Approval voting reconsidered

Summary. The merit of approval voting has been widely discussed in the past 25 years. The distinct property of this rule is the extent of flexibility it allows; any voter can approve as many alternatives/candidates as he wishes. Nevertheless, this advantage is the very reason for two drawbacks of approval voting: its extreme vulnerability to majority decisiveness (Theorem 1) and its extreme vulnerability to erosion of the majority principle (Theorem 2). On the one hand, under some feasible voting strategies any majority of more than 1/2 of the voters can guarantee the selection of its most favorable candidate, regardless of the preferences of the other voters. On the other hand, under alternative voting strategies even the largest majority cannot impose its common most preferred candidate. A simultaneous resolution of the two problems is possible by restricted approval voting (RAV), a voting rule that allows partial voter flexibility by restricting the minimal and maximal number of candidates that can be approved. Our main result (Theorem 3) clarifies how the foregone flexibility in voters sovereignty mitigates the above mentioned drawbacks under sincere and insincere coordinated voting. Our findings suggest a new possible justification of a particular voting rule which is based on the significance assigned to three considerations: the advantages of voters flexibility, immunity to majority decisiveness and immunity to erosion of the majority principle. Such justification can provide a possible explanation to the prevalent use of some special cases of RAV, notably, of the plurality rule and of approval voting.Received: 8 October 2003, Revised: 17 June 2004, JEL Classification Numbers: D71, D72. Correspondence to: Shmuel NitzanWe are indebted to Steve Brams, Remzi Sanver and an anonymous referee for their very useful comments and suggestions.     

Unsettling aspects of voting theory

Summary. Voting procedures are known to be plagued with a variety of difficulties such as strategic voting, or where a voter is rewarded with a better election outcome by not voting, or where a winning candidate can lose by receiving more support. Once we know that these problems can occur, the next objective should be to completely understand how, why, and where they arise. Namely, for each election procedure, the new goal is to determine when such problems can occur, all voter types who can cause these difficulties, and the actions they must take. This paper develops an easily used approach to handle all of these issues for standard voting methods. New intuitive explanations for these various oddities follow from this approach. Received: August 14, 2002 ; revised version: September 4, 2002 RID="*" ID="*"My thanks to Richard Barrett and, in particular, to Hannu Nurmi for corrections, several useful comments, and suggestions they made about earlier versions. Also, my thanks to a referee for calling some excellent references to my attention. An earlier draft was presented at the 2002 Public Choice Society meeting. This research was supported by an NSF grant.     

A live experiment on approval voting

This paper presents a large-scale experiment on the Approval Voting rule that took place during the 2002 French presidential election. We describe the experiment and its main results. The findings are as follows: (i) Such an experiment is feasible, and very well accepted by voters. (ii) The principle of approval voting is easily understood and accepted. (iii) Within the observed political context, compared to the official first-round vote, approval voting modifies the overall ranking of candidates. (iv) The candidates Le Pen and Chirac, more than the others, were able to convert approval votes into official first-round votes. JEL Classification C93, D70, D72     

On the inconsistent behavior in voting for incumbents and for term limitation

This paper considers the seemingly inconsistent behavior of individuals who simultaneously vote for incumbents and for limitations on their terms in office. We argue that such behavior may occur even if voters pursue their self-interests in both candidate and term-limitation elections. First, we formulate elections for Congressional candidates as a two-person game, where each candidate maximizes votes by proposing a distribution of benefits to voters. Then we discuss the term limitation at the state level, where voters in each district compare, over time, the average benefits obtained from two alternative series of campaign games: one with a longer tenure associated with no term limit and the other with a shorter tenure created by the introduction of a term limit. In elections of candidates for Congress, the incumbent is successful because he can generate more aggregate benefits for voters. We show, however, that at some critical point of the tenure, his behavior will be less beneficial to his core constituents. In term-limitation elections, those voters tend to support a term limit. In some cases, they represent a majority in the state, and term limits are enacted. Received: February 23, 1999; revised version: January 24, 2000     

Fully sincere voting

In a general social choice framework where the requirement of strategy-proofness may not be sensible, we call a social choice rule fully sincere if it never gives any individual an incentive to vote for a less-preferred alternative over a more-preferred one and provides an incentive to vote for an alternative if and only if it is preferred to the default option that would result from abstaining. If the social choice rule can depend only on the number of votes that each alternative receives, those rules satisfying full sincerity are convex combinations of the rule that chooses each alternative with probability equal to the proportion of the vote it receives and an arbitrary rule that ignores voters' preferences. We note a sense in which the natural probabilistic analog of approval voting is the fully sincere rule that allows voters maximal flexibility in expressing their preferences and gives these preferences maximal weight.     

