# The Negative Participation Paradox in Three-Candidate Instant Runoff
Elections

David McCune  David McCune, Department of Mathematics and Data Science,
William Jewell College, 500 College Hill, Liberty, MO, 64068
<EMAIL_ADDRESS>and  Jennifer Wilson  Jennifer Wilson, The New
School, Department of Natural Sciences and Mathematics, 66 West 12th Street
New York, NY 10011<EMAIL_ADDRESS>
###### Abstract.

In this paper, we provide theoretical and empirical estimates for the
likelihood of a negative participation paradox under instant runoff voting in
three-candidate elections. We determine the probability of the paradox and
related conditional probabilities based on the impartial anonymous culture and
impartial culture models for both complete and partial ballots. We compare
these results to the empirical likelihood of a negative participation paradox
occurring under instant runoff voting in a large database of voter profiles
which have been reduced to three candidates. Lastly, we analyze the relative
likelihood of this paradox in comparison to other well-known paradoxes.

##  1\. Introduction

The voting method of instant runoff voting (IRV) is famously susceptible to
many classical paradoxes. For example, under IRV an election can demonstrate
an upward or downward monotonicity paradox [4, 16, 18, 22], or a no-show
paradox [3, 4, 20]. IRV is also susceptible to paradoxes of negative
participation, a lesser-studied type of paradox. The purpose of this article
is to examine the likelihood that a three-candidate IRV election demonstrates
this paradox.

An election is said to demonstrate a negative participation paradox under a
given voting method if the addition of voters who rank candidate $A$ last
causes $A$ to go from losing to winning. This paradox is closely related to
the notion of a positive participation paradox [11], also referred to in the
language of “positive involvement” [25], where the addition of voters who rank
candidate $A$ uniquely first causes $A$ to go from winning to losing. Such
outcomes are sometimes referred to as strong no-show paradoxes [2, 24]. To the
best of our knowledge, positive participation paradoxes have received more
attention in prior literature than negative participation paradoxes, even
though IRV is not susceptible to the positive type. Prior research on positive
participation paradoxes focuses on other voting methods such as Condorcet
methods [2, 8, 9, 24] (this literature tends to use the term “strong no-show
paradox” instead of positive participation paradox). Much of the prior work
concerning negative participation paradoxes also examines the susceptibility
of Condorcet methods to this paradox [2, 24], while some focuses on scoring
runoff rules like IRV [11, 13]. Much previous work on these kinds of voting
paradoxes focuses on elections with a small number of candidates, often
analyzing elections with only three candidates. Our work similarly focuses on
the three-candidate case.

This paper makes one small and two substantive contributions regarding the
likelihood that an election demonstrates a negative participation paradox
under IRV in a three-candidate election. First, for our small contribution we
provide new theoretical results under the impartial anonymous culture (IAC)
model of voter behavior. Most of the interesting work in this vein has been
done in [11, 13], and our results in this area fill some small gaps left from
those papers. Second, we provide results under the impartial culture (IC)
model of voter behavior. We are able to obtain these new results using
geometric techniques developed in [10] and [26]. Third, we provide empirical
results using a large database of IRV elections from American and Scottish
political elections, as well as elections from the American Psychological
Association. Ours is the first study which provides such empirical results;
the only other previous work which searched for negative participation
paradoxes in real-world ballot data is [15], which focuses on multiwinner
single transferable vote elections from Scotland. Our general finding is that
negative participation paradoxes occur at a much higher rate than other
paradoxes.

Because we are interested in providing both theoretical and empirical analyses
of the negative participation paradox we consider situations when all voters
provide complete rankings and when voters are allowed to provide only partial
ballots. This allows for a better comparison between theoretical and empirical
results since partial ballots are extremely common in real-world election
data. On the theoretical side, we provide estimates that the likelihood of a
negative participation paradox occurs under IRV for 3 candidates under under
both the IAC model and IC models. For the IAC model, we analyze the complete
and partial ballot cases using the software package Normaliz and Monte Carlo
simulations, respectively. For the IC model we use geometric techniques for
both full and partial ballots.

The widespread use of the method of single transferable vote in Scotland and
other locations, as well as the recent rise of IRV in the US, has made
available a large collection of ranked data from real elections. When an
election contained more than three candidates, we reduce each election to
three candidates by running a number of initial rounds of IRV. Then we
determine the number of elections susceptible to a negative participation
paradox using the existing ballot data as well as data which we make more
“complete” using numerical techniques which we describe in Section 4.

The paper is organized as follows. Section 2 provides relevant definitions as
well as a motivating example. In Section 3 we provide theoretical results
under the IAC and IC models of voter behavior. As mentioned above, our primary
contribution is for the IC model, as the IAC model has been mostly analyzed
previously. In Section 4 we give results using a large database of real-world
elections. Section 5 provides a discussion and comparison of the likelihood of
negative participation paradoxes to other well-known paradoxes. We conclude in
Section 6.

##  2\. Preliminaries

Instant-runoff voting (IRV), often colloquially referred to as “ranked-choice
voting”, has become increasingly popular in the United States as an
alternative to the method of plurality when there are more than 2 candidates.
In such elections voters cast preference ballots with a (possibly partial)
linear ranking of the candidates. After an election, the ballots are
aggregated into a preference profile, which shows the number of each type of
ballot cast. For example, Table 1 shows a preference profile involving the
three candidates $J$, $S$, and $W$. The number 1430 denotes that 1430 voters
ranked $J$ first, $S$ second, and $W$ third. We use $\succ$ to denote when a
voter ranks one candidate over another, so that 1430 voters cast the ballot
$J\succ S\succ W$. An IRV election proceeds in rounds. In each round, the
number of first-place votes for each candidate is calculated. If a candidate
has a majority of the first-place votes, they are declared the winner. If no
candidate receives a majority of votes, the candidate with the fewest first-
place votes is eliminated and the names of the remaining candidates on the
affected ballots are moved up. The process repeats until a candidate is
declared the winner111The issue of ties does not arise in our work in this
paper, and thus we ignore the possibility of multiple winners.. In the case
when partial ballots are allowed, any ballot in which all candidates have been
eliminated is considered “exhausted” and plays no further role in the
determination of the winner.

When there are only three candidates, at most 2 rounds are required to
determine a winner; in general, several rounds may be required. In our
analysis of the real-world database in Section 4, we run a sufficient number
of rounds for each set of voter preferences until the number of candidates is
reduced to three and consider the resulting three-candidate election. We
demonstrate IRV in such a case in the following example.

