# The Spoiler Effect in Multiwinner Ranked-Choice 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 the popular debate over the use of ranked-choice voting, it is often
claimed that the method of single transferable vote (STV) is immune or mostly
immune to the so-called “spoiler effect,” where the removal of a losing
candidate changes the set of winners. This claim has previously been studied
only in the single-winner case. We investigate how susceptible STV is to the
spoiler effect in multiwinner elections, where the output of the voting method
is a committee of size at least two. To evaluate STV we compare it to numerous
other voting methods including single non-transferable vote, $k$-Borda, and
the Chamberlin-Courant rule. We provide simulation results under three
different random models and empirical results using a large database of real-
world multiwinner political elections from Scotland. Our results show that STV
is not spoiler-proof in any meaningful sense in the multiwinner context, but
it tends to perform well relative to other methods, especially when using
real-world ballot data.

##  1\. Introduction

Multiwinner voting methods are used in a wide variety of settings, including
elections of multi-seat districts or city councils, the selection of a short-
list of job applicants, or the choice of webpages output by a search engine
given an initial search query. The social choice literature often takes an
axiomatic approach to evaluating such voting rules, articulating fairness
criteria and identifying which methods satisfy which criteria. If a method
fails a given criterion, various techniques are employed to estimate the
frequency with which violations occur. In this article we focus on one
fairness criterion–that of spoiler-proofness–and compare how frequently
different multiwinner voting methods are susceptible to the presence of
spoiler candidates. Our primary voting rule of interest is the method of
single transferable vote (STV), which is the most commonly used voting method
for multiwinner political elections in which voters cast preference ballots.
We examine several other voting methods in order to provide context for the
performance of STV regarding the spoiler effect. Our general finding is that
STV performs well with respect to this criterion, although depending on how we
generate ballot data other methods perform significantly better. Our study is
of interest because the single-winner implementation of STV (known as instant
runoff voting, the Hare method, the alternative vote, and colloquially, in the
single-winner case, as “instant-runoff voting” or “ranked-choice voting”) is
widely-touted as being spoiler-proof or virtually spoiler-proof. Previous work
[27] has largely shown this to be the case; we show that the multiwinner
picture is less clear.

The formal study of multiwinner voting rules is a relatively recent phenomenon
in the field of social choice. The modern mathematical discussion around
single-winner methods dates to the 18th century debate between Borda and
Condorcet, while the social-choice oriented study of multiwinner methods
arguably did not truly begin until the late 20th century with studies such as
[10], [11], [16], and [17]. The majority of recent articles which analyze
properties of multiwinner voting rules focus on methods which are approval-
based (i.e., voters cast approval ballots) [2, 8, 24, 20, 21, 33]. Since our
work is motivated by the study of STV, we do not consider approval-based
rules. Instead, we analyze methods which are preference-based, where voters
cast preference ballots which express a (possibly partial) linear preference
ranking of the candidates. In this setting, much of the previous work has
studied various monotonicity properties [4, 14, 16, 26, 35], generalizations
of the single-winner Condorcet criterion [3, 12, 17, 22, 19, 32], and axioms
of proportional representation [4, 7, 11, 37].

As far as we are aware, this article provides the first formal study of the
spoiler effect for multiwinner voting rules. The only work related to ours is
[6], which analyzes the spoiler effect in the context of apportionment-based
multi-district party elections. This work focuses on “spoiler parties” which
can affect the apportionment of seats to stronger parties. The results are
only tangentially related to our work, which focuses on spoiler candidates
instead of parties. In this paper, we adopt the “standard” definition of a
spoiler candidate [27] from the single-winner for the multiwinner setting: a
candidate is a _spoiler_ if they are not a winner and, if we remove the
candidate from the election, then the winning committee changes. An election
which contains a spoiler candidate under a given voting method is said to
exhibit the _spoiler effect_ for that method. In the U.S., the quintessential
example of an election demonstrating the spoiler effect is the 2000
presidential election, where the presence of the third-party candidate, Ralph
Nader, was seen as changing the election outcome from the Al Gore to the
election winner, George W. Bush. Multiwinner elections are also susceptible to
spoilers; the presence of losing candidates may cause changes to the winning
committee.

