Susceptibility to strategic voting: a comparison of plurality and instant-runoff elections1
Andrew C. Eggers†
Tobias Nowacki†
Abstract
Reformers and researchers often speculate that some voting systems are more sus- ceptible to strategic voting than others, but existing measures are unsatisfying: they essentially ask how likely a voter is to regret a sincere vote, not how likely a voter is to anticipate that an insincere vote is beneficial. We propose a new approach to assessing susceptibility to strategic voting (and other voting system properties) under more realistic assumptions about uncertainty and voter behavior, and we use it to compare three-candidate plurality and instant-runoff (IRV) elections. Drawing on preference data from 160 surveys, we estimate that being a strategic voter (vs. always voting sincerely) is between 5 and 35 times more beneficial in plurality than in IRV. IRV is less susceptible to strategic voting both because insincere votes are more risky in IRV and because strategic behavior by other voters tends to reduce the benefit of being strategic.
1This version: January 17, 2020. Thanks to Steve Callander, Gary Cox, Justin Grimmer, Matias Iaryczower, Paul Klemperer, Dimitri Landa, Gregory Martin, David Myatt, Karine van der Straeten, and seminar audiences at Trinity College Dublin, the Toulouse School of Economics, the Democracy and Policy Lab at Stanford University, and the ECPR Standing Group on Strategic Interactions for valuable feedback. Thanks to Mats Ahrenshop, Tak Huen Chau, and Tom Robinson for research assistance. Earlier versions were circulated starting June 2019 under the title “Comparing strategic voting incentives in plurality and instant-runoff elections”. †Nuffield College and Department of Politics and International Relations, University of Oxford, UK. email: [email protected] †Stanford University. email: [email protected]
1 1 Introduction
What voting method should we use to choose a leader, a representative, or a policy from a set of multiple alternatives? This question is relevant not just to constitutional designers and voting theorists but also to voters who have in recent years been asked to evaluate electoral reforms in large-scale referendums in the U.K., British Columbia, and U.S. states and cities.2
While there are certainly other factors to consider, many voting theorists argue that we should prefer voting methods that are less susceptible to strategic voting (e.g. Dummett, 1984;
Chamberlin and Featherston, 1986; Brams and Fishburn, 2002; Tideman, 2018), because (among other reasons) strategic voting can make election results difficult to interpret (Satterthwaite,
1973; Green-Armytage, Tideman and Cosman, 2016) and requires some voters to forego the evident expressive benefits of a sincere vote (Hamlin and Jennings, 2011; Pons and Tricaud,
2018). Accordingly, a large body of research has attempted to measure the susceptibility of various voting methods to strategic voting. Following Gibbard(1973) and Satterthwaite(1975)’s proofs that any reasonably fair voting system might produce a manipulable election result (i.e. a result where at least one voter could get a better outcome by casting an insincere vote), researchers have sought to estimate how often each voting system would produce manipulable election results (e.g. Chamberlin and Featherston, 1986; Saari, 1990; Favardin and Lepelley,
2006; Ornstein and Norman, 2014; Green-Armytage, Tideman and Cosman, 2016; Tideman,
2018). Studies in this manipulability literature suggest, for example, that plurality elections
(as used in the U.K. or U.S.) are far more susceptible to strategic voting than elections held under instant-runoff rules (also known as the Hare system, the alternative vote, or ranked-choice voting, as used in Australia).
One key problem with previous efforts to measure susceptibility to strategic voting is that they ignore the uncertainty voters face when they make their voting decision. In essence, the manipulability literature asks how often a voting result could be manipulated by a voter
(or group of voters) who knows exactly how other voters have voted. Ignoring uncertainty simplifies the analysis, of course, and it is appropriate for estimating how often voters would
2The UK held a national referendum on electoral reform in 2011, British Columbia held three electoral reform referendums between 2005 and 2018, the U.S. state of Maine adopted instant-runoff voting for federal elections by referendum in 2016, and New York City did so for municipal primaries and special elections in 2019.
2 regret a sincere vote in a given voting system, which may be an important consideration. But there is good reason to care about the extent to which voters might anticipate the need to cast an insincere vote in a given voting system, not only whether they might regret a sincere vote.