######  Example 1.

The 2003 election for Minneapolis City Council in Ward 8 involved four not
write-in candidates: Andrea Jenkins, Soren Stevenson, Bob Sullentrop, and
Terry White. Sullentrop earned the fewest first-place votes and is eliminated
first, creating the three-candidate preference profile shown in Table 1. The
resulting first-place vote totals for Jenkins, Stevenson, and White are 3591,
3640, and 668, respectively. White is eliminated in the next round, causing a
transfer of 303 votes to Jenkins and 216 to Stevenson. As a result, Jenkins
defeats Stevenson 3894 votes to 3856.

Num. Voters | 1149 | 1430 | 1012 | 830 | 1989 | 821 | 149 | 303 | 216
---|---|---|---|---|---|---|---|---|---
1st choice | $J$ | $J$ | $J$ | $S$ | $S$ | $S$ | $W$ | $W$ | $W$
2nd choice |  | $S$ | $W$ |  | $J$ | $W$ |  | $J$ | $S$
3rd choice |  | $W$ | $S$ |  | $W$ | $J$ |  | $S$ | $J$
Table 1. Vote totals for the top 3 candidates in the 2023 election for
Minneapolis City Council Ward 8

It is well-known that IRV can produce strange outcomes when ballots are added
or removed from an election [3, 4, 5, 6, 11, 13, 15, 16], or when candidates
are added or removed from an election [7, 17, 19, 28]. The example below
illustrates a type of paradoxical outcome which can occur when ballots are
added to an election.

######  Example 2.

Suppose we add 4000 ballots of the form $W\succ J\succ S$ to the ballots from
in Table 1. What effect should this have on the electoral outcome? Since these
voters all prefer $W$, it would make sense for White to win after receiving
this large boost of support. Since these voters prefer Jenkins to Stevenson,
it would also make sense for these ballots to bolster Jenkins’ victory.
However, the addition of these ballots causes Jenkins to be eliminated before
White and Stevenson edges out White in the final round.

Thus, adding thousands of ballots in which Stevenson is ranked last can result
in a good outcome for him. Note that this outcome is also true in the original
four-candidate election including Sullentrop ($Su)$: the same result is
obtained if we add 4000 ballots of the form $W\succ J\succ Su\succ S$ or 4000
ballots of the form $W\succ Su\succ J\succ S$. That is, when we add ballots to
create a negative participation paradox in an IRV election which has been
reduced to three candidates, only the first and last rankings matter. We can
fill in the intermediate rankings in an arbitrary fashion; the original
election with all candidates also demonstrates a negative participation
paradox as long as sufficient votes are added so that the plurality loser
remains until eliminated in the final round.

This example motivates the primary definition of the paper.

######  Definition 1.

An election with preference profile $P$ is said to demonstrate a negative
participation paradox if there exists a losing candidate $A$ and a set of
identical ballots $\mathcal{B}$ with $A$ ranked last such that if we add the
ballots from $\mathcal{B}$ to the ballots from $P$, candidate $A$ wins the
resulting election.

That is, an election demonstrates this paradox when a losing candidate can be
made into a winner by adding ballots on which the loser is ranked last. We use
the language of “negative participation paradox” following [11], but there is
no standard vocabulary around the topic. For example, much of the literature
concerning these paradoxes and Condorcet methods [8, 9, 24] uses the the
language of no-show paradoxes and negative and positive involvement. The
previous example shows that IRV is susceptible to negative participation
paradoxes; we note that other voting methods such as positional scoring rules
cannot exhibit this type of paradox.

In the US, the most common voting method for political elections is the method
of plurality, in which the candidate with the most first-place votes wins the
election. Throughout the article, we refer to the candidate who receives the
most first-place votes in a three-candidate election as the _plurality
winner_. For example, Stevenson is the plurality winner of the preference
profile in Table 1. The _plurality loser_ is the candidate with the fewest
first-place votes. In the Minneapolis Ward 8 election Sullentrop was the
actual plurality loser; because we care only about the three-candidate case,
we ignore Sullentrop and say that White is the plurality loser of the election
in Table 1.

In a three-candidate election, a negative participation paradox can only occur
when several conditions are satisfied. First, the added votes cannot result in
the plurality loser being made into the IRV winner: if the added votes rank
the plurality loser last, the candidate will remain the plurality loser and be
eliminated in the first round. Second, the IRV winner must be different from
the plurality winner. To see this, suppose that the IRV winner is $A$ and the
candidates are ranked by first-place votes $A>B>C.$ Since $A$ is the IRV
winner, $B$ must be the losing candidate who can be made into a winner by the
addition of votes ranking $B$ last. But $B$ cannot be the winner in such a
fashion: if $B$ becomes the new plurality loser then they are eliminated in
the first round, and if $C$ remains the plurality loser than $C$ is eliminated
in the first round, leaving $B$ to be eliminated in the second round as they
were in the original election (the added votes increase the margin of $A$ over
$B$). Hence the IRV and plurality winners must be distinct, implying it is the
plurality winner who can be made to win under IRV by the addition of votes
ranking them last. Similar reasoning implies that the new votes must rank the
plurality loser first so that the original IRV winner is eliminated in the
first round of the new election.

We use this logic to determine the probability of a negative participation
paradox when there are three candidates in the next section.

##  3\. The Impartial Anonymous and Impartial Culture Models

In this section we provide probabilities that an election demonstrates a
negative participation paradox under the impartial anonymous culture (IAC) and
impartial culture (IC) models of voter behavior. Suppose the three candidates
are labeled $A$, $B$ and $C$, and there are $V$ voters. If voters must provide
complete rankings, there are six possible candidate rankings; if voters can
provide partial rankings, this number increases to 9. Assume the preference
profiles for complete and partial rankings are as shown in Tables 2 and 3
where in each case the numbers $a_{i},b_{i}$, and $c_{i}$ indicate the number
of voters with the corresponding ranking and that these numbers sum to $V.$
Note in Table 3 we equate the rankings $A\succ B$ and $A\succ B\succ C$ and
hence do not list them separately. For either model of voter behavior, we are
concerned with the _limiting probability_ that an election demonstrates a
negative participation paradox, where $V\rightarrow\infty$.

