Three-candidate Elections Research Articles

How frequently do different voting rules encounter voting paradoxes in three-candidate elections?

We estimate the frequencies with which ten voting anomalies (ties and nine voting paradoxes) occur under 14 voting rules, using a statistical model that simulates voting situations that follow the same distribution as voting situations in actual elections. Thus the frequencies that we estimate from our simulated data are likely to be very close to the frequencies that would be observed in actual three-candidate elections. We find that two Condorcet-consistent voting rules do, the Black rule and the Nanson rule, encounter most paradoxes and ties less frequently than the other rules do, especially in elections with few voters. The Bucklin rule, the Plurality rule, and the Anti-plurality rule tend to perform worse than the other eleven rules, especially when the number of voters becomes large.

Strong measures of group coherence and the probability that a pairwise majority rule winner exists

A Pairwise Majority Rule Winner (PMRW) exists for a voting situation if some candidate can defeat each of the remaining candidates by Pairwise Majority Rule. The PMRW would be very appropriate for selection as the winner of an election, but it is well known that such a candidate does not always exist. This paper concludes a series of studies regarding the probability that a PMRW should be expected to exist in three-candidate elections, by introducing the notion of a strong measures of mutually coherent group preferences. In order for voting situations to be reasonably expected to fail to have a PMRW in a three-candidate election, voters’ preferences must be generated in an environment that is far removed from the situation in which there is a strong-overall-unifying candidate. So far removed, that it is extremely unlikely that a PMRW will not exist in voting situations with large electorates for a small number of candidates.

Plurality Rule Works In Three-Candidate Elections

In the citizen–candidate approach each citizen chooses whether or not to run as candidate. In a single-peaked preference domain, we find that the strategic entry decision of the candidates eliminates one of the most undesirable properties of Plurality rule, namely to elect a poor candidate in three-candidate elections since as we show, the Condorcet winner among the self-declared candidates is always elected. We find that the equilibria with three candidates are basically 2-fold, either there are two right-wing candidates and a left-wing candidate who wins the elections (or its symmetric), or there is a right-wing candidate, a left-wing candidate, and a candidate located in between the two others who becomes winner. We also show that when four or more candidates enter the contest, Plurality rule can elect the Condorcet-loser among the self-declared candidates.

The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences

Niemi (Am Polit Sci Rev 63:488–497, 1969) proposed a simple measure of the cohesiveness of a group of n voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, k, reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW(n,IAC|k), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of k. Results show a very strong relationship between PASW(n,IAC|k) and k, such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.

Voting rules, manipulability and social homogeneity ∗

To what extent are some voting rules more vulnerable tostrategic manipulation than others? In order to answer thisquestion, representations are developed for the coalitionalmanipulability of eight voting rules under various assumptionsconcerning the likelihood that given voters' preferenceprofiles are observed on three alternatives. Of particularinterest is the impact that social homogeneity (defined as thetendency of voters' preference to be similar) has on themanipulability of voting rules. The results we obtain showthat the hierarchy of the voting rules that results from ourcomputations can crucially depend on the degree of socialhomogeneity. However, it turns out that, whatever the degreeof homogeneity, the Hare method (or two-stage plurality)minimizes susceptibility to strategic manipulation bycoalitions of voters in three-candidate elections.

On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, wj, to the alternative she ranks in jth position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases.

Condorcet winners on four candidates with anonymous voters

May’s Theorem is extended to four-candidate elections with the Impartial Anonymous Culture Condition (IAC) to obtain a closed form representation for the probability, P(4, n,IAC), that a Condorcet winner exists for odd n voters. Representations of this type are found to become much more cumbersome for four-candidate elections, as compared to the relatively elegant results that are obtained for three-candidate elections. The precise limiting probability P(4, ∞, IAC)= 1717 2048 =0.8384 is found to be slightly different than previously reported Monte-Carlo simulation results, and the limiting value is approached very quickly for n at all large.

Scoring Rules, Condorcet Efficiency and Social Homogeneity

In a three-candidate election, a scoring rule λ, λ∈[0,1], assigns 1,λ and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule λ has the greatest Condorcet efficiency? After recalling the known answer to this question, we investigate the impact of social homogeneity on the optimal value of λ. One of the most salient results we obtain is that the optimality of the Borda rule (λ=1/2) holds only if the voters act in an independent way.

On the probability that all decision rules select the same winner

Social choice literature proposes several decision processes to achieve a social compromise. We know that different procedures may yield different solutions. However, no study has been done to estimate the probability that they simultaneously pick the same outcome. We partially answer this question for three-candidate elections. Let P* be the probability that almost all the voting rules: the Condorcet procedures, the scoring rules and the runoff methods select simultaneously the same winner. We provide new techniques for the computation of the occurrence of voting outcomes when the number of voters approaches infinity and each voter selects independently a linear preference order on the candidates according to a uniform probability distribution. With these assumptions, the exact value of P* turns out to be 0.50116.