If (like Riker, 1982) we are concerned that strategic voting might make election results difficult to interpret, for example, we want to know whether voters (or the elites who mobilize them) would be likely to perceive ex ante an opportunity to benefit from an insincere vote; answering that question requires us to consider the strategic voting problem that voters face before the election, when results are uncertain. (As we discuss further below, there are circumstances when a voting system could appear highly manipulable ex post but not ex ante, so the two approaches may yield different answers.) For this and other reasons the manipulability literature offers an incomplete measure of voting systems’ susceptibility to strategic voting. What is missing is a measure of how much the typical voter would benefit ex ante from an insincere vote given reasonable assumptions about voter uncertainty and voter behavior.
In this paper we introduce a new way of assessing susceptibility to strategic voting that improves on previous work in three principal ways. First, to characterize typical voter prefer- ences we draw on 160 election surveys across 56 democracies; like Green-Armytage(2014) and
Green-Armytage, Tideman and Cosman(2016) we use voters’ numerical evaluations of parties as utility measures, but we draw on a larger set of preference distributions in order to produce more broadly representative results. Second, we consider the uncertainty voters face when they decide to vote in advance of elections. Instead of asking whether our survey respondents could benefit from an insincere vote given perfect knowledge of others’ votes, we compute for each respondent the expected utility of each possible vote given a reasonable model of beliefs over election outcomes; crucially, these beliefs are realistically uncertain, with the precision cali- brated from empirical work on voters and forecasters in recent UK elections (Fisher and Myatt,
2017; Eggers and Vivyan, 2020). The susceptibility of a voting system to strategic voting can then be measured by the average gain in expected utility each voter would receive by switch- ing from always voting sincerely to always casting the optimal strategic (i.e. highest expected utility) vote. Third, we consider how voters’ incentive to vote strategically depends on other voters’ strategic behavior. We begin by assuming (as does the manipulability literature) that
3 voters expect other voters to vote sincerely; we then use a novel iterative polling algorithm to compute strategic voting incentives when voters expect other voters to vote strategically to varying degrees. This allows us to check how strategic voting incentives are shaped by (beliefs about) strategic voting by other voters, which we argue may be an important determinant of a voting system’s susceptibility to strategic voting.
We use this approach to compare strategic voting incentives in elections held under plurality
(the most common single-winner voting system in the world) and instant-runoff voting (IRV), a comparison that is particularly relevant because of recent referendums in both the U.K. and
U.S. in which voters were asked to choose between the two systems (see footnote2). In contrast to plurality elections, in which each voter casts a ballot for a single candidate and the winner is the candidate with the most votes, voters in IRV elections rank the candidates and the winner is determined by successively eliminating less-popular candidates.3 IRV variants are used to elect legislators at the state and federal level in Australia, presidents of Sri Lanka, India, and Ireland, mayors of Sydney, London, San Francisco and several other American cities, and (as of 2018) all federal offices in the U.S. state of Maine. IRV’s advocates sometimes argue that it is immune to strategic voting,4 but critics are quick to point out circumstances in which a savvy voter could benefit from an insincere vote in an IRV election (e.g. Fishburn and Brams, 1983; Dummett,
1984). The manipulability literature finds IRV to be among the least ex post manipulable voting methods and plurality to be among the most ex post manipulable (e.g. Chamberlin and Featherston, 1986; Green-Armytage, Tideman and Cosman, 2016), but no previous study systematically compares the two voting methods in a way that takes into account uncertainty or that considers how other voters’ strategic behavior shapes strategic voting incentives.
3Most commonly, in each round the candidate receiving the fewest top rankings is eliminated from all ballots until one candidate remains (or, equivalently, until one candidate has a majority of top rankings). In the system used to elect the mayor of London and and the president of Sri Lanka (sometimes called the “contingent vote” or “supplementary vote” system), all candidates but two are eliminated immediately. The other variation within the family of instant-runoff systems is how many candidates voters may rank (up to two in the London mayoral election, up to three in San Francisco, as many as desired in some Australian states, and all in other Australian states). In the three-candidate case (our focus), the elimination procedures are identical and the number of candidates voters rank makes little or no difference. 4According to FairVote, a US advocacy organization, voters in IRV elections “can honestly rank candidates in order of choice without having to worry about how others will vote and who is more or less likely to win.” https://www.fairvote.org/rcv#rcvbenefits, visited 11 June 2019. Similarly, UK Deputy Prime Minister Nick Clegg testified in advance of the 2011 UK referendum that IRV “stops people from voting tactically and second- guessing how everybody else will vote in their area.” “The Coalition Government’s programme of political and constitutional reform: Oral and written evidence”, 15 July 2010, HC 358-i, published 22 October 2010 (link).
