Recovering non-monotonicity problems of voting rules

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.

Economics Letters, 2024

In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.

Preliminary and incomplete draft prepared for presentation at the 2002 Annual Meeting of the Public Choice Society, San Diego, March 22-24, 2002 MONOTONICITY FAILURE UNDER STV AND RELATED VOTING SYSTEMS It is now generally known that the single transferable vote (STV) and related voting systems that entail runoffs of one sort or other are subject to the paradoxical feature that enhanced support for a candidate can cost that candidate electoral victory (or, conversely, that reduced support can give the candidate electoral victory) — that is, that they are subject to so-called "monotonicity failure." However, it remains unclear how extensive this problem is. Is monotonicity failure merely a logical possibility that is highly unlikely to occur in practice, or is it a problem that may occur with considerable frequency? Is the possibility of monotonicity failure indicated by manifest characteristics of election results, or is it a problem that remains largely hidden from view? ...

Social Choice and Welfare, 2017

Scoring Elimination Rules (SER), that give points to candidates according to their rank in voters' preference orders and eliminate the candidate(s) with the lowest number of points, constitute an important class of voting rules. This class of rules, that includes some famous voting methods such as Plurality Runoff or Coombs Rule, suffers from a severe pathology known as monotonicity paradox or monotonicity failure, that is, getting more points from voters can make a candidate a loser and getting fewer points can make a candidate a winner. In this paper, we study three-candidate elections and we identify, under various conditions, which SER minimizes the probability that a monotonicity paradox occurs. We also analyze some strategic aspects of these monotonicity failures. The probability model on which our results are based is the Impartial Anonymous Culture condition, often used in this kind of study. 1 See also Miller (2012) for a (more or less) real-world example. second round is organized which confronts the two candidates with the highest number of votes in the first round and the one who obtains the majority of votes wins. Suppose there are 127 voters whose rankings of the three candidates, a, b and c, are as follows:

Review of Economic Design

In this paper we introduce the plurality kth social choice function selecting an alternative, which is ranked kth in the social ranking following the number of top positions of alternatives in the individual ranking of voters. As special case the plurality 1st is the same as the well-known plurality rule. Concerning individual manipulability, we show that the larger k the more preference profiles are individually manipulable. We also provide maximal non-manipulable domains for the plurality kth rules. These results imply analogous statements on the single non-transferable vote rule. We propose a decomposition of social choice functions based on plurality kth rules, which we apply for determining non-manipulable subdomains for arbitrary social choice functions. We further show that with the exception of the plurality rule all other plurality kth rules are group manipulable, i.e. coordinated misrepresentation of individual rankings are beneficial for each group member, with an appropr...

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.

TOP, 2013

A society has to choose within a set X of programs, each defining a decision regqrding a finite number D of yes-no issues. An X-profile associates with every program x in X a finite number of voters who support x. We prove that the outcome of the issue-wise simple majority rule Maj is an element of X at any X-profile where Maj is well-defined if and only if this is true when Maj is applied to any profile involving only 3 elements of X, each being supported by exactly one voter. We call this property triple-consistency. Moreover, we characterize the class of anonymous issue-wise choice functions that are triple-consistent. We discuss three applications of the results. First, interpreting X as a domain of preference relations over a finite set of alternatives, we argue that they generalize a well-known consequence of the value-restriction propery . Second, we can characterize the sets of approval ballots for which the strong version of paradox of multiple elections never occurs. Third,we can provide some new insights to the dynamics of club formation.

Mathematical Analyses of Decisions, Voting, and Games, eds. M. A. Jones, D. McCune, and J. Wilson, Contemporary Mathematics, American Mathematical Society, 2023, 2023

A fundamental principle of individual rational choice is Sen's γ axiom, also known as expansion consistency, stating that any alternative chosen from each of two menus must be chosen from the union of the menus. Expansion consistency can also be formulated in the setting of social choice. In voting theory, it states that any candidate chosen from two fields of candidates must be chosen from the combined field of candidates. An important special case of the axiom is binary expansion consistency, which states that any candidate chosen from an initial field of candidates and chosen in a head-to-head match with a new candidate must also be chosen when the new candidate is added to the field, thereby ruling out spoiler effects. In this paper, we study the tension between this weakening of expansion consistency and weakenings of resoluteness, an axiom demanding the choice of a single candidate in any election. As is well known, resoluteness is inconsistent with basic fairness conditions on social choice, namely anonymity and neutrality. Here we prove that even significant weakenings of resoluteness, which are consistent with anonymity and neutrality, are inconsistent with binary expansion consistency. The proofs make use of SAT solving, with the correctness of a SAT encoding formally verified in the Lean Theorem Prover, as well as a strategy for generalizing impossibility theorems obtained for special types of voting methods (namely majoritarian and pairwise voting methods) to impossibility theorems for arbitrary voting methods. This proof strategy may be of independent interest for its potential applicability to other impossibility theorems in social choice.

Journal of Mathematical Economics, 2011

Collective rationality of voting rules, requiring transitivity of social preferences (or quasi-transitivity, acyclicity for weaker notions), has been known to be incompatible with other standard conditions for voting rules when there is no prior information, thus no restriction, on individual preferences Sen, 1970). proposes two restricted domains of individual preferences where majority voting generates transitive social preferences; they are the domain consisting of preferences that have at most two indifference classes, and the domain where any set of three alternatives is partitioned into two non-empty subsets and alternatives in one set are strictly preferred to alternatives in the other set. On these two domains, we investigate whether majority voting is the unique way of generating transitive, quasi-transitive, or acyclic social preferences. First of all, we rule out non-standard voting rules by imposing monotonicity, anonymity, and neutrality. Our main results show that majority rule is the unique voting rule satisfying transitivity, yet all other voting rules satisfy acyclicity (also quasi-transitivity on the second domain). Thus we find a very thin border dividing majority and other voting rules, namely, the gap between transitivity and acyclicity.

Journal of Economic Theory, 1997