The spoiler effect has received some attention in the single-winner setting.
Depending on one’s definition of the spoiler effect, study of this topic dates
to Arrow’s classical axiom of the independence of irrelevant alternatives
(IIA) [1]. Miller [29] discusses the spoiler effect in the context of the
consistency property and its relationship to Arrow’s IIA axiom. Susceptibility
to spoilers has previously been studied by Tideman [36] with respect to
clones. A _clone_ is a candidate who is always just above or just below a
fixed second candidate in all voters’ rankings. A voting method is independent
of clones if the existence of a clone does not affect the outcome of the
election. The drawback of analyzing the spoiler effect through this lens is
that in real-world elections, particularly in the political context, clones do
not exist. McCune and Wilson [27] use simulations and empirical analysis based
on data from 170 municipal and statewide elections in the US to compare the
susceptibility of instant runoff voting and plurality to the spoiler effect.
They also analyze how frequently the spoiler is a “weak” candidate, defined as
a Condorcet loser. (This analysis was supplemented in the note [30]). In this
paper, we extend the work in [27] to the multiwinner setting, comparing STV to
a number of other multiwinner voting methods. Using Monte Carlo simulations
and data from 999 Scottish local government elections which used STV to elect
a winning committee, we examine the frequency with which the spoiler effect
occurs. We also measure how frequently weak candidates are spoilers based on
two notions of weakness: plurality losers and top-$k$ losers (candidates who
appear in the fewest number of voters’ top-$k$ rankings). Because partial
ballots occur frequently in our database of real-world elections, we include
comparisons assuming both complete and partial rankings.

Multiwinner voting methods for political elections fall loosely into two
categories based on the purpose of the election: excellence-based (or
_majoritarian_) and proportional [14, 23]. There is no standard criterion for
defining a proportional method; proposed criteria include Dummett’s
proportionality for solid coalitions [11], Woodall’s Droop proportionality
criterion [37], Aziz’s extended justified representation [2], and Brill and
Peters’ proportional justified representation [7]. In general, methods which
are designed for greater diversity, such as STV and the Chamberlin-Courant
method, are considered proportional. In contrast, methods which are designed
to select similar candidates—those who perform well on the majority of voters’
ballots—are considered excellence-based. These include single-winner voting
methods which are applied repeatedly to generate multiple winners, such as
$k$-Borda and sequential ranked-choice voting. In our analysis we include some
methods that are proportional and some that are majoritarian to investigate if
susceptibility to spoilers has any relationship to a method’s categorization.
We find that whether a method is proportional or majoritarian does not predict
its tendency to exhibit the spoiler effect.

In what follows, we compare the behavior of STV and other multiwinner voting
methods using both Monte Carlo simulations and empirical analysis. In Section
2 we provide notation and descriptions of the voting methods used in the
analysis which include bloc, $k$-Borda, single non-transferable vote,
sequential ranked-choice voting, a version of the Chamberlin-Courant rule (and
a greedy approximation), and STV. In Section 3 we compare the likelihood of an
election demonstrating a spoiler effect among these methods using three models
of voter behavior: impartial culture (IC), independent anonymous culture
(IAC), and a 1D-spatial model. We consider both complete and partial ballots.
We also examine the likelihood for the spoiler candidates to be weak. In
Section 4 we compare these methods empirically, based on their behavior on the
database of Scottish elections. To more closely mirror the analysis in Section
3, we conduct the analysis twice: once using the ballots directly from the
database (which includes a large number of partial ballots), and a second
time, extrapolating from the voters’ preferences to extend the lengths of the
ballots and create preferences that are closer to complete. As with all such
analyses, the results must be interpreted with caution: had methods other than
STV been used in practice, it is possible that voters’ preferences, as
expressed through their ballots, would be different. In Section 5, we
introduce two new methods, one which is related to STV and one which is
inspired by the Condorcet criterion, which are almost spoiler-proof when
tested against the Scottish database. We conclude in Section 6.

