏序排名来探究对象间的性能差异。