Before providing the limiting probabilities, we examine the conditions
(expressed as inequalities) under which an election demonstrates a negative
participation paradox.

Table 2. Preference profile with complete preferences. $a_{1}$ | $a_{2}$ | $b_{1}$ | $b_{2}$ | $c_{1}$ | $c_{2}$
---|---|---|---|---|---
A | A | B | B | C | C
B | C | A | C | A | B
C | B | C | A | B | A
Table 3. Preference profile with some partial preferences. $a_{0}$ | $a_{1}$ | $a_{2}$ | $b_{0}$ | $b_{1}$ | $b_{2}$ | $c_{0}$ | $c_{1}$ | $c_{2}$
---|---|---|---|---|---|---|---|---
A | A | A | B | B | B | C | C | C
| B | C |  | A | C |  | A | B
| C | B |  | C | A |  | B | A

In the complete ballot case, we can assume WLOG that

(1) $a_{1}+a_{2}>b_{1}+b_{2}>c_{1}+c_{2};$

similarly, in the partial ballot case we assume WLOG that

(2) $a_{0}+a_{1}+a_{2}>b_{0}+b_{1}+b_{2}>c_{0}+c_{1}+c_{2}.$

That is, we assume $A$ is the plurality winner and $C$ is the plurality loser.
As described in Section 2, a negative participation paradox can only occur
when the plurality ($A$) and IRV ($B$) winners are distinct and when
additional votes are added with the ranking $C\succ B\succ A.$ Hence,
initially

(3) $a_{1}+a_{2}+c_{1}<b_{1}+b_{2}+c_{2}.$

If $z$ votes are added, then we require $B$ to be eliminated in the first
round followed by $C$ in the second round. Hence,

$b_{1}+b_{2}<c_{1}+c_{2}+z,\quad a_{1}+a_{2}+b_{1}>b_{2}+c_{1}+c_{2}+z$

Solving for $z$ in each of these yields for $z$ yields

$b_{1}+b_{2}-(c_{1}+c_{2})<z<a_{1}+a_{2}+b_{1}-(b_{2}+c_{1}+c_{2})$

which implies

(4)
$b_{1}+b_{2}-(c_{1}+c_{2})<a_{1}+a_{2}+b_{1}-(b_{2}+c_{1}+c_{2})\quad\mbox{ or
}\quad a_{1}+a_{2}>2b_{2}.$

It is easily seen that (3) and (4) are also sufficient to ensure a negative
participation paradox can occur given that $A$ is the plurality winner and $C$
is the plurality loser.

In the partial rankings case, (3) and (4) are replaced by

(5) $b_{0}+b_{1}+b_{2}+c_{2}>a_{0}+a_{1}+a_{2}+c_{1}$

and we obtain

(6) $a_{0}+a_{1}+a_{2}>b_{0}+2b_{2}.$

Before defining and analyzing the two models, we note that they are widely
used in the social choice literature to provide _a priori_ estimates for the
probabilities of many phenomena in social choice, and they tend to provide
theoretical upper bounds for such probabilities. For example, in three-
candidate elections the probability that an election fails to contain
Condorcet winner is 6.25% under the IAC model (this is easily calculated using
Normaliz); the probability falls essentially to 0 in real-world elections [6,
21, 27]. Thus, the models are useful for providing theoretical “worst-case”
estimates for various election probabilities.

###  3.1. Impartial Anonymous Culture

Under the impartial anonymous culture (IAC) model, the vector of vote totals
at the top of the preference profile is assumed to be chosen at random. For
example, in the complete ballot case we choose at random a vector of the form
$(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2})$ subject to the constraints that each
entry is non-negative and the six vote totals sum to the number of voters.
Since we examine only limiting probabilities, we choose a vector of
proportions of length six in the complete ballot case or a vector of
proportions of length nine in the partial ballot case. Limiting probabilities
involving IAC can be determined using the theory of Ehrhart polynomials, which
has been implemented in the software package Normaliz [1]. The probabilities
presented in Proposition 1 were calculated using Normaliz. The probability of
a negative participation paradox using IRV assuming complete rankings appears
in [11]. In the following proposition, we include this result along with
related conditional probabilities for reference.

Assuming, as in the previous section, that first-place vote ranking is
$A>B>C,$ the probability that an election demonstrates a negative
participation paradox is equal to the probability that (3) and (4) are
satisfied. The results in (ii) and (iii) below are conditional probabilities.
If an election demonstrates a negative participation paradox then there cannot
be a majority candidate (since the plurality and IRV winners would coincide).
Thus, in (ii), the probability that an election demonstrates a negative
participation paradox winner assuming the election does not contain a majority
candidate is equal to the the probability of a negative participation paradox
divided by the probability that an election does not have a majority
candidate. Likewise, in (iii), the probability that an election demonstrates a
negative participation paradox assuming the IRV winner is not the plurality
winner is equal to the probability that an election demonstrates a negative
participation paradox divided by the probability the IRV winner is not the
plurality winner.

######  Proposition 1.

In a 3-candidate election in which all voters provide complete preferences,
under IAC

  1. i

the limiting probability that an election demonstrates a negative
participation paradox is $7/96=7.29\%$ [11].

  2. ii

the limiting probability that an election demonstrates a negative
participation paradox is $7/54=12.96\%$, assuming the election does not
contain a majority candidate.

  3. iii

the limiting probability that an election demonstrates a negative
participation paradox is $42/71=59.15\%$, assuming the IRV winner is not the
plurality winner.

We are unaware of analytical methods which extend the Ehrhart polynomial
techniques employed by Normaliz to the partial ballot setting where a
preference profile has nine parameters. Thus, to estimate the corresponding
IAC probabilities under partial ballots we use Monte Carlo simulation. For
each run, we choose a percentage distribution of length nine at random
(straightforward techniques exist for this task, see [12] for example) and
check for a paradox. Using 100,000 runs, in the partial ballot setting we
estimate the corresponding limiting probabilities from Proposition 1 to be
approximately 7.24%, 16.74%, and 62.49%.

###  3.2. Impartial Culture

Under the impartial Culture (IC) model, each voter selects uniformly and
independently among all possible rankings of the candidates. We can
approximate the likelihood of a negative participation paradox for a large
number of voters by calculating the limiting probability as
$V\rightarrow\infty$ using the geometric methods introduced in [26] and
subsequently elaborated on in [10]. Note that the probability that there is a
majority candidate is zero in the limiting case, thus we provide counterpoints
to only two of the probabilities identified in Proposition 1.