##  2\. Preliminaries

An election is a pair $E=(C,V)$ where $C=\\{C_{1},\ldots,C_{m}\\}$ is a set of
candidates, $V=\\{v_{1},\ldots,v_{n}\\}$ is a set of voters endowed with a set
$\leq$ of linear orders $\\{\leq_{i}\\}_{i\in V}$ over the set of candidates.
We assume, for each voter $v_{i}$, the order is strict for some $j<m$
candidates, and that the voter is indifferent among the remaining candidates,
who do not appear on the voter’s ballot. That is, we assume a framework in
which partial ballots are allowed, and candidates left off a ballot are all
ranked last by the voter. A multiwinner voting rule $F$ is a function that,
for each election $(C,V)$ and number of winners $k\leq m$, outputs a set of
winning committees, $F(C,V,k)=\\{S_{1},\ldots,S_{\alpha}\\}$, each of size
$k$. The voting rules we consider are generally resolute—that is, they output
a single winning committee, multiple winning committees being a result of
(infrequent) ties. When the voting rule is clear, we will denote the set of
winning committees by $\mathcal{S}=\\{S_{1},\ldots,S_{\alpha}\\}$, or when
uniquely defined, $S$.

To define a spoiler candidate, let $C_{-i}=C\setminus{C_{i}}$ be the candidate
set remaining after candidate $C_{i}$ leaves the election, and let
$F(C_{-i},V)$ be the resulting set of winning committees under voting rule
$F$. Then $C_{i}$ is a _spoiler candidate_ if $C_{i}$ is not in any winning
committee and $F(C,V)\neq F(C_{-i},V)$. An election exhibits the _spoiler
effect_ under some voting method if there exists a spoiler candidate under
that method. A voting method that is more likely to exhibit the spoiler effect
is less desirable than one that isn’t, particularly if the spoilers produced
tend to be weak in some sense. (We prefer to avoid weak spoiler candidates
because they suggest that the election outcome may be affected by the presence
or absence of a candidate who in some sense is irrelevant to the election.) In
this way, being susceptible to spoilers is similar to failing Arrow’s famous
IIA criterion. For single-winner voting methods, failure of IIA occurs if a
winner can become a loser when a voter’s preferences change without affecting
their relative ranking of the winner and the loser (the ranking of a third
candidate changes). In a spoiler situation, a winner becomes a loser when a
non-winning third candidate is removed entirely from voters’ preferences. As
with the IIA criterion, spoiler-proofness is virtually impossible for
reasonable voting methods to achieve. However, some voting methods are much
more susceptible to the spoiler effect than others.

Since spoiler candidates are considered especially egregious when they have
little chance of being among the winners, we identify two notions of weakness.
The first is the plurality loser, which is the candidate with the fewest
first-place votes. The second is the top$-k$ loser, which is the candidate who
ranks among the top $k$ candidates in the fewest number of ballots.

###  2.1. Multiwinner voting rules

We now define the voting methods which we analyze in the paper. Initially we
provide definitions which assume that every voter provides a complete ranking
of the candidates; we address how to adapt definitions to partial ballots at
the end of the section. Because ties are rare in our work, we do not discuss
how to break ties as part of our definitions.

