Optimal pairs of score vectors for positional scoring rules

Abstract

Letw=(w ₁,⋯,w _m ) andv=(v ₁,⋯,v _m-1 ) be nonincreasing real vectors withw ₁>w _m andv ₁>v _m-1 . With respect to a lista ₁,⋯,a _n of linear orders on a setA ofm⩾3 elements, thew-score ofa∈A is the sum overi from 1 tom ofw _i times the number of orders in the list that ranka inith place; thev-score ofa∈A∖{b} is defined in a similar manner after a designated elementb is removed from everya _j .

We are concerned with pairs (w, v) which maximize the probability that ana∈A with the greatestw-score also has the greatestv-score inA∖{b} whenb is randomly selected fromA∖{a}. Our model assumes that linear ordersa _j onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityP _m (w, v) forn→∞ that the element inA with the greatestw-score also has the greatestv-score inA∖{b}.

It is shown thatP _m (m,v) takes on its maximum value if and only if bothw andv are linear, so thatw _i −w _i+1=w _i+1−w _i+2 fori⩽m−2, andv _i −v _i+1 =v _i+1 −v _i+2 fori⩽m−3. This general result for allm⩾3 supplements related results for linear score vectors obtained previously form∈{3,4}.