$p$-orderings: From Slater to Kemeny-Young to Ranked Pairs

We introduce a family of ranking rules for preferential elections, called $p$-orderings, obtained by minimizing the $p$-norm of the pairwise majority margins that disagree with a given ranking.

This family is defined on the margin-of-victory matrix of the election and has the Slater orderings as its limit as $p \to 0^+$, includes the Kemeny-Young rule as the case $p=1$, and coincides with Ranked Pairs for all sufficiently large $p$.

We show that, under natural assumptions of scale invariance, dependence only on margin magnitude, and monotonicity with respect to margin size, the score function underlying this construction is uniquely of the form $c|x|^p$.

Thus Ranked Pairs arises as the eventual large-$p$ member of a canonical family of margin-based ranking rules.