Single transferable vote (STV) refers to a family of related voting rules. We
consider here the version used for the Scottish elections which uses the droop
quota $q=\lfloor\frac{n}{k+1}\rfloor+1$ and transferal of fractional votes.
The election proceeds in a series of rounds, in each of which either a
candidate is eliminated or one or more candidates is selected to be part of
the winning committee. In each round, the number of first-place votes for each
candidate is determined. Any candidate receiving at least $q$ votes is
selected as a winner and the number of votes over the quota they receive
(referred to as their surplus), is transferred to the candidates next on the
voters’ ballots in proportion to the votes those candidates receive (with the
caveat that no previously eliminated or elected candidate can receive
transferred votes). Specifically, if $C_{i}$ is selected as a winner after
receiving $x_{i}\geq q$ first-place votes, and if candidate $C_{j}$ appears
next in $x_{j}$ of these ballots, then the number of fractional votes
transferred to $C_{j}$ is equal to $\frac{x_{i}-q}{x_{i}}x_{j}$. If no
candidate receives at least $q$ votes then the candidate with the least first-
placed votes is eliminated. This process continues until either $k$ candidates
have met the quota or until some number $k^{\prime}<k$ candidates have been
selected and there are only $k-k^{\prime}$ candidates remaining who have not
been eliminated (and hence are selected). A complete description of the STV
ballot counting process as implemented in Scottish local government elections
can be found at https://www.legislation.gov.uk/sdsi/2007/0110714245.

Under the method of single non-transferable vote (SNTV), or $k$-plurality, the
winning committee consists of the $k$ candidates with the largest number of
first-place votes.

Sequential ranked-choice voting (SRCV) works by sequentially filling seats
using the singe-winner implementation of STV. The first seat is given to the
winner of the election when using STV but assuming $k=1$. That is, we
eliminate the candidate with the fewest first-place votes and transfer their
ballots as if using STV, until a candidate earns a majority of the first-place
votes. That candidate is then given the first seat, and is eliminated from the
ballot data. We repeat the process, again using STV assuming $k=1$, on the
modified ballot data (where the first winner has been removed) to assign the
second seat; the resulting winner is given the second seat and eliminated from
the ballot data. This process continues until all $k$ seats are filled. In
other words, SRCV is equivalent to running $k$ separate STV elections with
$k=1$, each time a seat is awarded to a candidate, eliminating that candidate
from the ballot data so that the next iteration of STV is based on on one
fewer candidate. SRCV is used rarely and is little studied; [28] compares SRCV
to STV, finding that its behavior is closer to excellence-based methods rather
than proportional methods such as STV. We include SRCV in our study because it
is a majoritarian method that has been used in real-world elections, most
notably in city council elections for several small cities in the state of
Utah in the US.

Under the $k$-Borda rule (Borda), candidates are assigned points based on
their position on individual voter’s ballots. After summing, the $k$
candidates with the highest total Borda score form the winning committee. For
a given ballot, a first-place ranking is worth $n-1$ points, a second-place
ranking is worth $n-2$ points, and so on.

Under Bloc voting (Bloc), the winning committee consists of the $k$ candidates
receiving the highest $k$-approval score. The $k$-approval score is the number
of voters who rank the candidate among their top $k$ candidates.

There are a number of variants of the Chamberlin-Courant rule (Cham-Cour) [9].
In each variant, voters are “assigned” a member of the winning set. This
assignment gives each voter a measure of individual satisfaction; these
measures are combined to create a measure of social satisfaction and the
winning committee is defined to be the set of candidates that maximizes this
social satisfaction. The version of Cham-Cour we consider here is the original
method proposed in [9] which uses the candidates’ Borda score and its sum to
determine the level of individual satisfaction and social satisfaction
respectively. Formally, let $r_{ic}$ denote the rank of candidate $c$ on voter
$i$’s ballot, so that this voter gives $m-r_{ic}$ points to $c$. For a fixed
committee $X$ of size $k$, let $V_{c}(X)$ denote the set of voters for whom
candidate $c$ is the most preferred candidate in the committee $X$ for all
voters in $V_{c}(X)$. Cham-Cour selects the committee $X$ of size $k$ which
maximizes the value

$\displaystyle\sum_{c\in X}\displaystyle\sum_{i\in V_{c}(X)}m-r_{ic}.$

Cham-Cour is computationally difficult to implement. (Determining the winning
committee is NP-hard; see, for instance, [25].) We thus also consider greedy
Chamberlin Courant (greedy-CC) [25], which approximates the outcome of Cham-
Cour. In the algorithm’s first step, the candidate with the highest Borda
score is selected. At each succeeding step, given a winning committee of size
$k-1$, a $k^{th}$-candidate is selected which maximizes the increase in social
satisfaction.