Weighted approval voting

To allow society to treat unequal alternatives distinctly we propose a natural extension of Approval Voting by relaxing the assumption of neutrality. According to this extension, every alternative receives ex-ante a strictly positive and finite weight. These weights may differ across alternatives. Given the voting decisions of every individual (individuals are allowed to vote for, or approve of, as many alternatives as they wish to), society elects the alternative for which the product of total number of votes times exogenous weight is maximal. If the product is maximal for more than one alternative, a pre-specified tie-breaking rule is applied. Our main result is an axiomatic characterization of this family of voting procedures.      

Candidate stability and nonbinary social choice

Summary. The requirement that a voting procedure be immune to the strategic withdrawal of a candidate for election can be formalized in different ways. Dutta, Jackson, and Le Breton (Econometrica, 2001) have recently shown that two formalizations of this candidate stability property are incompatible with some other desirable properties of voting procedures. This article shows that Grether and Plott's nonbinary generalization of Arrow's Theorem can be used to provide simple proofs of two of their impossibility theorems. Received: August 15, 2001; revised version: March 11, 2002 RID="*" ID="*" Parts of this article were previously circulated in somewhat different form in a working paper with the same title by the second author. We are grateful to Michel Le Breton and an anonymous referee for their comments. Correspondence to:J.A. Weymark     

The geometry of voting power: Weighted voting and hyper-ellipsoids

Suppose legislators represent districts of varying population, and their assembly's voting rule is intended to implement the principle of one person, one vote. How should legislators' voting weights appropriately reflect these population differences? An analysis requires an understanding of the relationship between voting weight and some measure of the influence that each legislator has over collective decisions. We provide three new characterizations of weighted voting that embody this relationship. Each is based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyper-ellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose–Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights.     

Strategy-proof and individually rational social choice functions for public good economies: A note

Summary. Serizawa [3] characterized the set of strategy-proof, individually rational, no exploitative, and non-bossy social choice functions in economies with pure public goods. He left an open question whether non-bossiness is necessary for his characterization. We will prove that non-bossiness is implied by the other three axioms in his characterization. Received: October 17, 1997; revised version: January 19, 1998     

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. Received: October 21, 1999; revised version: January 12, 2000     

The likelihood of dubious election outcomes

Summary. A disturbing phenomenon in voting, which causes most of the problems as well as the interest in the field, is that election outcomes (for fixed preferences) can change with the way the ballots are tallied. This causes difficulties because with each possible choice, some set of voters can be dubious about whether it is the “correct” one. But, how likely are these settings allowing multiple election outcomes? By combining properties of the geometry of voting developed by Saari with a analytic-geometric technique created by Schlafli, we determine the likelihood that a three candidate election can cause these potentially dubious outcomes. Received: April 11, 1997; revised version: November 12, 1997     

Geometry and impossibility

Summary. This note presents a simple proof of Arrow's impossibility theorem using Saari's [3, 4] “geometry of voting”. Received: March 5, 2001; revised version: August 16, 2001     

A comparison of Dodgson's method and the Borda count

Summary. In an election without a Condorcet winner, Dodgson's Method is designed to find the candidate that is “closest” to being a Condorcet winner. In this paper, we show that the winner from Dodgson's Method can occur at any position in the ranking obtained from the Borda Count, the plurality method, or any other positional voting procedure. In addition, we demonstrate that Dodgson's Method does not satisfy the Independence of Irrelevant Alternatives axiom. Received: January 12, 2001; revised version: June 7, 2001     

On the constancy of bribe-proof solutions

Summary. We consider the problem of choosing one point in a set of alternatives when monetary transfers are possible. In this context, Schummer (2000) shows that a social choice function must be a constant function if manipulation through bribes is ruled out. But he requires two kinds of domain-richness conditions. One is either smooth connectedness or the finiteness of the set of alternatives and the other is monotonical closedness. However, dispensing with the former condition, we alternatively prove the same result under a weaker condition than monotonical closedness. Received: April 11, 2000; revised version: February 25, 2002 RID="*" ID="*" This paper received the Osaka University Institute of Social and Economic Research Moriguchi Prize in January 2001. I am grateful to Prof. Ryoichi Nagahisa, Prof. Tatsuyoshi Saijo, Prof. Ken-ichi Shimomura, Prof. Ken Urai, and especially two anonymous referees for their useful and helpful comments and suggestions. I am a Research Fellow of the Japan Society for the Promotion of Science.     