4 Our analysis shows that plurality is indeed more susceptible to strategic voting than IRV, and that previous literature has understated the gap in susceptibility between the two systems by ignoring uncertainty and the effect of strategic behavior on election outcomes. In particular, a voter who switches from being a sincere voter to a strategic voter gains between 5 and 35 times more expected utility on average in plurality than in IRV, with the gap being smallest in the situation closest to that studied by the manipulability literature: the more prevalent strategic voting is in the electorate, and the less precise voters’ information is, the larger the ratio of average strategic voting incentives in plurality and IRV. IRV is less susceptible to strategic voting than plurality even when we assume (as might be the case) less strategic behavior and more uncertainty in IRV than in plurality. To the extent that the objective reward from strategic voting (as perceived either by voters or by elites who mobilize voters) is an important reason why voters engage in it, our analysis implies that strategic voting would be considerably less widespread on average under IRV than under plurality even if there were no difference in the complexity of the task or the quality of information available.
Beyond estimating the difference in average strategic voting incentives between plurality and IRV, we highlight an important qualitative difference in how strategic voting incentives interrelate in these two systems. In an environment where voters respond to public polling information, strategic voting in plurality elections is characterized by positive feedback: the more a candidate’s supporters are expected to desert that candidate, the greater the incentive for other supporters to do the same (on average). In IRV, by contrast, strategic voting is predominantly governed by negative feedback: the more voters with a given preference are expected to respond strategically to a poll, the lower the reward for like-minded voters from doing the same (on average). We argue that this negative feedback constrains the likely role of strategic voting in IRV elections, because once some voters are expected to engage in it the incentive for others to join in disappears. We speculate that this distinction between positive and negative feedback is more generally important in determining how strategic voting works in different voting systems.
We emphasize that the methods developed in this paper are useful not just for studying susceptibility to strategic voting in plurality, IRV, and other systems not examined here, but
5 also for other properties of voting systems. For example, many studies in the manipulability literature also compare voting systems according to the probability of electing the Condorcet winner or the aggregate utility from the election outcome (e.g. Chamberlin and Featherston,
1986; Green-Armytage, Tideman and Cosman, 2016). Crucially, these studies assume sincere voting, leaving open the possibility that a given voting system may perform better or worse if voters act strategically. The machinery for analyzing strategic voting we introduce in this paper
(including the iterative polling algorithm) offers a way to study these and other properties of voting systems while relaxing the assumption of sincere voting. We focus on susceptibility to strategic voting in this paper to keep our task manageable, but we plan to develop this approach to assess voting systems more fully in future work.
2 Why does susceptibility to strategic voting matter, and how
should we measure it?
Voting theorists have argued that we should prefer a voting system that is less susceptible to strategic voting (e.g. Saari, 1990; Tideman, 2018), meaning a system that is less likely to reward voters who manipulate the result by submitting an insincere preference. Two types of manipulability are potentially concerning.
First, one might be concerned about the ex post manipulability of a voting system, by which we mean the frequency with which voters might regret a sincere vote after the results are announced. It is natural and unavoidable for voters who favored unsuccessful candidates to feel disappointment with the outcome, but it seems desirable to avoid situations where voters regret having submitted a sincere ballot, as may have been the case for left-leaning Floridians who voted for Ralph Nader in the 2000 U.S. presidential election. Voters who recognize ex post a chance to improve the outcome with an insincere ballot may feel dismay about the vote they or others cast; they may also question the legitimacy of a result that could have been reversed by clever manipulation, particularly when that reversal operates in a perverse way, as when some voters could have obtained a better result by not voting (e.g. Fishburn and Brams, 1983) or by submitting an incomplete ranking of candidates (Fishburn and Brams, 1984).
6 One can also be concerned about the ex ante manipulability of a voting system, meaning the frequency with which voters could obtain a better expected election outcome by submitting an insincere vote at the polls. A system that is more ex ante manipulable raises several concerns.
