Characterizations of voting rules based on majority margins

SSRN Electronic Journal, 2000

The concept of distance rationalizability has several applications within social choice. In the context of voting, it allows one to define ("rationalize") voting rules via a consensus class (roughly, a set of elections in which it is obvious who should win) and a distance function: namely, a candidate is said to be an election winner if it is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be represented in this manner. In this paper, we provide new results on distance rationalizability of several well-known voting rules such as all scoring rules, Approval, Young's rule and Maximin. We also show that a previously published proof of distance rationalizability of Young's rule is incorrect: the consensus notion and the distance function used in that proof give rise to a voting rule that is similar to-but distinct from-the Young's rule. Finally, we demonstrate that some voting rules cannot be rationalized via certain notions of consensus. To the best of our knowledge, these are the first non-distance-rationalizability results for voting rules.

2011

Distance rationalizability is a framework for classifying voting rules by interpreting them in terms of distances and consensus classes. It also allows to design new voting rules with desired properties. A particularly natural and versatile class of distances that can be used for this purpose is that of votewise distances [12], which "lift" distances over individual votes to distances over entire elections using a suitable norm. In this paper, we continue the investigation of the properties of votewise distance-rationalizable rules initiated in . We describe a number of general conditions on distances and consensus classes that ensure that the resulting voting rule is homogeneous or monotone. This complements the results of , where the authors focus on anonymity, neutrality and consistency. We also introduce a new class of voting rules, that can be viewed as "majority variants" of classic scoring rules, and have a natural interpretation in the context of distance rationalizability.

Social Choice and Welfare, 2025

May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May’s axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three- alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.

Annals of the New York Academy of Sciences, 1990

Proponents of Condorcet voting face the question of what to do in the rare case when no Condorcet winner exists. Recent work provides compelling arguments for the rule that should be applied in three-candidate elections, but already with four candidates, many rules appear reasonable. In this paper, we consider a recent proposal of a simple Condorcet voting method for Final Four political elections. Our question is what normative principles could support this simple form of Condorcet voting. When there is no Condorcet winner, one natural principle is to pick the candidate who is closest to being a Condorcet winner. Yet there are multiple plausible ways to define closeness, leading to different results. Here we take the following approach: identify a relatively uncontroversial sufficient condition for one candidate to be closer than another to being a Condorcet winner; then use other principles to help settle who wins in cases when that condition alone does not. We prove that our principles uniquely characterize the simple Condorcet voting method for Final Four elections. This analysis also points to a new way of extending the method to elections with five or more candidates that is simpler than an extension previously considered. The new proposal is to elect the candidate with the most head-to-head wins, and if multiple candidates tie for the most wins, then elect the one who has the smallest head-to-head loss. We provide additional principles sufficient to characterize this simple method for Final Five elections.

2019

This dissertation focuses on voting with partial preferences in the form of top-k ballots. We study ways to minimize the amount of communication required to approximate or to output the correct winner from truncated ballots with respect to different voting rules and we propose and analyze different methods allowing a compromise between the accuracy of the result and the amount of communication required; some require only one round of communication, while others are interactive. First, we assume that voters give, in a single shot, their top-k alternatives (for a fixed k); we define a version of voting rules (Borda, Harmonic, Copeland, Maximin, ranked pairs and STV) that works for such k-truncated votes. We consider two measures of the quality of the approximation: the probability of selecting the same winner as the original rule, and the score ratio. We do a worst-case study (for the latter measure only), and for both measures, an average-case study and a study from real data sets. A...

2013

Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule's behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi-)coloring of the election graph - the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its "monochromatic components", i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. ...

arXiv (Cornell University), 2024

We study the setting of single-winner elections with ordinal preferences where candidates might be members of alliances (which may correspond to e.g., political parties, factions, or coalitions). However, we do not assume that candidates from the same alliance are necessarily adjacent in voters' rankings. In such case, every classical voting rule is vulnerable to the spoiler effect, i.e., the presence of a candidate may harm his or her alliance. We therefore introduce a new idea of alliance-aware voting rules which extend the classical ones. We show that our approach is superior both to using classical cloneproof voting rules and to running primaries within alliances before the election. We introduce several alliance-aware voting rules and show that they satisfy the most desirable standard properties of their classical counterparts as well as newly introduced axioms for the model with alliances which, e.g., exclude the possibility of the spoiler effect. Our rules have natural definitions and are simple enough to explain to be used in practice.

SIAM Journal on Applied Mathematics, 1978

Voting rules on many alternatives may be broadly divided into two classes: those that use a scheme of "weighting" the alternatives to determine their overall order of desirability, and those that use binary comparison t o ascertain whether there is an alternative (called a Condorcet alternative) that is able t o defeat every other alternative by a simple majority. The first approach is identified with Borda, the second with Condorcet. In this paper it is shown that the basic desirable property of weighting systems - namely "consistency" under aggregationcan be achieved without sacrificing the common-sense property of choosing a Condorcet alternative whenever one exists. In fact, these two properties, together with the requirement of "neutrality" on alternatives, essentially determine a unique rule known in the literature as Kerneny's rule.

The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin-Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties.