Variable Electorates Research Articles

Bivariate scoring rules: Unifying the characterizations of positional scoring rules and Kemeny's rule

This paper studies social preference functions (SPFs), which map the voters' ordinal preferences over a set of alternatives to a non-empty set of strict rankings over the alternatives. Maybe the most prominent SPFs are positional scoring rules and Kemeny's rule. While these two types of rules behave intuitively quite differently, they are axiomatically surprisingly similar and we thus provide a joint axiomatic characterization of these SPFs. To this end, we introduce the class of bivariate scoring rules which generalize Kemeny's rule by weighting comparisons between alternatives depending on their positions in the voters' preference relations. In particular, this class contains both Kemeny's rule and all positional scoring rules. As our main result, we then characterize the set of bivariate scoring rules as the only SPFs that satisfy—aside from some standard axioms—mild consistency conditions for variable electorates and variable agendas. As corollaries of this result, we also derive variants of the well-known characterizations of positional scoring rules by Smith (1973) and of Kemeny's rule by Young (1988). Thus, our result unifies the independent streams of research on positional scoring rules and Kemeny's rule by giving a joint axiomatic basis of these SPFs.

Refined Characterizations of Approval-Based Committee Scoring Rules

In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, for which voters give points to each committee and the committees with maximal total points are chosen. While the set of ABC scoring rules has recently been characterized in a model where the output is a ranking of all committees, no full characterization of these rules exists in the standard model where a set of winning committees is returned. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result for ABC ranking rules to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.

New characterizations of strategy-proofness under single-peakedness

We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule.

Monotonicity Violations by Borda’s Elimination and Nanson’s Rules: A Comparison

This paper compares the vulnerability of Borda Elimination Rule (BER) and of Nanson Elimination Rule (NER) to monotonicity paradoxes under both fixed and variable electorates. It is shown that while NER is totally immune and BER is vulnerable to monotonicity failure in 3-candidate elections, neither of these two rules dominates the other in n-candidate elections (n > 3) when no Condorcet Winner exists. When the number of competing alternatives is larger than three and no Condorcet Winner exists, we find profiles where NER violates monotonicity while BER does not, profiles where BER violates monotonicity while NER does not, as well as profiles where both NER and BER violate monotonicity. These findings extend to both fixed and variable electorates, as well as to situations where the initial winners under both rules are the same, as well as to situations where the initial winners under both rules are different. So, which of the two rules should be preferred in terms of monotonicity in n-candidate elections (n > 3) where no Condorcet Winner exists, depends on the kind of profiles one can expect to encounter in practice most often. Nevertheless, in view of the results of 3-candidate elections under other scoring elimination rules, we conjecture that inasmuch as BER and NER exhibit monotonicity failures, it is more likely to occur in closely contested elections.

Two types of participation failure under nine voting methods in variable electorates

This paper expands the illustration and analysis regarding the susceptibility of nine voting procedures to two types of what are generally known as No-Show paradoxes. Following the article by Felsenthal and Tideman (Theory and Decision 75:59–77, 2013), the two paradoxes are denoted as P-TOP and P-BOT paradoxes. According to the P-TOP paradox it is possible that if candidate x has been elected by a given electorate then, ceteris paribus, another candidate, y, may be elected if additional voters join the electorate who rank x at the top of their preference ordering. Similarly, according to the P-BOT paradox it is possible that if candidate y has not been elected by a given electorate then, ceteris paribus, y may be elected if additional voters join the electorate who rank y at the bottom of their preference ordering. Voting procedures that are susceptible to these paradoxes are considered to be afflicted with a particularly serious defect because instead of encouraging voters to participate in an election and vote according to their true preference orderings, they may inhibit voters from participating in an election and thereby undermine the rationale for conducting elections.

Consistent Probabilistic Social Choice

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (1984). Fishburn's function returns so-called maximal lotteries, that is, lotteries that correspond to optimal mixed strategies in the symmetric zero-sum game induced by the pairwise majority margins. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming. [web URL: http://onlinelibrary.wiley.com/doi/10.3982/ECTA13337/abstract]

Consistency without neutrality in voting rules: When is a vote an average?

