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Public Choice (2014) 161:1–9

DOI 10.1007/s11127-013-0118-2

Frequency of monotonicity failure under Instant Runoff

Voting: estimates based on a spatial model of elections

Joseph T. Ornstein ·Robert Z. Norman

Received: 5 October 2012 / Accepted: 7 September 2013 / Published online: 17 October 2013

© Springer Science+Business Media New York 2013

Abstract It has long been recognized that Instant Runoff Voting (IRV) suffers from a defect

known as nonmonotonicity, wherein increasing support for a candidate among a subset of

voters may adversely affect that candidate’s election outcome. The expected frequency of

this type of behavior, however, remains an open and important question, and limited access

to detailed election data makes it difﬁcult to resolve empirically. In this paper, we develop

a spatial model of voting behavior to approach the question theoretically. We conclude that

monotonicity failures in three-candidate IRV elections may be much more prevalent than

widely presumed (results suggest a lower bound estimate of 15 % for competitive elections).

In light of these results, those seeking to implement a fairer multi-candidate election system

should be wary of adopting IRV.

Keywords Voting theory ·Instant Runoff Voting ·Agent-based modeling ·Monotonicity

JEL Classiﬁcation D72

1 Introduction

Instant Runoff Voting (IRV) is a ranked-ballot voting system that selects a single winner

by successively eliminating the candidate with the fewest ﬁrst place votes until a single

candidate receives a majority. IRV, like many systems that employ a successive-elimination

procedure, violates the monotonicity criterion (Smith 1973), meaning there exist some con-

ditions under which increasing the support for a candidate (without changing the voters’

rankings for any of the other candidates) would be harmful to that candidate.

J.T. Ornstein (B)

Department of Political Science, University of Michigan, 5700 Haven Hall, 505 South State Street,

Ann Arbor, MI 48109-1045, USA

e-mail: ornstein@umich.edu

R.Z. Norman

Department of Mathematics, Dartmouth College, 340 Kemeny Hall, Hanover, NH 03755, USA

2 Public Choice (2014) 161:1–9

Fishburn and Brams (1983) describe a paradox that may result from this defect, which

they term the “more-is-less paradox”. This paradox, which we will refer to as a monotonicity

failure, is characterized as a situation in which the IRV winner would lose if ranked higher

by some subset of voters. The converse, an election in which the IRV loser would win if

ranked lower by some subset of voters, is also a type of monotonicity failure. We term the

former paradox an upward monotonicity failure, and the latter a downward monotonicity

failure. In this paper, we will focus exclusively on the prevalence of upward monotonicity

failures under IRV, leaving downward monotonicity failure as a topic for future research.

How often could we reasonably expect to observe this behavior in real-world IRV elec-

tions? One study (Allard 1996) suggests that, if the United Kingdom adopted IRV for its

general elections, this paradox would occur only about once per century. If this were true,

then the monotonicity criterion would be cause for little practical concern. However, more

recent theoretical work on the topic (Lepelley et al. 1996;Norman2006,2012; Miller 2012)

ﬁnds that the proportion of possible IRV elections that exhibit a monotonicity paradox is

non-trivial.

Here we build on this body of work by estimating IRV’s monotonicity failure rate in the

context of a spatial model of voter behavior. This method allows us to avoid the assumption

that all electoral outcomes are equiprobable, and to explore the conditions under which IRV

may be more or less vulnerable to monotonicity failure.

2 The 2009 Burlington mayoral election: a case study

Before introducing the model, we present a case study from the 2009 mayoral election in

Burlington, VT, which illustrates the key features of an upward monotonicity failure. Table 1

reports the ballot results for the three candidates remaining before the ﬁnal elimination—the

Republican (R), Democrat (D), and Progressive (P).

The Republican candidate had the most ﬁrst-place votes, with 3,297. The Democrat had

2,554 ﬁrst-place votes, and the Progressive incumbent had 2,982. Although the Democrat

was the Condorcet winner (a majority of voters preferred him in all two way contests), he

received the fewest ﬁrst-place votes and so was eliminated. After the Democrat’s votes were

redistributed to the other two candidates, the Progressive won the election 4,314 to 4,067.

