On the selection of the same winner by all scoring rules

Different scoring rules can result in the selection of any of the k competing candidates, given the same preference profile, (Saari DG 2001, Chaotic elections! A mathematician looks at voting. American Mathematical Society, Providence, R.I.). It is also possible that a candidate, and even a Condorcet winning candidate, cannot be selected by any scoring rule, (Saari DG 2000 Econ Theory 15:55–101). These findings are balanced by Saari’s result (Saari DG 1992 Soc Choice Welf 9(4):277–306) that specifies the necessary and sufficient condition for the selection of the same candidate by all scoring rules. This condition is, however, indirect. We provide a sufficient condition that is stated directly in terms of the preference profile; therefore, its testability does not require the verdict of any voting rule.     

A majority-rule characterization with multiple extensions

When May's necessary and sufficient conditions for majority rule as a binary voting rule are extended in a natural way to decisions over more than two options, the resulting conditions are consistent with the Borda and Black voting rules, but not with a variety of other voting rules for more than two options. This paper presents an alternative set of necessary and sufficient conditions for binary majority rule, which permits the plurality, Condorcet and (simplified) Dodgson rules, as well as the Borda and Black rules, but not the Copeland or Nanson rules, to be classified as extensions of binary majority rule to decisions over more than two options.I am indebted to Amartya Sen and anonymous referees for helpful suggestions     

Modeling large electorates with Fourier series, with applications to Nash equilibria in proximity and directional models of spatial competition

 In this paper we introduce harmonic analysis (Fourier series) as a tool for characterizing the existence of Nash equilibria in two-dimensional spatial majority rule voting games with large electorates. We apply our methods both to traditional proximity models and to directional models. In the latter voters exhibit preferences over directions rather than over alternatives, per se. A directional equilibrium can be characterized as a Condorcet direction, in analogy to the Condorcet (majority) winner in the usual voting models, i.e., a direction which is preferred by a majority to (or at least is not beaten by) any other direction. We provide a parallel treatment of the total median condition for equilibrium under proximity voting and equilibrium conditions for directional voting that shows that the former result is in terms of a strict equality (a knife-edge result very unlikely to hold) while the latter is in terms of an inequality which is relatively easy to satisfy. For the Matthews [3] directional model and a variant of the Rabinowitz and Macdonald [7] directional model, we present a sufficiency condition for the existence of a Condorcet directional vector in terms of the odd-numbered components of the Fourier series representing the density distribution of the voter points. We interpret our theoretical results by looking at real-world voter distributions and direction fields among voter points derived from U.S. and Norwegian survey data. Received: 7 July 1995 / Accepted: 14 May 1996     

Voting rules as statistical estimators

We adopt an ‘epistemic’ interpretation of social decisions: there is an objectively correct choice, each voter receives a ‘noisy signal’ of the correct choice, and the social objective is to aggregate these signals to make the best possible guess about the correct choice. One epistemic method is to fix a probability model and compute the maximum likelihood estimator (MLE), maximum a posteriori (MAP) estimator or expected utility maximizer (EUM), given the data provided by the voters. We first show that an abstract voting rule can be interpreted as MLE or MAP if and only if it is a scoring rule. We then specialize to the case of distance-based voting rules, in particular, the use of the median rule in judgement aggregation. Finally, we show how several common ‘quasiutilitarian’ voting rules can be interpreted as EUM.     

Characterizing best–worst voting systems in the scoring context

An increasing body of theoretical and empirical work on discrete choice considers a choice design in which a person is asked to select both the best and the worst alternative in an available set of alternatives, in contrast to more traditional tasks, such as where the person is asked to: select the best alternative; select the worst alternative; rank the alternatives. Here we consider voting systems motivated by such “best–worst” choice; characterize a class of “best–worst” voting systems in terms of a set of axioms in the context of scoring rules; and discuss briefly possible extensions to approval–disapproval systems.     

Stable sets and cores for multi-criteria simple games and for their extensions

In this article we deal with multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. We generalize two closely related solution concepts for these games, the stable sets and the core. A relationship between stable sets and minimal winning coalitions is established. Two generalizations of the core for multi-criteria simple games are characterized in terms of veto players. In addition, extended multi-criteria simple games obtained from different aggregation operations, such as union, intersection, marginalization, and composition, are introduced. It is shown that a voting system can be established when these operations are applied to complex voting systems.     

