Inverted orders for monotone scoring rules

AbstractWhen each of n judges ranks a set A of m objects from best to worst, and s=(s1,…,sm) is a decreasing sequence of real numbers, the collective ranking determined by s orders the objects in A according to their total scores. The total score of x equals sp times the number of judges who rank x in pth place, summed over p.For normalization purposes, let Sm denote the set of all decreasing s=(s1,…,sm) for which sm − 1=1 and sm=0. Given any m ⩾ 3, we show firstthat if s and s′ in Sm are not identical, then some profile of judges' rankings yields a linear collective order for s′ that is the reverse or dual of the linear collective order for s.We then consider reversals in collective rankings when one object is removed from A. Suppose s is in Sm and t is in Sm − 1, with m ≥ 3. A simple constructive proof shows that there is a profile of judges' rankings on A which yields a collective linear order for s such that, when any pre-specified object in A is removed, t yields the reverse ranking on the remaining m − 1 objects. More detailed results are derived for m=3, and shown to depend on the nature of s=(s1, 1, 0). In particular, the sum-of-ranks procedure with s1=2 permits fewer reversals than any other s1>1