######  Proposition 2.

In a 3-candidate election in which all voters provide complete preferences,
under IC

  1. i

the limiting probability that an election demonstrates a negative
participation paradox is approximately $0.151084\approx 15.11\%.$.

  2. ii

the limiting probability that an election demonstrates a negative
participation paradox is $0.617525\approx 61.75\%$, assuming the IRV winner is
not the plurality winner.

######  Proposition 3.

In a 3-candidate election in which voters provide partial preferences, under
IC

  1. i

the limiting probability that an election demonstrates a negative
participation paradox is approximately $0.143861\approx 14.39\%$.

  2. ii

the limiting probability that an election demonstrates a negative
participation paradox is approximately $0.672803\approx 67.28\%$, assuming the
IRV winner is not the plurality winner.

The proof of Proposition 3 is similar to that of Proposition 2. It can be
found in the Appendix. We prove Proposition 2 below.

###### Proof.

Proof of (i)

Step 1 First, we find the probability that $A$ and $B$ are the plurality and
IRV winners respectively. Following the arguments of [26], the probability
that (1) and (3) are satisfied is equal to the area of the spherical simplex
$S$ defined by these inequalities on the surface of the unit sphere in
$\mathbb{R}^{3}$, divided by the area of this sphere. Let

$\displaystyle\mathbf{v}_{1}$ $\displaystyle=(1,1,-1,-1,0,0)$
$\displaystyle\mathbf{v}_{2}$ $\displaystyle=(0,0,1,1,-1,-1)$
$\displaystyle\mathbf{v}_{3}$ $\displaystyle=(-1,-1,1,1,-1,1).$

be the normal vectors of the three hyperplanes bounding $S$. By the Gauss-
Bonet Theorem, $Vol_{2}(S)=\alpha_{12}+\alpha_{13}+\alpha_{23}-\pi$ where
$\alpha_{ij}$ is the angle between vectors $\mathbf{v}_{i}$ and
$\mathbf{v}_{j}$ and $Vol_{n}$ is the $n-$dimensional volume.

Using the law of cosines,

$\displaystyle\alpha_{12}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{2})=\cos^{-1}(\frac{-\mathbf{v}_{1}\cdot\mathbf{v}_{2}}{\lVert\mathbf{v}_{1}\rVert\lVert\mathbf{v}_{2}\rVert})=\cos^{-1}(\frac{2}{2\cdot
2})=\cos^{-1}(\frac{1}{2})$ $\displaystyle\alpha_{13}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{3})=\cos^{-1}(\frac{-\mathbf{v}_{1}\cdot\mathbf{v}_{3}}{\lVert\mathbf{v}_{1}\rVert\lVert\mathbf{v}_{3}\rVert})=\cos^{-1}(\frac{4}{2\cdot\sqrt{6}})=\cos^{-1}(\frac{1}{\sqrt{6}})$
$\displaystyle\alpha_{23}$
$\displaystyle=\angle(\mathbf{v}_{2},\mathbf{v}_{3})=\cos^{-1}(\frac{-\mathbf{v}_{2}\cdot\mathbf{v}_{3}}{\lVert\mathbf{v}_{2}\rVert\lVert\mathbf{v}_{3}\rVert})=\cos^{-1}(\frac{-2}{2\cdot\sqrt{6}})=\cos^{-1}(\frac{-1}{\sqrt{6}}).$

Thus the probability is equal to

$\displaystyle\frac{1}{4\pi}Vol_{2}(S)=\frac{1}{4\pi}[\alpha_{12}+\alpha_{12}+\alpha_{12}-\pi]=0.04077671.$

Step 2: Next, we find the probability that additionally (4) is satisfied.
Following [10], we introduce the modified inequality with parameter
$t\in[0,1],$

(7) $x_{1}+x_{2}>x_{3}+x_{4}+t(x_{4}-x_{3}).$

Inequality (7) replaces (4). If $t=0$, (7) reduces to the first inequality in
(1); if $t=1$, we recover (4). Similarly to Step 1, the probability that these
conditions are met for a large number of voters $N$ is equal to the volume of
the spherical simplex $S$ defined by these inequalities on the surface of the
unit sphere in $\mathbb{R}^{4}$, divided by the volume of the surface of this
sphere, $2\pi^{2}$. To find the $Vol_{3}(S)$, we use the Schäfli formula

$dVol_{n}=\frac{1}{n-1}\sum_{1\leq j<k\leq n}Vol_{n-2}(S_{j}\cap
S_{k})d\alpha_{jk}.$

where $n=3$ and $S_{i}$ are the hyperplanes bounding $S$. Let $S_{i}$ be the
hyperplane with normal vector $\mathbf{v}_{i}$. Since (7) has normal vector
$\mathbf{v}_{4}=(1,1,t-1,-t-1,0,0)$ and
$\lVert\mathbf{v}_{4}\rVert=\sqrt{4+2t^{2}}$, we have,

$\displaystyle\alpha_{14}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{4})=\cos^{-1}(\frac{-4}{2\cdot\sqrt{4+2t^{2}}})=\cos^{-1}(\frac{-2}{\sqrt{4+2t^{2}}})$
$\displaystyle\alpha_{24}$
$\displaystyle=\angle(\mathbf{v}_{2},\mathbf{v}_{4})=\cos^{-1}(\frac{2}{2\cdot\sqrt{4+2t^{2}}})=\cos^{-1}(\frac{1}{\sqrt{4+2t^{2}}})$
$\displaystyle\alpha_{34}$
$\displaystyle=\angle(\mathbf{v}_{3},\mathbf{v}_{4})=\cos^{-1}(\frac{4}{\sqrt{6}\cdot\sqrt{4+2t^{2}}}).$

Differentiating, gives

$\displaystyle d\alpha_{14}=\frac{-\sqrt{2}}{2+t^{2}}\quad
d\alpha_{24}=\frac{t}{\sqrt{3+2t^{2}}\cdot(2+t^{2})}\quad\mbox{and}\quad
d\alpha_{34}=\frac{2t}{\sqrt{2+3t^{2}}\cdot(2+t^{2})}.$