First and second best voting rules in committees

A committee of people with common preferences but different abilities in identifying the best alternative (e.g., a jury) votes in order to decide between two alternatives. The first best voting rule is a weighted voting rule that takes the different individual competences into account, and is therefore not anonymous, i.e., the voters’ identities matter. Under this rule, it is rational for the committee members to vote according to their true opinions, or informatively. This is not necessarily true for an anonymous voting rule, under which members may have an incentive to vote non-informatively. Thus, strategic, sophisticated voters may vary their voting strategies according to the voting rule rather than naively voting informatively. This paper shows that the identity of the best anonymous and monotone (i.e., quota) voting rule does not depend on whether the committee members are strategic or naive or whether some are strategic and some are naive. One such rule, called the second best rule, affords the highest expected utility in all cases.

“Wasn’t he sweet?” said Yossarian. “Maybe they should give him three votes.” Joseph Heller, Catch-22

     

On the equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.     

On the structure of minimal winning coalitions in simple voting games

According to Coleman’s index of collective power, a decision rule that generates a larger number of winning coalitions imparts the collectivity a higher a priori power to act. By the virtue of the monotonicity conditions, a decision rule is totally characterized by the set of minimal winning coalitions. In this paper, we investigate the structure of the families of minimal winning coalitions corresponding to maximal and proper simple voting games (SVG). We show that if the proper and maximal SVG is swap robust and all the minimal winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism) system. We also provide examples of proper SVGs to show that the number of winning coalitions is not monotone with respect to the intuitively appealing system parameters like the number of blockers, number of non-dummies or the size of the minimal blocking set.     

Sophisticated voting and equilibrium refinements under plurality rule

In this paper we show in the context of voting games with plurality rule that the “perfect” equilibrium concept does not appear restrictive enough, since, independently of preferences, it can exclude at most the election of only one candidate. Furthermore, some examples show that there are “perfect” equilibria that are not “proper”. However, also some “proper” outcome is eliminated by sophisticated voting, while Mertens' stable set fully satisfies such criterium, for generic plurality games. Moreover, we highlight a weakness of the simple sophisticated voting principle. Finally, we find that, for some games, sophisticated voting (and strategic stability) does not elect the Condorcet winner, neither it respects Duverger's law, even with a large number of voters. Received: 16 March 1999/Accepted: 25 September 1999     

On the manipulability of approval voting and related scoring rules

We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of candidates: by comparison of worst candidates, best candidates, or by comparison based on stochastic dominance. We perform a similar exercise for k-approval rules, where voters approve of a fixed number k of candidates. These results can be used to compare (k-)approval rules with respect to their manipulability. Analytical results are obtained for the case of two voters, specifically, the values of k for which the k-approval rule is minimally manipulable—has the smallest number of manipulable preference profiles—under the various preference extensions are determined. For the number of voters going to infinity, an asymptotic result is that the k-approval rule with k around half the number of candidates is minimally manipulable among all scoring rules. Further results are obtained by simulation and indicate that k-approval rules may improve on the approval rule as far as manipulability is concerned.     

The traveling group problem

This paper combines social choice theory with mathematical optimization by applying various group decision concepts to a classical problem of combinatorial optimization, namely the famous traveling salesperson (salesman) problem. The aim of the latter is to find a tour through all vertices of a given graph along edges of minimal total cost. In this contribution we replace the measure of additive edge costs by the social acceptance of different edges and the resulting tours. In particular, for four different voting rules, the Borda rule, Approval voting, Plurality rule and Simple Majority rule, we will investigate the social acceptance of tours derived from global and local decisions. It will be shown that these two decision approaches can lead to widely varying results.     

In quest of the banks set in spatial voting games

The Banks set (1(4):295–306, 1985) is one of the more important concepts in voting theory since it tells us about the sophisticated outcomes of standard amendment voting procedures commonly in use throughout the English speaking world (and elsewhere as well). While the properties of the Banks set for finite voting games have been extensively studied, little is known about how to find members of this set for majority rule spatial voting games involving possibly infinite agendas. We look at this question for two-dimensional games where voters have Euclidean preferences, and offer a variety of new results that delimit areas of the space that can be shown to lie within the Banks set, such as the Schattschneider set, the tri-median set, and the Banks line set—geometric constructs which we show to be nested within one another.     