To the extent that voters respond to the opportunity to manipulate by submitting insincere votes, the submitted ballots may become more disconnected from the sincere preferences of the electorate, making the election difficult to interpret (Satterthwaite, 1973; Riker, 1982; Green-
Armytage, Tideman and Cosman, 2016): did the voters who voted for candidate A really prefer that candidate, or did they vote for A for strategic reasons? Voters also evidently derive ex- pressive benefits from voting for candidates they sincerely support (Hamlin and Jennings, 2011;
Pons and Tricaud, 2018), so a system that induces widespread insincere voting also deprives many voters of these expressive benefits. Furthermore, there is evidence that some types of voters (e.g. poorer ones) are less able or inclined to vote strategically (Eggers and Vivyan, 2020;
Eggers, Rubenson and Loewen, 2019), which suggests that a system that is more ex ante ma- nipulable disadvantages these voters to a greater extent.5 Finally, a voting system that is more ex ante manipulable creates stronger incentives for voters to pay attention to polls, for media outlets to conduct polls, and for parties to spread polling information (and misinformation); all of this effort might be better spent on other activities, such as scrutinizing candidates’ track records or policy proposals (though see Dowding and Van Hees, 2008). Interpretability, expres- sive utility, fairness, and efficiency arguments thus all favor voting systems that are less ex ante manipulable.
Perhaps surprisingly, previous research on susceptibility to strategic voting focuses exclu- sively on ex post manipulability. The Gibbard-Satterthwaite Theorem states that any reason- able voting system is ex post manipulable in some circumstance; the manipulability literature
(e.g. Chamberlin and Featherston, 1986; Saari, 1990; Favardin and Lepelley, 2006; Ornstein and
Norman, 2014; Green-Armytage, Tideman and Cosman, 2016; Tideman, 2018) quite reasonably builds on this result by measuring the proportion of likely election outcomes that are ex post manipulable (assuming sincere voting). More precisely, both the manipulability literature and
5It remains to be seen how heterogeneity in strategic behavior varies across more and less ex ante manipulable voting systems, but a voting system in which essentially no one submits insincere votes naturally would have smaller heterogeneity in observed strategic behavior.
7 the social choice literature from which it emerges assume perfect information about the elec- tion result, which describes the situation after the election takes place but (apart from special circumstances in small committees6) is clearly unrealistic as a description of the environment in which voters typically decide how to vote before the result is announced. The manipulability literature is also unsatisfying because it examines (ex post) manipulability assuming that every- one votes sincerely, which is unrealistic at least in plurality elections (where strategic voting is known to be prevalent) even when the assumptions about underlying preferences are realistic.
The lack of research into the ex ante manipulability of voting systems would not be a problem if we knew that the ex ante and ex post approaches produced the same conclusions, but this is not the case. For example, consider a plurality election in which candidates A, B, and C receive nearly equal support, with candidate A defeating candidate B by just one vote and C finishing slightly behind B. Clearly ex post manipulation is possible: for example, two supporters of candidate C who prefer B over A could elect B by switching their votes from C to B. But given a poll predicting the same near three-way tie, there is little ex ante reason for these voters to switch to B: each pair of candidates is approximately equally likely to be tied for
first, so switching from C to B is just as likely to backfire (in the event of a B-C tie) or to lead to a wasted vote (in the event of an A-C tie) as it is to pay off (in the event of an A-B tie). More generally, a system that creates conflicting voting incentives (i.e. that rewards an insincere vote in one set of circumstances but punishes it in another similar set of circumstances) may look more manipulable ex post than ex ante, essentially because the ex ante perspective considers both the benefits and risks of an insincere vote.
To remedy this gap in previous research, we seek to estimate the prevalence and strength of the ex ante incentive to vote strategically in different voting systems given realistic assumptions about voters’ preferences and beliefs. How much could voters expect to benefit from being strategic (i.e. consulting their beliefs about likely outcomes and voting accordingly) rather than simply voting according their true preferences in each system? What proportion of voters would optimally submit an insincere vote?
Ideally, we would address these questions with a large randomized control trial in which
6Even in committees where voters communicate with each other extensively there may be uncertainty due to voters’ incentive for untruthful communication.
8 we randomly assign voting systems to a large number of polities, allow time for voters and candidates to adapt to their assigned system, and then compare strategic voting incentives across systems given voters’ preferences and reasonable beliefs. Obviously this heroic experiment is not possible.