To calculate $Vol_{1}(S_{j}\cap S_{k})$, we must determine where the
hyperplanes intersect. Let $P_{ijk}$ be the vertex lying at the intersection
of $S_{i}$, $S_{j}$ and $S_{k}$. We identify a basis for the subspace
orthogonal to that spanned by $\mathbf{v}_{1}\ldots,\mathbf{v}_{4}$. One such
basis is $\mathbf{v}_{5}=(1,1,0,0,0,0)$ and $\mathbf{v}_{6}=(1,1,1,1,1,1).$

Next, we find points $P_{ijk}$ lying at the intersection of the hyperplanes
with normals given by the vectors $\mathbf{v}_{i}$, $\mathbf{v}_{j}$ and
$\mathbf{v}_{k}$. Vertex $P_{124}$, for instance, can then be found by solving
the conditions

$\displaystyle n_{1}+n_{2}-n_{3}-n_{4}$ $\displaystyle=0$ $\displaystyle
n_{3}+n_{4}-n_{5}-n_{6}$ $\displaystyle=0$ $\displaystyle-
n_{1}-n_{2}+n_{3}+n_{4}-n_{5}+n_{6}$ $\displaystyle>0$ $\displaystyle
n_{1}+n_{2}+(t-1)n_{3}+(-t-1)n_{4}$ $\displaystyle=0$ $\displaystyle
n_{1}-n_{2}$ $\displaystyle=0$ $\displaystyle
n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$ $\displaystyle=0.$

This yields $P_{124}=(0,0,0,0,-1,1).$ Similarly, we have
$P_{134}=(1,1,1,1,-2,-2)$ and
$P_{234}=(1,1,\frac{-3-t}{2t},\frac{3-t}{2t},-2,1).$

Since $\lVert P_{124}\rVert=\sqrt{2}$, $\lVert P_{134}\rVert=2\sqrt{3}$ and
$\lVert P_{234}\rVert=\frac{1}{t}\sqrt{\frac{3}{2}}\sqrt{5t^{2}+3}$, we have

$\displaystyle Vol(S_{1}\cap C_{4})$
$\displaystyle=\angle(P_{124},P_{134})=\cos^{-1}(\frac{0}{\sqrt{2}\cdot
2\sqrt{3}})=\frac{\pi}{2}$ $\displaystyle Vol(S_{2}\cap C_{4})$
$\displaystyle=\angle(P_{124},P_{234})=\cos^{-1}(\frac{\sqrt{3}t}{\sqrt{5t^{2}+3}})$
$\displaystyle Vol(S_{3}\cap C_{4})$
$\displaystyle=\angle(P_{134},P_{234})=\cos^{-1}(\frac{\sqrt{2}t}{\sqrt{5t^{2}+3}})$

By the Schäfli formula, $dVol_{3}=\frac{1}{2}\sum_{1\leq j<k\leq
4}Vol_{1}(S_{j}\cap S_{k})d\alpha_{jk}$, so

$\displaystyle Vol_{3}(S_{t=1})$
$\displaystyle=Vol_{3}(S_{t=0})+\int_{0}^{1}dVol_{3}S$
$\displaystyle=Vol_{3}(S_{t=0})+\frac{1}{2}\int_{0}^{1}Vol_{1}(S_{1}\cap
S_{4})d\alpha_{14}+Vol_{1}(S_{2}\cap S_{4})d\alpha_{24}+Vol_{1}(S_{3}\cap
S_{4})d\alpha_{34}dt$
$\displaystyle=Vol_{3}(S_{t=0})+\frac{1}{2}\left[I_{1}+I_{2}+I_{3}\right]$

where $I_{1}=\int^{1}_{0}\frac{\pi}{2}\cdot\frac{-2}{4+2t^{2}}dt=-0.99679$ and
similarly $I_{2}=0.11231$ and $I_{3}=0.268772$

By Step 1,
$Vol_{3}(S_{t=0})=Vol_{2}(S_{t=0})\frac{2\pi^{2}}{4\pi}=0.04077671(2\pi^{2})$.
Hence the probability that all these conditions are met is equal to

$\frac{1}{2\pi^{2}}\left[0.04077671(2\pi^{2})+\frac{1}{2}(-0.99679+0.11231+0.268772)\right]=0.02518064.$

Since the choice of $A$ and $B$ as plurality and IRV winners was arbitrary,
the probability of a negative participation paradox is $6\times
0.02518064\approx 0.151084.$

Proof of (ii) Since a negative participation paradox can only occur when the
IRV and plurality winners are distinct, the probability that an election
demonstrates a negative participation paradox given that these winners are
different is equal to the probability of negative participation paradox
divided by the probability that the IRV and plurality winners are distinct.
The latter was determined in Step 1; hence this probability is equal to
$0.02518386/0.04077671\approx 0.617525.$ ∎

###  3.3. Analysis of IAC and IC Results

Overall, a negative participation paradox is twice as likely under the IC than
the IAC model, and somewhat more likely assuming the IRV and plurality winners
are distinct. This is to be expected, as the IAC model frequently produces
preference profiles containing a majority candidate while this is impossible
under IC in the limiting case. Under both models, the probability of a
negative participation paradox is slightly higher for complete voter
preferences than for partial voter preferences; the relationship is reverse
when conditioned on the elections having distinct IRV and plurality winners.
This is possibly because the probability that these winners are distinct is
somewhat lower for partial preferences than for complete preferences (in the
IC case, 21% versus 24%), With partial ballots, it is less likely that a
candidate ranking second in first-place votes will gain sufficient additional
votes, after the plurality loser is eliminated, to overcome the vote total of
the plurality winner. We compare these results with an empirical analysis in
the next section.

##  4\. Empirical Results

In this section we present empirical results using a large database of real-
world IRV elections. Our data comes from three sources. First, we use single-
winner political elections from the United States with at least three (not
write-in) candidates. This data is available at the FairVote data repository
[23], and contains ballot data from municipal elections in cities such as San
Francisco, CA and Minneapolis, MN. The repository also contains ballot data
from federal elections in Alaska and Maine. Second, we use single-winner
elections from the American Psychological Association, which generally is
willing to share ballot data for ranked-choice presidential elections and
elections for the Board of Directors. Some of this data is available on
preflib.org [14], and some was shared directly with the first author. Third,
we use single-winner political elections from Scotland from a database of
Scottish local government elections. The ballot data for these elections is
available at https://github.com/mggg/scot-elex.

In total, we have access to the preference profiles for 361 single-winner IRV
elections with at least three candidates. Many of these elections contain more
than three candidates; because we focus on the three-candidate case, for each
election we run the IRV algorithm until three candidates remain and use the
resulting profile. As with Example 1, any paradox we find in the resulting
three-candidate profile will also be demonstrated in the original election.
The reason is that when we create the ballots which produce the paradox only
the first and last rankings matter; the intermediate rankings can be filled in
arbitrarily using the remaining candidates from the original candidate set. Of
course, it is possible that an election with four or more candidates
demonstrates a paradox which cannot be found when we reduce down to three
candidates; our methodology would not find such elections.

In Section 3 we analyze the likelihood of negative participation paradoxes in
two cases: when all voters provide complete preferences and when some voters
cast partial ballots. To provide comparisons to those results, with our real
data we also give results using fully complete preferences and using partial
ballots. In real elections voters often provide partial rankings of the
candidates, and thus our results from partial ballots represent the “actual”
results, the results obtained from using the actual ballot data. To create
complete preferences, we proportionally fill in partial ballots using the
available complete ballots to create hypothetical results for the case that
all voters submit complete preferences.

To see how we fill in the ballots, consider the preferences profile from Table
1. We need to take the bullet votes for $J$ and extend them to complete
ballots over the three candidates. Since $1430+1012=2442$ voters rank $J$
first and provide complete rankings, we extend $1430/2442*100\%$ of the 1149
bullet votes for $J$ to ballots of the form $J\succ S\succ W$ and
$1012/2442*100\%$ of the bullet votes for $J$ to ballots of the form $J\succ
W\succ S$, rounding to the nearest integer. As a result, 673 of the 1149
bullet votes are extended to $J\succ S\succ W$ and 476 are extended to $J\succ
W\succ S$. This methodology is used on every three candidate profile to create
complete preferences.

Jurisdiction | USA | APA | Scotland
---|---|---|---
Elections | 304 | 27 | 30
No Majority Cand | 191 | 23 | 30
IRV Winner $\neq$ plurality winner | 18, 30 | 3, 3 | 5, 8
Paradoxes | 7, 17 | 3, 3 | 2, 3
Table 4. Results from real-world elections. In the final two rows, the first
number is obtained using the actual ballots and the second is obtained from
ballots which have been completed.

Our results are summarized in Table 4. The top row shows the number of
elections from each jurisdiction and the second row shows the number of
elections without a majority candidate. The third row shows the number of
elections in which the IRV winner is different from the plurality winner. The
first number is the number of such elections when using the actual data with
partial ballots; the second number is the number of such elections when using
complete preferences, created as described above. The fourth row shows the
number of participation paradoxes, using the same format. For example, there
are 191 American political elections without a majority candidate. Of these,
the IRV winner is not the plurality winner in 18 (respectively 30) elections
when using the actual ballot data (respectively, complete preference data).
Seven American elections demonstrate a paradox when using the actual data,
which rises to 17 when we complete the ballots.

Election | Num. Voters | Ballot Range
---|---|---
2020 San Francisco Board of Supervisors D7 | 38321 | 798-1605
2021 Minneapolis City Council Ward 2 | 8907 | 80-224
2023 Minneapolis City Council Ward 8 | 7899 | 2973-4090
2010 Oakland Mayor | 113217 | 2315-3951
2016 Oakland School Director D5 | 12950 | 3151-4267
2022 Oakland School Director D4 | 26432 | 38-598
2021 New York City Rep Primary D5 | 8182 | 773-830
2005 APA President | 14079 | 2275-2780
2007 APA President | 12925 | 2397-2717
2020 APA Board of Directors Race 2 | 6227 | 1093-1883
2021 Argyll Bute By-Election Isle of Bute Ward† | 1804 | 26-38
2021 Highland By-Election Aird Ward† | 3321 | 266-266
Table 5. Each election which demonstrates a negative participation paradox
when using the actual ballot data.

Table 5 provides a complete list of elections which demonstrate a paradox when
using the actual ballots. The table also provides the range of the number of
ballots which can be added to produce the paradox. For example, the third row
shows that we can add anywhere from 2973 to 4090 ballots of the form $W\succ
J\succ S$ to create the paradox in Example 1. Note that the last election in
the table just barely produces a paradox, as the paradox can be demonstrated
using only 266 ballots exactly.

When using the actual ballot data, in total we find 12 elections which
demonstrate this paradox. From Table 4, we obtain an empirical probability of
$12/361=3.3\%$ that the paradox occurs. When we condition on the non-existence
of a majority candidate we obtain a probability of $12/244=4.9\%$, which rises
to $12/26=46.2\%$ when we restrict to elections when the IRV and plurality
winners are different. The corresponding three probabilities for complete
preferences are 6.4%, 9.4%, and 56.1%, respectively. As with the IAC and IC
models we obtain higher probabilities for complete preferences for the two
conditional probabilities. Unlike those models, we obtain higher probabilities
for complete preferences for the unconditioned probabilities as well. Overall,
the frequency with which the paradox of negative participation occurs in the
database is lower than that which is predicted by the IAC and IC models, as
expected since these models tend to provide upper bounds for the probabilities
of paradoxical behavior. It is interesting, though, that some of the empirical
probabilities are not significantly lower than the theoretical probabilities.
Often, an empirical estimate of the probability that an election demonstrates
a particular voting paradox is much lower than what is predicted by a model
like IAC or IC. When we assume the IRV and plurality winners are different,
the empirical probability for negative participation paradoxes is surprisingly
close to the probabilities reported in Section 3.

##  5\. Comparison to other paradoxes

In this section we compare the conditions and likelihood that an IRV election
demonstrates a negative participation paradox to other more well-known (and
well-studied) paradoxes: upward and downward monotonicity paradoxes, and no-
show paradoxes (see e.g. [4] and [18] for definitions and discussions of these
paradoxes). In the three-candidate complete ballot setting, for an upward
(resp. downward) monotonicity paradox to occur, the plurality loser must earn
at least 25% (respectively $16\frac{2}{3}$%) of the first-place votes [18]. In
either the complete or partial ballot setting, the plurality loser must earn
at least $16\frac{2}{3}$% of the first-place votes for a no-show paradox to
occur [5]. By contrast, for a large enough electorate there is no such
positive lower bound for the plurality loser regarding negative participation
paradoxes. That is, it is possible for the plurality lower to receive an
arbitrarily small amount of first-place votes and yet the election
demonstrates a negative participation paradox.

To see this, consider Table 6. The top row shows the percentage of the first-
place vote controlled by each candidate. Suppose these percentages satisfy
$p_{C}<p_{B}<p_{A}<0.5$. Note that $B$ is the IRV winner, but if we add enough
$C\succ B\succ A$ ballots so that the adjusted percentages satisfy
$p^{\prime}_{B}<p^{\prime}_{C}<p^{\prime}_{A}$ (which is possible for a large
enough electorate), then $A$ wins. Thus, it is possible for the plurality
loser to be arbitrarily weak and yet observe this paradox. In particular, as
long as no candidate earns 50% or more of the first-place votes, it is
possible to find a preference profile which demonstrates a negative
participation paradox for any distribution of first-place votes.

% Voters | $p_{A}$ | $p_{B}$ | $p_{C}$
---|---|---|---
1st choice | $A$ | $B$ | $C$
2nd choice | $B$ | $A$ | $B$
3rd choice | $C$ | $C$ | $A$
Table 6. A table illustrating that an election can demonstrate a negative
participation paradox even if the plurality loser controls a small percentage
of the first-place votes.

Furthermore, to create such a paradox we can control the type and number of
ballots added to the original ballot data. Thus, we expect that such paradoxes
are exhibited more frequently in real-world data than other paradoxes. Prior
empirical research bears this out. Using essentially the same database of
American elections that we use, [6] found four elections which demonstrate an
upward monotonicity paradox, three which demonstrate a downward monotonicity
paradox, and one which demonstrates a no-show paradox. The authors did not
include the APA elections in that study, but they evaluated those elections
using the same methodology as in [6] and found no paradoxes of any kind. The
study in [16] included the 30 Scottish elections we evaluate as part of a much
larger study of single transferable vote elections, and of these 30 elections
found only one which demonstrated a paradox (this election demonstrated both
an upward monotonicity paradox and a no-show paradox). In total, of the 361
elections in our database there are five documented elections which
demonstrate an upward monotonicity paradox, three which demonstrate a downward
monotonicity paradox, and two which demonstrate a no-show paradox. As in this
paper, all of the previously documented paradoxes occurred at the “three
candidate level” in the sense that the paradox can be demonstrated after
eliminating all but the final three candidates.

Based on prior empirical work, IRV three-candidate elections demonstrate a
negative participation paradox 2-6 times more frequently than other classical
paradoxes, an unsurprising result given the above discussion.

##  6\. Conclusion

We conclude with two final comments. First, IRV proponents sometimes claim
that a strength of IRV is that the method can choose a different (and
presumably more deserving) winner than the plurality winner. This is a natural
point to make, since if the two winners always coincide then there is no
reason to use IRV, and arguably the plurality winner sometimes does not
deserve to win an election (e.g. if the plurality winner is a Condorcet
loser). However, in some sense our work shows that this feature of IRV comes
with a price: when the IRV and plurality winners differ, it is often the case
that the election demonstrates a negative participation paradox. That is, when
IRV “works properly” by choosing a more deserving candidate than the plurality
winner, often the plurality winner could have been the IRV winner if only the
turnout of some voters who don’t like the plurality winner were increased.

Second, we make no claim regarding the normative status of negative
participation paradoxes. Social choice theorists care about paradoxes such as
monotonicity and participation paradoxes in part because such paradoxes show
there is some mathematical irrationality built into the mechanics of IRV. Put
simply, sometimes IRV does not behave rationally in response to changes in the
ballot data, and this seems normatively undesirable. On the other hand,
paradoxes in this vein are hypothetical–negative participation paradoxes
especially so. In Example 2, it is extremely unlikely that there were
thousands of potential voters in Minneapolis’ eighth ward whose favorite
candidate was White and whose least favorite was Stevenson, and these voters
abstained from the actual election. Example 2 is built using thousands of
people who probably do not exist, which perhaps ultimately says nothing about
whether IRV is a “good” or “appropriate” voting method.

Regardless of one’s normative stance on these paradoxes, the results in this
paper contribute to the literature on voting paradoxes by providing limiting
probabilities regarding negative participation paradoxes in the three-
candidate case under the impartial culture model, empirical probabilities
using a large database of real-world elections, and limiting probabilities in
the three-candidate case for some cases left over from [11] under the
impartial anonymous culture model.

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## Appendix

The proof of Proposition 3 is similar to that of Proposition 2.

Proof of Proposition 3 (i)

Step 1 First, we find the probability that $A$ and $B$ are the plurality and
IRV winners respectively. The inequalities (2) and (5) have normal vectors

$\displaystyle\mathbf{v}_{1}$ $\displaystyle=(1,1,1,-1,-1,-1,0,0,0)$
$\displaystyle\mathbf{v}_{2}$ $\displaystyle=(0,0,0,1,1,1,-1,-1,-1)$
$\displaystyle\mathbf{v}_{3}$ $\displaystyle=(-1,-1,-1,1,1,1,0,-1,1).$

So

$\displaystyle\alpha_{12}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{2})=\cos^{-1}(\frac{-\mathbf{v}_{1}\cdot\mathbf{v}_{2}}{\lVert\mathbf{v}_{1}\rVert\lVert\mathbf{v}_{2}\rVert})=\cos^{-1}(\frac{3}{\sqrt{6}\cdot\sqrt{6}})=\cos^{-1}(\frac{1}{2})$
$\displaystyle\alpha_{13}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{3})=\cos^{-1}(\frac{-\mathbf{v}_{1}\cdot\mathbf{v}_{3}}{\lVert\mathbf{v}_{1}\rVert\lVert\mathbf{v}_{3}\rVert})=\cos^{-1}(\frac{6}{\sqrt{6}\cdot\sqrt{8}})=\cos^{-1}(\frac{\sqrt{3}}{2})$
$\displaystyle\alpha_{23}$
$\displaystyle=\angle(\mathbf{v}_{2},\mathbf{v}_{3})=\cos^{-1}(\frac{-\mathbf{v}_{2}\cdot\mathbf{v}_{3}}{\lVert\mathbf{v}_{2}\rVert\lVert\mathbf{v}_{3}\rVert})=\cos^{-1}(\frac{-3}{\sqrt{6}\cdot\sqrt{8}})=\cos^{-1}(\frac{-\sqrt{3}}{4}).$

Thus, the probability is equal to

$\displaystyle\frac{1}{4\pi}Vol_{2}(S)=\frac{1}{4\pi}[\alpha_{12}+\alpha_{12}+\alpha_{12}-\pi]=0.03563737.$

Step 2: Next, we find the probability that additionally (6) is satisfied. We
introduce the inequality $x_{1}+x_{2}+x_{3}>x_{4}+x_{5}+x_{6}+t(x_{6}-x_{5})$.
When $t=0$ this reduces to a (2); when $t=1$ we get the required extra
condition.

Since this inequality has normal vector equal to
$\mathbf{v}_{4}=(1,1,1,-1,t-1,-t-1,0,0,0)$, we have

$\displaystyle\alpha_{14}$
$\displaystyle=\angle(\mathbf{v}_{1},\mathbf{v}_{4})=\cos^{-1}(\frac{-6}{\sqrt{6}\cdot\sqrt{6+2t^{2}}})=\cos^{-1}(\frac{-\sqrt{3}}{\sqrt{3+t^{2}}})$
$\displaystyle\alpha_{24}$
$\displaystyle=\angle(\mathbf{v}_{2},\mathbf{v}_{4})=\cos^{-1}(\frac{3}{\sqrt{6}\cdot\sqrt{6+2t^{2}}})=\cos^{-1}(\frac{\sqrt{3}}{2\sqrt{3+t^{2}}})$
$\displaystyle\alpha_{34}$
$\displaystyle=\angle(\mathbf{v}_{3},\mathbf{v}_{4})=\cos^{-1}(\frac{6}{\sqrt{8}\cdot\sqrt{6+2t^{2}}})=\cos^{-1}(\frac{3}{2\sqrt{3+t^{2}}})$

leading to

$\displaystyle d\alpha_{14}=\frac{-\sqrt{3}}{(3+t^{2})}\quad
d\alpha_{24}=\frac{\sqrt{3}t}{\sqrt{9+4t^{2}}(3+t^{2})}\quad\mbox{and}\quad
d\alpha_{34}=\frac{3t}{\sqrt{3+4t^{2}}(3+t^{2})}.$

Next, we require a basis for the space in $\mathbb{R}^{9}$ that is orthogonal
to $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ and $\mathbf{v}_{4}.$ One
such basis is

$\displaystyle\mathbf{v}_{5}$ $\displaystyle=(1,1,-2,0,0,0,0,0,0)$
$\displaystyle\mathbf{v}_{6}$ $\displaystyle=(1,-1,0,0,0,0,0,0,0)$
$\displaystyle\mathbf{v}_{7}$ $\displaystyle=(0,0,0,-2,1,1,0,0,0)$
$\displaystyle\mathbf{v}_{8}$ $\displaystyle=(0,0,0,0,0,0,-2,1,1)$
$\displaystyle\mathbf{v}_{9}$ $\displaystyle=(1,1,1,1,1,1,1,1,1).$

We find $P_{124}$, $P_{134}$ and $P_{234}$ as before to get
$P_{124}=(0,0,0,0,0,0,0,-1,1),$ $P_{134}=(1,1,1,1,1,1,-2,-2,-2)$ and
$P_{234}=(4t,4t,4t,-2t,-9-2t,9-2t,-2t,-11t,7t).$ Since $\lVert
P_{124}\rVert=\sqrt{2}$, $\lVert P_{134}\rVert=3\sqrt{2}$ and $\lVert
P_{234}\rVert=\sqrt{48t^{2}+8t^{2}+121t^{2}+49t^{2}+162+8t^{2}}=\sqrt{234t^{2}+162}=3\sqrt{2(13t^{2}+9)}$,
we have

$\displaystyle Vol(S_{1}\cap C_{4})$
$\displaystyle=\angle(P_{124},P_{134})=\cos^{-1}(\frac{0}{\sqrt{2}\cdot
2\sqrt{18}})=\frac{\pi}{2}$ $\displaystyle Vol(S_{2}\cap C_{4})$
$\displaystyle=\angle(P_{124},P_{234})=\cos^{-1}(\frac{18t}{\sqrt{2}\cdot
3\sqrt{2(13t^{2}+9)}})=\cos^{-1}(\frac{3t}{\sqrt{13t^{2}+9}})$ $\displaystyle
Vol(S_{3}\cap C_{4})$
$\displaystyle=\angle(P_{134},P_{234})=\cos^{-1}(\frac{18t}{3\sqrt{2}\cdot
3\sqrt{2(13t^{2}+9)}})=\cos^{-1}(\frac{t}{\sqrt{13t^{2}+9}})$

So

$\displaystyle Vol_{3}(S_{t=1})$
$\displaystyle=Vol_{3}(S_{t=0})+\int_{0}^{1}dVol_{3}S$
$\displaystyle=Vol_{3}(S_{t=0})+\frac{1}{2}\int_{0}^{1}Vol_{1}(S_{1}\cap
S_{4})d\alpha_{14}+Vol_{1}(S_{2}\cap S_{4})d\alpha_{24}+Vol_{1}(S_{3}\cap
S_{4})d\alpha_{34}dt$
$\displaystyle=Vol_{3}(S_{t=0})+\frac{1}{2}\left[I_{1}+I_{2}+I_{3}\right]$

where
$I_{1}=\int^{1}_{0}\frac{\pi}{2}\cdot\frac{-\sqrt{3}}{(3+t^{2})}dt=-0.82247$
and similarly $I_{2}=0.0806918$ and $I_{3}=0.28144178$

By Step 1,
$Vol_{3}(S_{t=0})=Vol_{2}(S_{t=0})\frac{2\pi^{2}}{4\pi}=0.03563737(2\pi^{2})$.
Hence the probability that all these conditions are met is equal to

$\frac{1}{2\pi^{2}}\left[0.03563737(2\pi^{2})+\frac{1}{2}(-0.82247+0.0806918+0.28144178)\right]=0.023976912.$

Since the choice of $A$ and $B$ was arbitrary, the probability of a negative
participation paradox is is $6\times 0.023976912\approx 0.143861$

Proof of (ii) As with the case with complete ballots, the probability that an
election demonstrates a negative participation paradox given that these
winners are different is equal to the probability of negative participation
paradox divided by the probability that the IRV and plurality winners are
distinct. The latter was determined in Step 1; hence this probability is equal
to $0.023976912/0.03563737\approx 0.672803.$