If voters cast partial ballots then we must state how partial preferences are
handled by any method which uses Borda scores (for the other methods defined
above, their adaptation to a setting with partial ballots is natural). We use
two models of processing partial ballots, an optimistic and pessimistic model
[5]. Suppose a partial ballot of voter $i$ has length $l$, so that $l<m$
candidates are ranked on the ballot. Under the optimistic model (OM), if
candidate $c$ appears on the ballot then they receive $m-r_{ic}$ from voter
$i$; if $c$ does not appear on the ballot then they receive $m-l-1$ points
from voter $i$. Under the pessimistic model (PM), if candidate $c$ appears on
the ballot then they receive $m-r_{ic}$ from voter $i$; if $c$ does not appear
on the ballot then they receive zero points from voter $i$. When using a
Chamberlin-Courant-based rule, the two models naturally extend when assigning
points to subsets. Under OM (respectively PM), a fixed committee $X$ receives
$m-l-1$ points (respectively zero points) from a voter if no candidate from
$X$ appears on that voter’s ballot.

Thus, for each of the three methods Borda, Cham-Cour, and greedy-CC, we have
OM and PM versions when an election contains partial ballots.

##  3\. Simulation Results

To establish a theoretical baseline for the frequency of the spoiler effect,
we ran Monte Carlo simulations under three different models of voter behavior.
Under each model we generated ballot data at random and for each election we
checked if the spoiler effect was demonstrated. In this section we describe
our models and present and analyze the results of the simulations.

###  3.1. About the Models

We use three models to generate ballot data for our simulations: impartial
culture (IC), independent anonymous culture (IAC), and a one-dimensional
spatial model (1D-spatial). The IC and IAC models are used frequently to
investigate voting method properties in the absence empirical information.
These models are commonly used to provide _a priori_ probability estimates
when there is no real-world information available, and often the models
provide theoretical upper bounds for probabilities of various phenomena in
voting theory. The IC model assumes each voter’s preference is chosen
independently and uniformly from among all possible rankings of the
candidates. The IAC culture assumes that voters are anonymous
(indistinguishable) and each voter profile occurs with equal probability.
Because the voter profiles contained in the Scottish database are based on
partisan elections, we also include a single-peaked 1D-spatial model that
reflects voter preferences along a political spectrum. We assume candidates
and voters are randomly assigned a point along $\mathbb{R}$ using a standard
normal distribution. Voters’ preferences are based on the Euclidean distance
from their point to each of the candidates (see [13] for a similar 2D-spatial
model).

In comparison to the database analyzed in Section 4, none of the simulation
models appear to be strong predictors of voter behavior. One reason for this
is that voter profiles are much less close (under any reasonable measure of
“closeness”) in the real-world election data than in any of the three models.
The second reason is that partial ballots are extremely common in real-world
data. In fact, partial ballots are prevalent in any database of real-world
political ranked-choice elections (see the ballot data at [31], for example).
In the Scottish case, only about 13.2% of the ballots in the database contain
a complete ranking of the candidates. This is lower than the percentage of
ballots which rank only a single candidate, 14.0%, and much lower than the
percentage of ballots which rank fewer candidates than the number of available
seats, 58.0%. That is, a “typical” voter in a Scottish local government does
not even rank $k$ candidates on their ballot. Thus, to better compare the
simulations with the empirical results, we also consider simulations based on
partial ballots. This is accomplished differently for each model. Under IC,
each voter is assumed to select independently and uniformly from all possible
partial ballots. Under the IAC model, voter profiles are chosen uniformly over
all possible distributions of complete and partial rankings. Under the
1D-spatial model, each voter is assumed to decide independently and uniformly
the number of candidates to rank, based on their individual preference order.
Since a ballot which ranks $m$ candidates conveys the same ranking information
as a ballot which ranks $m-1$ candidates, we omit ballots which rank $m$
candidates to avoid double counting.

