Mathematical structure of voting paradoxes

Summary. A theory is developed to explain all positional voting outcomes that can result from a single but arbitrarily chosen profile. This includes all outcomes, paradoxes, and disagreements among positional procedure outcomes as well as all discrepancies in rankings as candidates are dropped or added. The theory explains why each outcome occurs while identifying all illustrating profiles. It is shown how to use this approach to derive properties of methods based on pairwise and positional voting outcomes. Pairwise voting is addressed in the preceding companion paper [15]; the theory for positional methods is developed here.     

Winning probabilities in a pairwise lottery system with three alternatives

Summary. The pairwise lottery system is a multiple round voting procedure which chooses by lot a winner from a pair of alternatives to advance to the next round where in each round the odds of selection are based on each alternatives majority rule votes. We develop a framework for determining the asymptotic relative likelihood of the lottery selecting in the final round the Borda winner, Condorcet winner, and Condorcet loser for the three alternative case. We also show the procedure is equivalent to a Borda lottery when only a single round of voting is conducted. Finally, we present an alternative voting rule which yields the same winning probabilities as the pairwise lottery in the limiting case as the number of rounds of the pairwise lottery becomes large.Received: 5 June 2003, Revised: 17 June 2004, JEL Classification Numbers: D71. Correspondence to: Jac C. HeckelmanWe thank Keith Dougherty and Andrew Yates for their comments.     

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     

The Borda rule,Condorcet consistency and Condorcet stability

Summary. The Borda rule is known to be the least vulnerable scoring rule to Condorcet inconsistency, Saari (2000). Such inconsistency occurs when the Condorcet winner (the alternative which is preferred to any other alternative by a simple majority) is not selected by the Borda rule. This note exposes the relationship between the Borda rule and the Condorcet q-majority principle as well as the Condorcet q-majority voting rule. The main result establishes that the Borda rule is Condorcet q-majority consistent when where k is the number of alternatives. The second result establishes that is the minimal degree of majority decisiveness corresponding to the Borda rule under sincere voting. The same majority is required to ensure decisiveness under the Borda rule and to ensure that a q-rule (the generalized q-majority Condorcet rule) is a voting rule. Received: April 8, 2002; revised version: July 17, 2002 Correspondence to:S. Nitzan     

Manipulation through political endorsements

We study elections with three candidates under plurality voting. A candidate is a Condorcet loser if the majority of the voters place that candidate at the bottom of their preference rankings. We first show that a Condorcet loser might win the election in a three-way race. Next we introduce to the model an endorser who has private information about the true probability distribution of the preferences of the voters. Observable endorsements facilitate coordination among voters who may otherwise split their votes and lead to the victory of the Condorcet loser. When the endorser has an ideological bias towards one of the candidates, the coordination impact of endorsements remains unaltered, moreover the endorser successfully manipulates the outcome of the election in favor of his bias, even if his ideological bias is known by the voters. The results are true for any endorsement cost and any magnitude of bias as long as the electorate is large enough.     

The Borda method is most likely to respect the Condorcet principle

Summary We prove that in the class of weighted voting systems the Borda Count maximizes the probability that a Condorcet candidate is ranked first in a group election. A direct result is that the Borda Count maximizes the probability that a transitive, binary ranking of the candidates is preserved in a group election. A preliminary result, but one of independent interest, is that the Borda Count maximizes the probability that a majority outcome betweenany two candidates is reflected by the group election. All theorems are valid when there is a uniform probability distribution on the voter profiles and can be generalized to other uniform-like probability distributions. This work extends previous results of Fishburn and Gehrlein from three candidates to any number of candidates.This work is a portion of my doctoral dissertation The Geometric Investigation of Voting Techniques: A Comparison of Approval Voting, Positional Voting Techniques and the Borda Count written under Don Saari at Northwestern University, Evanston, Illinois.     

Multi-stage voting, sequential elimination and Condorcet consistency

A class of voting procedures based on repeated ballots and elimination of one candidate in each round is shown to always induce an outcome in the top cycle and is thus Condorcet consistent, when voters behave strategically. This is an important class as it covers multi-stage, sequential elimination extensions of all standard one-shot voting rules (with the exception of negative voting), the same one-shot rules that would fail Condorcet consistency. The necessity of repeated ballots and sequential elimination are demonstrated by further showing that Condorcet consistency would fail in all standard voting rules that violate one or both of these conditions.     

Approval voting in subset elections

Summary. Approval voting is designed to be “insensitive to numbers” of voters, and likely to elect a Condorcet candidate. However, the result of an election among one group of candidates gives no information about the results of elections among any other groups, even if every voter follows the recommended utility-maximizing strategy, which places strong restrictions on the individual voter's subset ballots. Thus the addition of a single candidate could completely reverse the outcome of an election, or a Condorcet candidate could finish last. Received: November 5, 1998; revised version: November 30, 1998     

How many voters are needed for paradoxes?

Summary. This paper presents a general procedure for finding profiles with the minimum number of voters required for many important paradoxes. Borda's and Condorcet's classic examples are revisited as well as generalizations. Using Saari's procedure line, we obtain an upper bound for the minimum number of voters needed for a profile for which the Condorcet winner is not strictly top ranked for all weighted positional procedures. Also we give a simple upper bound on the minimum number of voters needed for a set of prescribed voting outcomes. In contrast to situations wherein small numbers of voters are needed, we consider paradoxes requiring arbitrarily large numbers of voters as well as large numbers of alternatives. Finally we indicate connections with statistical rank based tests. Received: April 18, 2001; revised version: May 25, 2001     

The Costs of Implementing the Majority Principle: The Golden Voting Rule

In a context of constitutional choice of a voting rule, this paper presents an economic analysis of scoring rules that identifies the golden voting rule under the impartial culture assumption. This golden rule depends on the weights β and (1−β) assigned to two types of costs: the cost of majority decisiveness (‘tyranny’) and the cost of the ‘erosion’ in the majority principle. Our first main result establishes that in voting contexts where the number of voters n is typically considerably larger than the number of candidates k, the golden voting rule is the inverse plurality rule for almost any positive β. Irrespective of n and k, the golden voting rule is the inverse plurality rule if β ≥ 1/2 .. This hitherto almost unnoticed rule outperforms any other scoring rule in eliminating majority decisiveness. The golden voting rule is, however, the plurality rule, the most widely used voting rule that does not allow even the slightest ‘erosion’ in the majority principle, when β=0. Our second main result establishes that for sufficiently “small size” voting bodies, the set of potential golden rules consists at most of just three rules: the plurality rule, the Borda rule and the inverse plurality rule. On the one hand, this finding provides a new rationalization to the central role the former two rules play in practice and in the voting theory literature. On the other hand, it provides further support to the inverse plurality rule; not only that it is the golden rule in voting contexts, it also belongs, together with the plurality rule and the Borda method of counts, to the “exclusive” set of potential golden voting rules in small committees. We are indebted to Jim Buchanan, Amichai Glazer, Noa Nitzan, Ken Shepsle, and an anonymous referee for their useful comments.     

The probability of violating Arrow’s conditions

Arrow’s impossibility theorem shows that all preference aggregation rules (PARs) must violate a specific set of normative conditions (transitivity, Pareto, IIA, nondictatorship) over an unrestricted domain of preference profiles. However, the theorem does not address which PARs are more likely to violate those conditions across preference profiles. We compare the probabilities that thirteen PARs (anti-plurality, Hare, Nanson, plurality, plurality runoff, Simpson–Kramer, Baldwin, Borda, Coombs, Copeland, Dowdall, pairwise majority, and ranked pairs) violate Arrow’s conditions. We prove that Baldwin, Borda, Coombs, Copeland, Dowdall, and ranked pairs are less likely to violate IIA than the first six PARs, and they are less likely to violate Arrow’s conditions jointly. In contrast, pairwise majority never violates IIA but can violate transitivity. Simulations with three alternatives reveal that among the PARs studied, pairwise majority is the most likely to satisfy Arrow’s conditions jointly. Our results suggest pairwise majority violates transitivity with a small probability, while the other PARs violate IIA with much larger probabilities.     

Would letting people vote for multiple candidates yield policy moderation?

We investigate whether letting people vote for multiple candidates would yield policy moderation. We do so in a setting that takes three key features of elections into account, namely, strategic voting, endogenous candidacy and policy motivation on the part of the candidates. We consider two classes of voting rules. One class consists of the voting rules where each voter casts several equally-weighed votes for the different candidates. The other class consists of the voting rules where each voter rank-orders the candidates. We identify conditions under which these voting rules yield policy moderation. We also show that these voting rules may yield policy extremism instead of policy moderation if one (or several) of the conditions is not satisfied! Finally, we find that amongst these voting rules the extent of policy moderation is maximal under the Borda Count if we consider only equilibria where all candidates are serious contenders. However, this result does not carry over to spoiler equilibria, where Approval Voting can yield more moderate policy outcomes than the Borda Count.     

Voting over selfishly optimal tax schedules: Can Pigouvian tax redistribute income?

This paper studies pairwise majority voting over selfishly optimal nonlinear income tax schedules proposed by a continuum of individuals who differ in privately observable skills and make consumption comparisons, which creates a negative positional externality. It shows that the tax schedule preferred by the median skill type will win the voting contest. Given a reference consumption defined as the average consumption in the population, all skills face the same Pigouvian tax rate in the utilitarian optimum, whereas in selfish optima high skills face a Pigouvian tax rate larger than that facing low skills, generating a novel income redistributive effect. Under a constant elasticity of labor supply, two more results are obtained. First, for Pareto, Champernowne, Weibull, and lognormal skill distributions, the selfishly optimal tax schedule facing high (low) skills tends to be more progressive when the bottom‐skill's (top‐skill's) status concern intensifies. Second, it identifies the conditions under which, in the voting equilibrium, high skills face higher marginal tax rates while low skills face lower ones than what they face in the utilitarian optimum.     

Non-manipulable domains for the Borda count

Summary. We characterize the preference domains on which the Borda count satisfies Arrows independence of irrelevant alternatives condition. Under a weak richness condition, these domains are obtained by fixing one preference ordering and including all its cyclic permutations (Condorcet cycles). We then ask on which domains the Borda count is non-manipulable. It turns out that it is non-manipulable on a broader class of domains when combined with appropriately chosen tie-breaking rules. On the other hand, we also prove that the rich domains on which the Borda count is non-manipulable for all possible tie-breaking rules are again the cyclic permutation domains.Received: 24 November 2003, Revised: 12 December 2004, JEL Classification Numbers: D71. Correspondence to: Clemens PuppeThe third author gratefully acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG), Graduiertenkolleg 629 at the University of Bonn and from the Hungarian Scientific Research Fund (OTKA F 043496).     

Condorcet Jury Theorem: An example in which informative voting is rational but leads to inefficient information aggregation

Recent research on the Condorcet Jury Theorem has proven that informative voting (that is, voting according to one’s signal) is not necessarily rational. With two alternatives, rational voting typically leads to the election of the correct alternative, in spite of the fact that not all voters vote informatively. We prove that with three alternatives, there are cases in which informative voting is rational and yet leads to the election of a wrong alternative.     

Comparison of Scoring Rules in Poisson Voting Games

Scoring rules are compared by their equilibria in simple voting games with Poisson population uncertainty, using new techniques for computing pivot probabilities. Best-rewarding rules like plurality voting can generate discriminatory equilibria where the voters disregard some candidate as not a serious contender, although he may be universally liked, or may be symmetric to other candidates as in the Condorcet cycle. Such discriminatory equilibria are eliminated by worst-punishing rules like negative voting, but then even a universally disliked candidate may have to be taken seriously. In simple bipolar elections, equilibria are always majoritarian and efficient under approval voting, but not other scoring rules. Journal of Economic Literature Classification: D72.     

Redistributive policies with heterogeneous social preferences of voters

There is growing evidence on the roles of fairness and other-regarding preferences as fundamental human motives. Call voters with fair preferences, as in Fehr and Schmidt (1999), fair-voters. By contrast, traditional political economy models are based on selfish-voters who derive utility solely from “own” payoff. In a general equilibrium model with endogenous labor supply, a mixture of fair and selfish voters choose optimal policy through majority voting. First, we show that majority voting produces a unique winner in pairwise contests over feasible policies (the Condorcet winner). Second, we show that a preference for greater fairness leads to greater redistribution. An increase in the number of fair voters can also lead to greater redistribution. Third, we show that in economies where the majority are selfish-voters, the decisive policy could be chosen by fair-voters, and vice versa. Fourth, while choosing labor supply, even fair voters behave exactly like selfish voters. We show how this apparently inconsistent behavior in different domains (voting and labor supply) can be rationalized within the model.     

Counterexamples on the Superiority of Approval versus Plurality

We present a simple voting environment with three candidates where the Condorcet winner exists. Under plurality rule, the derived game has a stable set where such a candidate is elected with probability one. However, no stable set of the approval game elects the Condorcet winner with positive probability. We also analyze the robustness of such an example to changes in the number of voters and their preferences. To conclude, we present a generic four‐candidate voting environment with the same properties.     

一种新的组合评价方法

本文根据Borda法和Copeland法,提出一种新的组合评价方法——Football（足球）法。与Bor山法和Copeland法相比,该方法通过进一步区分“优”和“劣”的程度,使组合评价结论更为精确,区分度更好,且与各综合评价方法的结论具有很好的一致性。最后本文通过实例说明该方法的优越性。     

Information and strategic voting

We theoretically and experimentally study voter behavior in a setting characterized by plurality rule and mandatory voting. Voters choose from three options. We are interested in the occurrence of strategic voting in an environment where Condorcet cycles may occur and focus on how information about the preference distribution affects strategic behavior. We also vary the relative importance of the second preferred option. Quantal response equilibrium analysis is used to analyze the game and derive predictions. Our results indeed show that strategic voting arises. Its extent depends on (i) information availability; (ii) the relative importance of the intermediate candidate; (iii) the electorate’s relative support for one’s preferred candidate; (iv) the relative position of the plurality-supported candidate in one’s preference ordering. Our results show that information serves as a coordination device where strategic voting does not harm the plurality-preferred candidate’s chances of winning.