Condorcet Efficiency of Borda Rule under the Dual Culture Condition

The Condorcet winner in an election is a candidate who could defeat each of the other available alternatives by majority rule in a series of pairwise votes. The Condorcet efficiency of a voting procedure is the conditional probability that it will elect the Condorcet winner, given that a Condorcet winner exists. The study considers the Condorcet efficiency of Borda Rule, which assigns weights of (1, 12, 0) respectively to each voter's first, second, and third ranked candidates, and elects the candidate with the most total points. A closed form representation is obtained for the Condorcet efficiency of Borda Rule with large electorates under the dual culture condition for three-candidate elections. Borda Rule is shown to perform very well over the range of dual culture probability vectors.

A Note on the Probability of Having a Strong Condorcet Winner

In an election, a strong Condorcet winner is a candidate who is top-ranked by more than 50% of the voters. The purpose of this note is to provide some algebraic representations for the probability of having a strong Condorcet winner in three-candidate elections. Three alternative procedures for generating voting situations are considered: the Impartial Culture condition, the Impartial Anonymous Culture condition and the Maximal Culture condition. It turns out that the conclusions we obtain strongly depend on the way for generating voting situations.

Approval Voting, Borda Winners, and Condorcet Winners: Evidence from Seven Elections

We analyze 10 three-candidate elections (and mock elections) conducted under approval voting (AV) using a method developed by Falmagne and Regenwetter (1996) that allows us to construct a distribution of rank orders from subset choice data. The elections were held by the Institute of Management Science, the Mathematical Association of America, several professional organizations in Britain, and the Institute of Electrical and Electronics Engineers. Seven of the 10 elections satisfy the conditions under which the Falmagne-Regenwetter method is suitable. For these elections we recreate possible underlying preferences of the electorate. On the basis of these distributions of preferences we find strong evidence that AV would have selected Condorcet winners when they exist and would have always selected the Borda winner. Thus, we find that AV is not just simple to use, but also gives rise to outcomes that well reflect voter preferences. Our results also have an important implication for the general study of social choice processes. They suggest that transitive majority orderings may be expected in real-world settings more often then the formal social choice literature suggests. In six out of seven data sets we find social welfare orders; only one data set generates cycles anywhere in the solution space.

Unified Models of Turnout and Vote Choice for Two-Candidate and Three-Candidate Elections

In this article I use a theory of individual utility maximization to derive a unified model of electoral behavior that includes both candidate choices and turnout decisions. Compared to this new unified model, existing specifications for jointly considering turnout and vote choice are found to be theoretically or empirically lacking. I provide methods for testing my model in elections with two or three candidates, and I show that the parameters of these models can be estimated without difficulty using maximum likelihood techniques. Application of these unified models to the 1988 and 1992 American presidential elections illustrates the potential contrasts between unified models and models that consider only candidate choices.

An experimental study of voting rules and polls in three-candidate elections

We report the results of elections conducted in a laboratory setting, modelled on a threecandidate example due to Borda. By paying subjects conditionally on election outcomes, we create electorates with (publicly) known preferences. We compare the results of experiments with and without non-binding pre-election polls under plurality rule, approval voting, and Borda rule. We also refer to a theory of voting “equilibria,” which makes sharp predictions concerning individual voter behavior and election outcomes. We find that Condorcet losers occasionally win regardless of the voting rule or presence of polls. Duverger's law (which asserts the predominance of two candidates) appears to hold under plurality rule, but close three-way races often arise under approval voting and Borda rule. Voters appear to poll and vote strategically. In elections, voters usually cast votes that are consistent with some strategic equilibrium. By the end of an election series, most votes are consistent with a single equilibrium, although that equilibrium varies by experimental group and voting rule.

Optimal sophisticated voting strategies in single ballot elections involving three candidates

A model of a rational, sophisticated voter is developed based on the assumption that a voter in a three-candidate election acts to maximize the weighted value of the available voting strategies. The model is used to analyze strategic voting behavior in various traditional and more modern single-ballot systems that employ plurality as a decision rule. In each case, the analysis proceeds by first determining the subset of admissable strategies which are undominated by other valid strategies under the assumption that a voter is not risk-averse. The analysis next involves determining isopreference contours that describe the conditions under which a voter is indifferent between a pair of admissable strategies. These contours are interpreted as decision thresholds which mark the boundaries where a voter shifts from one strategy to another and, hence, govern voting behavior in a particular system. Based on these analyses, an insincerity index, which calibrates the extent to which sophistication produces voting that is inconsistent with actual preferences, is computed. In addition, it is determined for each system whether there are conditions where a group of like-minded, independent voters will vote as a block even though no explicit coalition is formed. Finally, assuming that voting is sophisticated and not risk-averse, we analyze the properties of formal equivalence and symmetry between pairs of systems and, in addition, determine whether a system is formally a mixture of other systems.