The sale of small firms: a multidimensional analysis

Summary. This paper endogeneizes the security voting structure in an auction mechanism used to sell a small firm. The design of security voting structure allows the seller to choose between two objectives which are not mutually consistent. If the seller wants to maximize his revenue, he should retain some shares to benefit from the future dividends generated by the acquirer. At the opposite, if he wants to sell his firm to the most efficient candidate, he should sell all the shares. Received: July 4, 2001; revised version: October 31, 2002 RID="*" ID="*" The paper has benefited from a number of comments from the anonymous referees. Correspondence to: C. At     

A new approach to measure tactical voting: evidence from the British elections

Although tactical voting attracts a great deal of attention, it is very hard to measure as it requires knowledge of both individuals’ voting choices as well as their unobserved preferences. In this article, we present a simple empirical strategy to nonparametrically identify tactical voting patterns directly from balloting results. This approach allows us to study the magnitude and direction of strategic voting as well as to verify which information voters and parties take into account to determine marginal constituencies. We show that tactical voting played a significant role in the 2010 election, mainly for Liberal–Democratic voters supporting Labour. Moreover, our results suggest that voters seem to form their expectations based on a national swing in vote shares rather than newspaper guides published in the main media outlets or previous election outcomes. We also present some evidence that suggests that campaign spending is not driving tactical voting.     

The welfare consequences of strategic behaviour under approval and plurality voting

This paper studies the welfare consequences of strategic behaviour under approval and plurality voting by comparing the utilitarian efficiencies obtained in simulated voting under two behavioural assumptions: expected utility-maximising behaviour and sincere behaviour. Under approval voting utilitarian efficiency is relatively high irrespective of the behavioural assumption, and under the plurality rule strategic voting significantly increases utilitarian efficiency.     

Public goods,voting, and growth

We study a parametric politico‐economic model of economic growth with productive public goods and public consumption goods. The provision of public goods is funded by a proportional tax. Agents are heterogeneous in their initial capital endowments, discount factors, and the relative weights of public consumption in overall private utility. They vote on the shares of public goods in gross domestic products (GDP). We propose a definition of voting equilibrium, prove the existence and provide a characterization of voting equilibria, and obtain a closed‐form solution for the voting outcomes. Also we introduce a “fictitious” representative agent and interpret the outcome of voting as a choice made by a central planner for his benefit. Finally, we undertake comparative static analysis of the shares of public goods in GDP and of the rate of balanced growth with respect to the discount factors and the preferences for public consumption. The results of this analysis suggest that the representative‐agent version of our model is capable of capturing the interaction between many voting heterogeneous agents only if the heterogeneity is one‐dimensional.     

Dictatorial domains

Summary. In this paper, we introduce the notion of a linked domain and prove that a non-manipulable social choice function defined on such a domain must be dictatorial. This result not only generalizes the Gibbard-Satterthwaite Theorem but also demonstrates that the equivalence between dictatorship and non-manipulability is far more robust than suggested by that theorem. We provide an application of this result in a particular model of voting. We also provide a necessary condition for a domain to be dictatorial and use it to characterize dictatorial domains in the cases where the number of alternatives is three. Received: July 12, 2000; revised version: March 21, 2002 RID="*" ID="*" The authors would like to thank two anonymous referees for their detailed comments. Correspondence to: A. Sen     

Preference profiles sustaining Arrow's theorem

Summary. Arrow's theorem is proved on a domain consisting of two types of preference profiles. Those in the first type are “almost unanimous": for every profile some alternative x is such that the preferences of any two individuals merely differ in the ranking of x, which is in one of the first three positions. Profiles of the second type are “appropriately heterogeneous”, with preferences similar to those generating the “paradox of voting”. Received: March 9, 2000; revised version: June 7, 2001