Optimal pairs of score vectors for positional scoring rules

Letw=(w1,⋯,wm) andv=(v1,⋯,vm-1) be nonincreasing real vectors withw1>wm andv1>vm-1. With respect to a lista1,⋯,an of linear orders on a setA ofm⩾3 elements, thew-score ofa∈A is the sum overi from 1 tom ofwi 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 everyaj.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 ordersaj onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityPm(w, v) forn→∞ that the element inA with the greatestw-score also has the greatestv-score inA∖{b}.It is shown thatPm(m,v) takes on its maximum value if and only if bothw andv are linear, so thatwi−wi+1=wi+1−wi+2 fori⩽m−2, andvi−vi+1=vi+1−vi+2 fori⩽m−3. This general result for allm⩾3 supplements related results for linear score vectors obtained previously form∈{3,4}.