Policy convergence under approval and plurality voting: the role of policy commitment

Proponents of Approval Voting argue that this electoral rule leads to more centrist outcomes compared to Plurality Voting. This claim has been substantiated by scholarly work using spatial models of political competition. We revisit this issue in the context of a model of political competition in which (1) candidates are policy-motivated; (2) candidacy decisions are endogenous; and (3) candidates can credibly commit to implementing any policy. Under these assumptions we find the opposite to be true – Plurality Voting yields convergence to the median voter’s ideal policy but Approval Voting may not. We argue that this result is driven by the differential incentives for candidate entry under the two voting rules. Our results suggest that whether Approval Voting yields more centrist outcomes vis-á-vis Plurality Voting depends on the possibility of policy commitment on the part of the candidates. In an election held under PV each citizen is given one vote he can cast for one (and only one) candidate, and the candidate who gets the most votes wins the election.     

Subgame perfect implementation of voting rules via randomized mechanisms

We characterize completely ordinal and onto choice rules that are subgame perfect of Nash equilibrium (SPE) implementable via randomized mechanisms under strict preferences. The characterization is very operationalizable, and allows us to analyse SPE implementability of voting rules. We show that no scoring rule is SPE implementable. However, the top-cycle and the uncovered correspondences as well as plurality with runoff and any strongly Condorcet consistent voting rule can be SPE implemented. Therefore our results are favourable to majority based voting rules over scoring rules. Nevertheless, we show that many interesting Condorcet consistent but not strongly Condorcet consistent rules, such as the Copeland rule, the Kramer rule and the Simpson rule, cannot be SPE implemented.     

Multiple votes,ballot truncation and the two-party system: an experiment

Duverger’s law states that Plurality Voting tends to favor a two-party system. We conducted laboratory experiments to study whether voting procedures other than Plurality Voting could favor a two-party system as well. Plurality Voting is compared with Approval Voting and Dual Voting, both of which allow to vote for multiple candidates, but differ in whether voters are required to cast all their votes. We find that in most elections held under Plurality Voting and Approval Voting, at most two candidates are viable. By contrast, three candidates are viable in a sizable number of elections held under Dual Voting due to strategic voting. Our evidence suggests that Approval Voting may encourage a two-party system, whereas Dual Voting may encourage multipartism. The voters’ ability to truncate ballots (i.e., not cast all their votes) is essential for supporting the two-party system under Approval Voting.     

Equilibrium in abstract political economies: with an application to a public good economy with voting

Consider an abstract political economy which has a collective choice rule together with strategic interactions among players. We prove that there exists an equilibrium in such an economy by synthesizing an equilibrium existence theorem in generalized games by Shafer and Sonnenschein (1975) and a voting core existence theorem in simple games by Schofield (1984, 1989). The theorem can be applied to a public good economy where public good provisions are determined by a class of voting rule.Thanks are due to David Austen-Smith, Jeffrey Banks, Marcus Berliant, Steve Ching, Ryo-Ichi Nagahisa, Mary Beth Savio, Norman Schofield, and Tomoichi Shinotsuka. Detailed comments from an anonymous referee of the journal are gratefully acknowledged. Errors are, of course, my own.     

Borda’s Paradox with weighted scoring rules

Representations are obtained for the probabilities that a Strict Borda Paradox and a Strong Borda Paradox are observed for large electorates with three candidates under the standard assumptions of Impartial Culture and Impartial Anonymous Culture. These representations are obtained for general weighted scoring rules (WSRs), and the probabilities are found to be maximized for voting rules like plurality rule and negative plurality rule. It is found that these paradox probabilities are not reduced for every scoring rule with the introduction of some degree of dependence among voters’ preferences with IAC. It is concluded that actual observances of a Strict Borda Paradox should be extremely rare, and that while observances of a Strong Borda Paradox should also be rare, they might occasionally be witnessed.     

On the positive association of parliamentary social choice functions

Intuitively, a voting rule satisfies the condition of positive association if it guarantees that an improvement for an alternative in the preferences expressed by voters results in a change, if there is any, of the social status of that alternative in the same direction. In this article, we consider two interpretations of this notion, and for parliamentary voting procedures, we provide results showing under the impartial anonymous culture (IAC) hypothesis how often the positive association is (not) satisfied. Furthermore, our analysis also permits us to investigate the relationships between these notions.