It was a very competitive election, as the winner’s margin of victory was 247 votes (2.8 %

of the electorate). A small shift in support from the Progressive to the Republican would have

resulted in a Republican victory. But what if Burlington’s electorate had been composed of

even more Progressive voters? If we shift the Progressive candidate up in 750 rankings, we

can construct the following election proﬁle (Table 2, changes in bold).

Tab le 1 Ballot results from the 2009 Burlington, VT mayoral election (Laatu and Smith 2009)

Ranking 1513 495 1289 1332 767 455 2043 371 568

1st R RR D DDP P P

2nd DP P R DR

3rd P D R P R D

Each column represents a unique rank-order ballot cast by voters, and the top row denotes the number of vot-

ers who submitted that ballot. For instance, the ﬁrst column denotes that 1513 voters ranked the Republican

ﬁrst, the Democrat second, and the Progressive third. In the Burlington election, voters were permitted to sub-

mit incomplete rankings (a “truncated ballot”). Though this form of IRV is also susceptible to monotonicity

failure, for simplicity of analysis we do not include it in the model.

Public Choice (2014) 161:1–9 3

Tab le 2 Hypothetical ballot results for Burlington election

Ranking 1513 195 839 1332 767 455 2043 1121 568

1st R RRDDDP PP

2nd D PPR DR

3rd P DRP RD

As a result of this mass shift in voting, the ﬁrst-place vote totals would nowbe: R—2,547,

D—2,554, and P—3,732. If the Progressive won the election in which those 750 voters

supported the Republican instead of him, then a monotonic voting system would award him

this election as well. However, in this hypothetical election, the Republican is eliminated

instead of the Democrat, and after redistributing the Republican’s votes, the Progressive

loses to the Democrat 3,927 to 4,067.

The Burlington election offers a compelling illustration of monotonicity failure’s prac-

tical importance, but such detailed IRV ballot data are rare. Therefore, in order to estimate

the frequency of its occurrence, we rely in this paper on a spatial model of elections. In the

following section, we describe the conditions under which a three-candidate election will

exhibit an upward monotonicity failure, which we use for the analysis of the model.1

3 Necessary and sufﬁcient conditions for monotonicity failure

The outcome of any election conducted using Instant Runoff Voting can be represented by

a vector P, termed an election proﬁle. Each element in Pis a non-negative integer denoting

the number of voters who cast a particular rank-ordered ballot. The sum of all elements in

Pis V, the number of voters. For three-candidate elections, we denote the candidates A,B,

and C,whereAis the IRV winner and Cis the candidate with the fewest ﬁrst-place votes.

Therefore the elements in Pare a1,a

2,b

1,b

2,c

1,c

2as in Table 3.

An election proﬁle Pexhibits an upward monotonicity failure if there exists a proﬁle

Pthat is identical to Pexcept that candidate Ais ranked higher by a subset of voters, but

candidate Ais not the IRV winner. Formally, there must exist non-negative integers λ1and

λ2such that we can construct an election proﬁle

P=a1+λ1+λ2,a

2,b

1−λ1,b

2−λ2,c

1,c

2

in which candidate Ais not a majority winner, candidate Bis eliminated, and candidate

Cis the IRV winner. According to our deﬁnitions from Sect. 1,proﬁlePwill exhibit a

downward monotonicity failure. There are two conditions which, jointly, are necessary and

sufﬁcient for the existence of P,shownin(1)and(2)below.

The ﬁrst is that Pmust be a competitive election, deﬁned as a proﬁle in which2

c>V+2

4(1)

1Our analysis here closely parallels work by Lepelley et al. (1996) and Miller (2012), but differs in its con-

struction of condition (1). In our analysis of the spatial model (Sect. 4), we ignore cases where two candidates

tie for fewest ﬁrst-place votes, so we include a stronger version of condition (1) than in these previous papers.