Alternatively, we could examine strategic voting incentives in actual elections.7 Here the main problem is that (particularly for voting systems not in wide use) it is difficult to know whether apparent differences in strategic voting incentives or behavior are due to the voting system itself or other factors that differ across settings employing different voting systems. In the case of IRV, for example, one can examine the large number of Australian elections, but we may wonder whether measurements from an essentially two-party system are appropriate for inferring likely strategic voting incentives under IRV in other settings.8 Research on two- round elections in France (Blais, 2004; Dolez and Laurent, 2010) indicates that very few (if any) voters attempt to elect a preferred candidate by supporting someone else in the first round (a potentially important type of strategic behavior in any runoff system, including IRV), but it is unclear whether this is because (i) such a vote is not optimal (given information available before the election), (ii) voters fail to perceive the consequentialist benefits of such a vote, or (iii) voters have non-consequentialist concerns (e.g. valuing honesty) that override these consequentialist benefits.
Instead, we study strategic voting incentives in plurality and IRV using a thought experi- ment that, as noted in the introduction, resembles manipulability research in some ways while departing in key respects. Our objective is to measure, for voters with realistic preferences and reasonable beliefs about likely election results, the expected benefit of voting strategically
(i.e. casting the expected utility maximizing vote) vs. voting sincerely. We use party ratings from recent election surveys as the basis for preferences in both plurality and IRV, building on work in the manipulability literature that draws on similar data (Chamberlin and Featherston,
1986; Green-Armytage, 2014; Green-Armytage, Tideman and Cosman, 2016).9 We then mea-
7One can also turn to the lab (e.g. Blais et al., 2016; Hix, Hortala-Vallve and Riambau-Armet, 2017). 8Apart from instances where ex post manipulability is evident discussed in Farrell and McAllister(2006) and blog posts (e.g. Kevin Bonham, “Oh Yes We Do Have Strategic Voting In Australia (Sometimes)”, 19 October 2018 http://kevinbonham.blogspot.com/2018/10/oh-yes-we-do-have-strategic-voting-in.html), there is surprisingly little research on strategic voting in Australian elections. 9See also Plassmann and Tideman(2014)’s empirically calibrated spatial models (and Chamberlin(1985)’s
9 sure strategic voting incentives for each voter in these surveys given beliefs that are reasonable in two key respects. First, the precision of these beliefs is calibrated from recent research on voter behavior (Fisher and Myatt, 2017) and forecasting errors (Eggers and Vivyan, 2020).
Second, the location of these beliefs is consistent with the preferences in the survey and a range of assumptions about the prevalence of strategic voting among other voters. Specifically, our iterative polling algorithm traces out a sequence of polls, where the first poll reports voters’ sincere preferences, the second poll is a mix of the first poll and voters’ best responses to the
first poll, the third poll is a mix of the second poll and voters’ best responses to the second poll, etc. This sequence can be viewed as (i) a model of how expectations would evolve over several polls or elections, (ii) a model of beliefs held by voters with different levels of strategic sophistication (where poll k represents the beliefs of level-k agents), or (iii) a way of locating a strategic voting equilibrium. By computing strategic voting incentives at each step in the sequence of beliefs we can show how susceptibility to strategic voting depends on how strategic other voters are expected to be.
Beyond the manipulability literature described above, this paper speaks to three other strands of research on strategic voting in related fields. Beginning with Bartholdi, Tovey and
Trick(1989), computer scientists have studied the complexity of manipulation in various voting systems (see Faliszewski and Procaccia, 2010, for a review), focusing on how the time to com- pute the optimal strategic vote increases with more voters and candidates. Bartholdi and Orlin
(1991) showed that computing the optimal vote in IRV is uniquely complex (in that sense), which Cox(1997, p. 94) interpreted as evidence that voters need more information to vote strategically in IRV. But complexity research (like previous manipulability research) assumes that others’ votes are known (though see Conitzer, Sandholm and Lang, 2007), so it speaks most directly to the possibility of ex post manipulation, i.e., the difficulty of determining whether to regret one’s vote after the election takes place.10 Computer scientists have also written on much earlier work), which have the advantage of being parameterized. By holding preferences fixed across voting systems, we assume away possible effects of the voting system on the distribution of preferences, a shortcoming shared by the entire manipulability literature 10Of course, if working out the optimal vote is hard in the complexity sense even when voters know others’ votes with certainty, it should be even harder when there is uncertainty, but Conitzer, Sandholm and Lang(2007)’s Proposition 1 shows that manipulation given perfect information is not hard even in IRV with a fixed number of candidates, so the “even in the best case” argument loses force.