An experiment on coordination in multi-candidate elections: The importance of polls and election histories

Do polls simply measure intended voter behavior or can they affect it and, thus, change election outcomes? Do candidate ballot positions or the results of previous elections affect voter behavior? We conduct several series of experimental, three-candidate elections and use the data to provide answers to these questions. In these elections, we pay subjects conditionally on election outcomes to create electorates with publicly known preferences. A majority (but less than two-thirds) of the voters are split in their preferences between two similar candidates, while a minority (but plurality) favor a third, dissimilar candidate. If all voters voted sincerely, the third candidate — a Condorcet loser — would win the elections. We find that pre-election polls significantly reduce the frequency with which the Condorcet loser wins. Further, the winning candidate is usually the majority candidate who is listed first on the poll and election ballots. The evidence also shows that a shared history enables majority voters to coordinate on one of their favored candidates in sequences of identical elections. With polls, majority-preferred candidates often alternate as election winners.

Strategic Electoral Choice in Multi-Member Districts: Approval Voting in Practice?

The chief aim of this paper is to elucidate the theory of electoral choice in an important but previously neglected electoral context: multi-member districts. Taking the simplest case of a threecandidate election in a double-member district, the paper develops a model of electoral choice as a standard decision under risk. It is found that this simplest case is essentially equivalent to approval voting with three candidates; thus, voting in double elections with three candidates is in the same sense that approval voting is, and a unique sincere strategy exists when preferences are dichotomous. Nonetheless, it is shown that electoral choice in double (and approval) elections is inherently strategic in the sense that voters' beliefs about how others will vote affect their decisions (when preferences are not dichotomous). This point is important because it has been claimed that sincerity of choice is an important property, guaranteeing that choices reflect preferences directly. This paper shows that there are fairly severe restrictions on the sense in which this is true, due to the strategic nature of choice. In the later sections of the paper, an examination of actual voting behavior in double-member districts is made to see if theoretically predicted strategic behavior is evident in practice. The evidence is positive.

U.S. Presidential Elections 1968-80: A Spatial Analysis

A methodology is presented for empirical estimation of spatial models of voting in mass elections. The basic choice model posits utility functions that depend on spatial distance and a random error term. The parameters of the utility functions are estimated by a polytomous logit model, which is applied to spatial maps of voters and candidates for all presidential elections since 1968. These maps indicate that presidential candidates take positions at the periphery of the distribution of voters. Results based on these maps support the hypothesis of sincere, spatial voting in two-candidate elections. In three-candidate elections, the coefficients of the voter utility function adjust to proxy for the lesser viability of the minor party candidate. In contrast to choice, turnout shows only weak spatial effects. The data are inconsistent with the proposition that turnout is low because voters are not offered distinct alternatives. While the utility function used by voters appears stable through time and candidate and voter positions appear stable in the last few months prior to an election, voter and candidate positions exhibit substantial change over longer time periods.

Deducing majority candidates from election data

An election among m ⩾ 3 candidates is conducted by system s ⊆ {1, 2, …, m −1} when each voter can vote for any k candidates so long as k ∈ s and the winner is the candidate who receives the most votes. Suppose the available data for an election conducted by s consists of the total number n of voters plus the number of votes n i received by candidate a i for i = 1, …, m, and for definiteness assume that n 1> n 2 ⩾ … ⩾ n m so that a 1 wins the election. The winner a 1 is defined to be a majority candidate if for each i > 1 more voters prefer a 1 to a i than prefer a i to a 1. Question: Given m and s, what must be true of the available election data so that the winner is certain to be a majority candidate? Assuming that a voter can have any weak preference order on the candidates but is not indifferent among all m candidates, and assuming that each voter might vote in any manner that is not clearly contrary to his own interests, the following answers to the Question are derived for m = 3 and m = 4. The only s in a three-candidate election that can guarantee that a 1 is a majority candidate is the approval voting system s = {1, 2}, and it can make this guarantee if and only if n 1 > n 2 + min{n 2, n 1 + n 2 + n 3 − n} . There are two systems for four-candidate elections that can guarantee that a 1 is a majority candidate, namely s = {1, 2} and s = {1, 2, 3}, and for each of these a 1 must be a majority candidate if and only if n 2 + n 3 + n 4 < n 2 . By implication, the plurality system s = {1} can never assure one that the winner is a majority candidate. Stronger assumptions than those used in the main analysis, which are able to ensure that the plurality winner is a majority candidate, are identified.

Social homogeneity and Condorcet's paradox

This paper has examined the relationship between social homogeneity measured by σ(p) = p12 + ... + p62 and the likelihood of Condorcet's paradox. Attention was restricted to three-candidate elections. It was shown first that the most general restriction on p vectors that produces a definite inverse relationship between σ(p) and the limit-in-voters probability P∞(p) of Condorcet's paradox is the dual culture restriction. We then deleted this restriction to allow any p vector and considered the relationship between σ(p) and the paradox probability when Abrams' positioning effect was removed by averaging the Pn(p) values all over rearrangements of the components of p. The resultant averaged probability of Condorcet's paradox with n voters was denoted as Qn(p). Theorem 1 showed that there are p vectors for all odd n ≥ 3 which deny a definite inverse relationship between Qn(p) and σ(p). However, Theorem 2 verified for n = 3 that the intervals of possible Qn(p) values for fixed values of σ(p) decrease as σ(p) increases. It was shown also that the latter relationship does not hold for large n although there is a partial tendency for Qn(p) to decrease as σ(p) increases for large n.