2For convenience, we deﬁne c=c1+c2.

4 Public Choice (2014) 161:1–9

Tab le 3 A three-candidate

election proﬁle Ranking a1a2b1b2c1c2

1st choice A A B B C C

2nd choice B C A C A B

3rd choice C B C A B A

The second condition necessary for Pto exhibit an upward monotonicity failure is that

candidate Cmust be preferred over Aby a majority of voters. This implies that, absent a

tie, either candidate Cis the Condorcet winner, or that there is no Condorcet winner (the

election exhibits a majority cyclic triple). This condition can be expressed by:

c+b2>V

2(2)

The necessity of condition (1) follows from the requirement that candidate Acannot receive

a majority of ﬁrst-place votes or tie under P. Formally

(b1−λ1)+(b2−λ2)+c>V

2(3)

Because we’ve also speciﬁed that candidate Bis eliminated under P, we can combine

(b1−λ1)+(b2−λ2)≤c−1(4)

with (3), yielding equation (1).

The necessity of condition (2) follows from the requirement that candidate Cwins under

Pwith c+b2−λ2votes. Since λ2is non-negative, it is necessary that Psatisfy (2).

To prove the sufﬁciency of conditions (1)and(2), we will show that when Psatisﬁes

(1)and(2), there must exist an election proﬁle Pin which candidate Breceives fewer

ﬁrst-place votes than C, and candidate Creceives a majority following B’s elimination.

Formally, it will sufﬁce to show that under these conditions there exist non-negative integers

λ1and λ2that satisfy (4)and

c+b2−λ2>V

2(5)

Let λ1=b1and λ2be the largest integer that satisﬁes (5). It follows by deﬁnition that

λ1is non-negative and from (2)thatλ2is non-negative. It follows by contradiction that

these values for λ1and λ2will also satisfy (4), because when (b2−λ2)≥c,(1) implies

c+b2−λ2>V+2

2,soλ2is not the largest integer satisfying (5). This completes the proof.

In the model presented in the following section, we use (1)and(2) to assess the mono-

tonicity failure rate of a set of simulated elections.

4 The model

The model we develop here is a two-dimensional spatial model, a method used widely to

analyze behavior in elections and performance among voting systems (Downs 1957; Merrill

1988; Kenny and Lotﬁnia 2005). Such modeling has also been used to inform disputes about

the merits of IRV in particular (Fraenkel and Grofman 2004; Horowitz 2004).

The positions of candidates and voters are represented by points on a two-dimensional

issue space, and each voter’s preference ranking is constructed by taking the reverse order

Public Choice (2014) 161:1–9 5

of the Euclidean distance to each candidate within the space.3Formally, let nrepresent the

number of issues (n=2 for this paper), let xji represent the ideal point for the jth voter’s ith

issue, and let yirepresent candidate y’s position on the ith issue. Voters rank each candidate

yin increasing order of utility, given by

uj(y) =n

i=1

(xji −yi)2−1/2

(6)

Voters’ ideal points are randomly seeded across the issue space using one of four stylized

distributions. These distributions are not drawn from data, but are constructed to mirror

features from plausible real-world electorates. Such non-empirical distributions allow us to

gauge the model’s behavior over a wider range of scenarios than empirical data alone would

permit.

In the base case of the model, the ideal points on each axis are drawn from a Gaussian

distribution with mean 0 and standard deviation 0.25. In the “polarized” case, ideal points

are drawn at random from one of two bivariate Gaussian distributions, one centered at point

(−0.25,−0.25)and the other centered at (0.25,0.25), each with standard deviation 0.1.

This conﬁguration mimics an election in which voters are split into two polarized camps. In

Sect. 5, we present the model results from two instantiations of this distribution: one where

voters are equally likely to be in either camp (Balanced Polarized), and one where voters

are 1.5 times more likely to position themselves in one camp than the other (Unbalanced

Polarized).

The ﬁnal, “multiparty”, distribution randomly draws half of the voters’ ideal points from

the polarized distributions described above, one-quarter of the voters from distributions cen-

tered at (0.25,−0.25)and (−0.25,0.25), and one-quarter of the voters from a distribution

centered at (0.0,0.0). Each distribution has standard deviation 0.1. This conﬁguration rep-

resents an election in which there are three major camps and two smaller camps. An illus-

tration of these three scenarios is in Fig. 1.

We model candidates as boundedly rational adaptive agents, following (Kollman et al.

1992; De Marchi 1999;Laver2005). Each candidate in the model attempts to maximize the

number of ﬁrst-place votes received, but does not know his or her optimal location given

the locations of other candidates, and receives information only through periodic polling.

All three candidates begin the election at random positions in the voter distribution,4and

each period they determine how many ﬁrst-place votes they would receive if the election

were held immediately. Candidates then change their positions on one or both issues by

a ﬁxed increment if that adjustment would result in a higher ﬁrst-place vote count. The

size of this ﬁxed increment is 0.01, sufﬁciently small such that agents will not “overstep”

an advantageous position. The positions of voters do not change during the course of the

simulation.

Since outcomes are stochastic, we model 5,000 elections for each type of voter distribu-

tion (varying by the number of periods that candidates may move during each election, L).

Each election is parameterized with 1,001 voters, though outcomes are robust to halving

or doubling this value, or setting Veven or odd. The simulation code is available from the

authors on request.

3The qualitative results presented in this paper are robust to alternate speciﬁcations of utility, including city

block distance and squared Euclidean distance.

4In the polarized case, we ensure that there is at least one candidate in each “camp”, and two in the larger

camp.

6 Public Choice (2014) 161:1–9

Fig. 1 Examples of the three voter distributions utilized in the model. The balanced and unbalanced polarized

distributions are visually indistinguishable, and so only one case is illustrated here

5Results

The results from the model suggest that upward monotonicity failures are likely to occur

with signiﬁcant frequency under IRV, and that this frequency increases with competitive-

ness. Depending on the type of voter distribution and length of the simulation, the simulated

elections exhibit monotonicity failure in anywhere from 0.7 % to 51 % of all cases, and

between 15 % and 51 % of competitive elections (excluding ties, which account for approx-

imately 1.1 % of simulated elections).

Monotonicity failure rates for each voter distribution appear to stabilize at higher values

of L(Fig. 2). The Unbalanced Polarized distribution exhibits the highest rate of monotonic-

ity failure (approximately 50 % at L>40), the Base Case and Balanced Polarized distribu-

tions exhibit monotonicity failures in 9 % to 12 % of simulated elections, and the Multiparty

distribution exhibits the most infrequent monotonicity failures (a lower bound of 0.7 %).

The simulation length parameter (L) appears to have a varied effect on the rate of mono-

tonicity failure. As Lincreases, the Base Case and Polarized distributions exhibit compet-

itive elections as deﬁned by (1) more frequently, which in turn results in a higher overall

monotonicity failure rate. By contrast, the proportion of competitive elections tends to de-

crease in the Multiparty distribution as Lincreases, as illustrated by Fig. 3.

In the base case and polarized distributions, candidates will tend to settle on a local equi-

librium given enough time. In the base case, candidates position themselves near the yolk of

the distribution (McKelvey 1986) centered on (0,0). This central positioning increases the

chance of a three-way competitive election. Candidates in the two polarized distributions

tend to locate near the yolks of their respective camps. This invariably leads to a competitive

election in the Unbalanced Polarized case, where the two candidates in the larger camp each

take 30 % of the vote, but rarely results in a competitive election in the Balanced Polarized

case, where two candidates must split roughly 50 % of the vote between them. On average,

50 % of competitive elections in either polarized distribution result in the elimination of the

Condorcet winner, and therefore exhibit a monotonicity failure (Fig. 4).