10 “iterative voting”, which (like our iterative polling algorithm) refers to a procedure in which agents first vote sincerely and then make myopic strategic adjustments (see Meir, 2018, for a review); the difference is that our agents adjust their voting intention based on imprecise beliefs centered at previous poll results, whereas their agents know exactly how others have voted and adjust their votes to achieve a (known) better result. Finally, our work speaks to game theo- retic models of elections, which unlike the previously mentioned research does consider voter uncertainty (Myerson and Weber, 1993; Myatt, 2007; Laslier, 2009; Bouton, 2013; Bouton and
Gratton, 2015). While we do not produce a general characterization of equilibria, our itera- tive polling algorithm yields reasonable predictions (including at an equilibrium) from arbitrary preference inputs under realistic aggregate uncertainty, which permits comparison across voting systems even when multiple equilibria (or no equilibria) exist.11
3 A new way to measure susceptibility to strategic voting
In this section we formally introduce the approach described above. Readers not interested in the technical details may want to skip ahead to Section4, where we present results.
The paradigmatic strategic voter forms preferences about candidates, develops beliefs about likely election outcomes, computes pivot probabilities (Myerson and Weber, 1993) based on these beliefs, and finally computes the expected utility of each possible ballot (where the opti- mal strategic vote is the one that yields the highest expected utility). Our presentation follows these steps in reverse: we show how to compute the expected utility of each ballot in each vot- ing system assuming preferences and pivot probabilities; then we show how to compute pivot probabilities given beliefs about likely election outcomes; next we introduce a new iterative polling algorithm that we use to infer reasonable beliefs for a given voting system from a pref- erence distribution and assumed level of belief precision; and finally we describe the preference distributions and precision parameter that are the empirical inputs of our analysis.
11Our algorithm focuses on the plurality equilibrium with the lowest strategic voting incentives. In that sense it yields the lower bound on strategic voting incentives in plurality, which is informative given that the strategic voting incentive is higher in plurality across all parameter values we investigate.
11 3.1 Computing the expected utility of each ballot from preferences and pivot probabilities
Suppose n voters participate in an election to choose a winner from a set of candidates denoted
C. We assume that these voters have Von Neumann-Morgenstern utility functions defined over the candidates, with ui,c denoting the utility of voter i from the election of candidate c ∈ C. We can organize these utilities into a utility matrix U with one row per voter and one column per candidate; for example, given candidates {A, B, C}, U is
u1A u1B u1C u2A u2B u2C U = . . . . . . . unA unB unC
We also assume that each voter is uncertain about how other voters will vote but all voters share a common belief about the probability of each possible election result, including the election results in which a single ballot could be decisive in various ways, i.e. pivot events. Let
B be the set of all permissible ballots (i.e. distinct votes that can be cast) in the voting system.
Let pc,b be the probability that candidate c is elected given one additional ballot b ∈ B is submitted, and organize these probabilities into an election probability matrix P with one row per candidate and one column per ballot (so its dimensions are |C| × |B|). Then the expected utility of each voter from submitting each possible ballot is the expected utility matrix U = UP with n rows (one row per voter) and one column per ballot. From the expected utility matrix we can compute the optimal (strategic) ballot for each of our n voters as well as the difference in each voter’s expected utility between casting the optimal ballot and casting a sincere ballot, which is a measure of the voter’s strategic voting incentive. Studying strategic voting incentives in any voting system given voters’ preferences and beliefs is thus essentially a problem of assembling the election probability matrix P. We now show how to do this in three-candidate plurality and IRV elections given the probability of pivot events (i.e. pivot probabilities); later we show how to compute these pivot probabilities given beliefs.
12 The P matrix in plurality
In plurality, voters submit ballots naming one candidate, so the set of admissible ballots is the set of candidates. Given candidates {A, B, C}, the P matrix is
pA,A pA,B pA,C P = p p p , B,A B,B B,C pC,A pC,B pC,C
where e.g. pB,A indicates the probability B is elected when one votes for A. Let an election result be written as a vector v = (vA, vB, vC ), with e.g. vA indicating the share of ballots naming candidate A. We assume throughout that voters consider v to be a continuous random variable, with beliefs summarized by pdf f(v); this simplifies the analysis by eliminating the possibility of ties. Given a total electorate of size N,12 the probability that a single ballot could change the plurality winner from candidate j to candidate i is then