Smith [J.H. Smith, Aggregation of preferences with variable electorate, Econometrica 41 (1973) 1027–1041] and Young [H.P. Young, A note on preference aggregation, Econometrica 42 (1974) 1129–1131; H.P. Young, Social choice scoring functions, SIAM J. Appl. Math. 28 (1975) 824–838] characterized scoring rules via four axioms: consistency, continuity, anonymity, and neutrality. In their context a ballot consists of a strict ranking of alternatives, and an election outcome is either a set of (winning) alternatives (Young) or a weak ordering of alternatives (Smith). Many rules fail to fit this context, yet intuitively satisfy one’s notion of a generalized scoring rule ; this very broad class GSR includes the Kemeny rule, approval voting, and certain grading systems. We show that GSR is identical with the class MPR of mean proximity rules loosely, rules in MPR are those for which the “average voter” determines the outcome. The techniques in the proof allow us to make some surprisingly direct comparisons between rules (for example, between Kemeny and Borda) that might initially seem to be of completely different sorts. The abstract anonymous voting rules provide the context for GSR, which is of necessity too general to admit a neutrality axiom. A natural question arises: “What happens to the Smith and Young characterizations in the absence of neutrality?” We discuss one answer in the form of a characterization of the rational mean neat voting rules (a class closely related to GSR) as those that are consistent and connected. Connectedness is a strong form of continuity that implies a discrete analogue to the Intermediate Value Theorem.

Scoring run-off paradoxes for variable electorates

Summary. A no-show paradox occurs each time a single voter or a group of voters can manipulate the outcome by not participating to the election process. Among other voting procedures, the scoring run-off methods, which eliminate progressively the alternatives on the basis of scoring rules, suffer from this flaw. We here estimate how frequent this paradox is for three candidate elections under the classical Impartial Culture and Impartial Anonymous Culture assumptions, for different population sizes. The conditions under which this paradox occurs are also described, as well as the relationships with manipulations for a fixed number of voters.

Dominant strategies and restricted ballots with variable electorate

This paper considers dominant (uniformly best) voting strategies in a variable-electorate model for social choice with restricted ballot sets. A voter can have any weak preference order on the alternatives. A voter who partitions the alternatives into three or more indifference classes never has a dominant strategy. The paper specifies conditions on a social choice function that are necessary and sufficient for a voter who partitions the alternatives into two indifference classes to have a dominant strategy.

A proof technique for social choice with variable electorate

A proof technique for social choice with variable electorate

Aggregation of Preferences with Variable Electorate

IN THIS PAPER we consider procedures for going from several individual preferences among several alternatives, called candidates, to something which may be called a collective preference. The individual preferences take the form of (total) orderings of the alternatives, and the collective preference is to take the form of a (total) weak ordering (i.e., ties allowed). We consider certain properties which seem desirable in such and investigate which have these properties. The of view taken here differs from that of other work in this area (e.g., [1, 2, 3, 4]) chiefly in asking that the procedure work for all possible sizes of the voting population, rather than for a fixed population, given in advance. This permits us to require, for example, that if each of two bodies of voters prefers candidate A to candidate B under a given procedure, then the combination of these bodies should prefer A to B under the same procedure. In Section 1 we give the formal definitions of an aggregation procedure and discuss certain desirable features, namely neutrality (treats candidates symmetrically), (the condition mentioned above), monotonicity, and an Archimedean property which says, roughly, that a sufficiently large body with a given distribution of preferences can impose its will on any body of fixed size. In Section 2 we introduce certain procedures: point systems and systems (roughly, allowing infinitesimal points), which are neutral and separable. They are monotonic if and only if the points are arranged in the natural order, and the are, in addition, Archimedean. In Section 3 we prove a converse, namely that any neutral and separable procedure can be realized by a generalized system and, if it is Archimedean, by a system. This part requires some familiarity with the notions of least upper bound of a set of real numbers and bases of vector spaces. In Section 4 (which is largely independent of Section 3), we consider point which use in a succession of eliminations. Such are neither separable nor monotonic but do satisfy some very weak separability and monotonicity conditions. While these probably do not characterize runoff systems, we know of no other satisfying them.