By contrast, candidates in the Multiparty distribution never settle into a local equilibrium

near the yolk, regardless of simulation length. This is likely because candidates at the center

of the distribution who have captured the vote of one of the smaller peripheral camps have

an incentive to compete for one of the larger peripheral camps instead. This sets off a race

between two candidates to gain the support of the periphery, destabilizing the equilibrium

at the yolk, where elections are three-way competitive. As indicated by Figs. 3and 4,very

few elections conducted with the Multiparty voter distribution are competitive, but those

Public Choice (2014) 161:1–9 7

Fig. 2 Nonmonotonic rate (%) by simulation length (L). Values derived from 5,000 simulated elections for

each value of L(ties excluded)

Fig. 3 Percentage of elections that are competitive as a function of L. Values derived from 5,000 simulated

elections for each value of L(ties excluded)

Fig. 4 Nonmonotonic rate (%) of competitive elections by simulation length (L). Values derived from 5,000

simulated elections for each value of L(ties excluded)

8 Public Choice (2014) 161:1–9

Fig. 5 Nonmonotonic rate (%) of elections by competitive ratio. This chart is derived from 500,000 runs of

the simulation for each distribution. Each point illustrates the rate of monotonicity failure for elections with

competitive ratio between xand (x+0.05)

that are competitive exhibit monotonicity failure roughly 20 % of the time. Indeed, com-

petitive simulated elections exhibited montonicity failure at least 15 % of the time for all

parameterizations (Fig. 4).

In addition to reporting overall monotonicity failure rates, we investigate whether an

election’s degree of competitiveness has an effect on its monotonicity failure rate. To do

so, we construct 500,000 simulated elections for each voter distribution and plot the rate

of monotonicity failure as a function of an election’s competitive ratio (deﬁned as the ra-

tio of ﬁrst-place votes received by candidate Cto ﬁrst-place votes received by candidate A;

elections with higher ratios are more competitive). Figure 5illustrates how the rate of mono-

tonicity failure increases with competitive ratio for all four voter distributions.

Finally, it is notable that very few of the model’s simulated proﬁles exhibited majority

cyclic triples. Of elections run with the Base Case distribution, only 0.4 % exhibited a major-

ity cyclic triple, 0.05 % in the Multiparty distribution, and 0.01 % for each of the Polarized

distributions. Since monotonicity failures can occur only when the election proﬁle exhibits

a majority cyclic triple or when IRV fails to elect the Condorcet winner, this result indi-

cates that the monotonicity failures simulated here occur primarily due to IRV’s Condorcet

inefﬁciency in competitive elections.

6Conclusion

We have demonstrated here in a spatial model of voter behavior that upward monotonicity

failures arise in a non-trivial percentage of simulated elections. The lower bound estimate of

15 % in competitive elections represents a testable prediction of the model, and suggests that

three-way competitive races will exhibit unacceptably frequent monotonicity failures under

IRV. We also ﬁnd that the rate of monotonicity failure increases with an election’s degree of

competitiveness, a ﬁnding that holds true for all of the distributions studied. We restrict our

attention in this paper to the three-candidate case for largely pragmatic reasons; the closed-

form method for determining which proﬁles exhibit monotonicity failure (Sect. 3) greatly

Public Choice (2014) 161:1–9 9

reduces the computational complexity of our model. The general case with more than three

candidates is a promising topic for future research.

Of course, upward monotonicity failure is not the only major defect of IRV, and future

work will need to examine the frequency of other paradoxes to which IRV is subject. Perhaps

the only deﬁnitive way these questions can be resolved is by examining a broad body of

data from real IRV elections. Such a body does not yet exist, though it is telling that out of

only two IRV elections in Burlington, VT, there has already been one recorded instance of

nonmonotonicity. If widespread use of Instant Runoff Voting continues, then we can expect

to see many more.

Acknowledgements The authors thank Ross A. Hammond and the anonymous reviewers for insightful

comments and suggestions. Any errors remain the responsibility of the authors.

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