Characterizations of voting rules based on majority margins
Yifeng Ding, Wesley H. Holliday, Eric Pacuit
TL;DR
This work axiomatizes margin-based (pairwise) voting rules by linking the margin-based invariant property to intuitive normative principles. On the domain of linear profiles, it proves that margin-based rules are exactly those satisfying Preferential Equality and Neutral Reversal, using a McGarvey-style construction to realize margin graphs. Extending to broader domains, the authors show that, under homogeneity, margin-basedness on all profiles is equivalent to Preferential Equality plus Tiebreaking Compensation and Neutral Reversal, with variants for head-to-head rules characterized by Nonlinear Neutral Reversal and Neutral Indifference. The results clarify the normative content of margin-based invariance and illuminate how Condorcet-type rules behave under incomplete rankings or ties, offering a framework for designing robust voting rules with predictable head-to-head implications.
Abstract
In the context of voting with ranked ballots, an important class of voting rules is the class of margin-based rules (also called pairwise rules). A voting rule is margin-based if whenever two elections generate the same head-to-head margins of victory or loss between candidates, then the voting rule yields the same outcome in both elections. Although this is a mathematically natural invariance property to consider, whether it should be regarded as a normative axiom on voting rules is less clear. In this paper, we address this question for voting rules with any kind of output, whether a set of candidates, a ranking, a probability distribution, etc. We prove that a voting rule is margin-based if and only if it satisfies some axioms with clearer normative content. A key axiom is what we call Preferential Equality, stating that if two voters both rank a candidate $x$ immediately above a candidate $y$, then either voter switching to rank $y$ immediately above $x$ will have the same effect on the election outcome as if the other voter made the switch, so each voter's preference for $y$ over $x$ is treated equally.