Robustness of positional scoring over subsets of alternatives

Positional score vectorsw=(w ₁,,w _m ) for anm-element setA, andv=(v ₁,,v _k ) for ak-element proper subsetB ofA, agree at a profiles of linear orders onA when the restriction toB of the ranking overA produced byw operating ons equals the ranking overB produced byv operating on the restriction ofs toB. Givenw ₁>w _mandv ₁>v _k , this paper examines the extent to which pairs of nonincreasing score vectors agree over sets of profiles. It focuses on agreement ratios as the number of terms in the profiles becomes infinite. The limiting agreement ratios that are considered for (m, k) in {(3,2),(4,2),(4,3)} are uniquely maximized by pairs of Borda (linear, equally-spaced) score vectors and are minimized when (w,v) is either ((1,0,,0),(1,,1,0)) or ((1,,,1,0),(1,0,,0)).This research was supported by the National Science Foundation, Grants SOC 75-00941 and SOC 77-22941.