###  3.2. Analysis

Separate runs of 100,000 simulations were completed for each model with: (i)
$m=4$ and $k=2$; (ii) $m=5$ and $k=2$; and (iii) $m=5$ and $3$. Due to limits
of computation time, we did not run simulations for $m>5$. For $m=4$, $k=2$,
we have included results for both Cham-Cour (OM) and Cham-Cour (PM), as well
as greedy-CC (OM) and greedy-CC (PM). Because of the computation time, for
$m=5$ only the greedy-CC methods have been simulated. Each simulation was
conducted assuming $1001$ voters. For comparison, we also ran some simulations
using 601 and 801 voters and the results did not meaningfully change. This
agrees with previous work in the single-winner case, where increasing the
number of voters does not materially affect the results [27]. Thus, our
results appear to be robust with respect to the number of voters as long as
the number is sufficiently high. (We do not care about small electorate sizes
since we are motivated by comparisons with actual elections.) Simulations
resulting in a tie, either in the original election or in the resulting
election when a candidate was removed, were thrown out. Since these occasions
were few in number, they also do not materially affect the analysis.

The results are provided in Tables 10 through 18 in Appendix B. Each table
indicates the probability (expressed as a percentage) that an election
contains a singe spoiler or multiple spoilers. We also indicate the percentage
of elections with spoiler candidates involving either the plurality loser or
the top-$k$ loser. Probabilities are indicated for both complete ballots (left
entry in each column) and partial ballots (right entry in each column).

Several observations can be drawn from the simulation results. Chief among
them is that STV generally performs very well in comparison to other voting
methods. Under the IC model, STV is the least susceptible to spoilers for
either complete or partial ballots. STV is also least susceptible under the
IAC model, with the exception of complete ballots for $m=4$, $k=2$, where SRCV
returns the lowest probability. Interestingly, STV does not perform well under
the 1D-spatial model, in some cases corresponding to the second-highest
probability. Generally, though, STV dramatically outperforms methods like
greedy-CC or SNTV. Both SRCV and Borda also perform well under all models.

On the other end of the spectrum, Bloc voting is the most susceptible to the
spoiler effect, followed by SNTV. This makes sense because both methods are
similar in spirit to plurality elections: information about voters’
preferences beyond the top $k$ rankings is disregarded. The other methods fall
somewhere in the middle. The exception to this is the 1D-spatial model, where
Bloc actually returns the lowest probability. In fact, the zeros corresponding
to complete ballots in Tables 16 and 18 are exact zeros, unlike the “0.0” for
STV in Table 15, for instance, where the probability of a spoiler is
vanishingly small. This is because it is not possible to have a spoiler under
Bloc if voters express complete preferences when $m=4$, $k=2$ and $m=5$,
$k=3$. Under the 1D-spatial model, a winning committee must be composed of
candidates who are adjacent on the number line; if a non-winning candidate is
eliminated, it is easy to see that the the change in votes will not be
sufficient to affect the winners. (This is not the case when $m=5$, $k=2$. If
the winning committee consists of the two left-most candidates and the right-
most candidate is eliminated, the left-most candidate may no longer be a
winner.)

Overall, the simulation results suggest that immunity to spoilers is not
directly linked to whether a method is proportional or excellence-based. Both
SRCV and Borda are majoritarian methods, yet their susceptibility to the
spoiler effect is wildly different. Similarly, STV, Cham-Cour and greedy CC
can be categorized as proportional, and return much different spoiler
probabilities.

In general, the probability of a spoiler increases with $m$ and, for fixed
$m$, decreases as $k$ increases. This accords with intuition: we would expect
the probability of a spoiler to rise with the number of candidates and, given
a fixed number of candidates, to decrease when the number of potential
spoilers (the number of non-winners $m-k$) is reduced. In addition, the
probability of there being multiple spoilers is quite small. Of course, when
$m$ and $k$ differ by only 2, an election will have more than one spoiler only
if both non-winning candidates are spoilers which, based on the Tables 10 and
18, happens relatively infrequently except under Bloc or SNTV. The
probabilities of multiple spoilers under all methods increase somewhat for
$m=5$ and $k=2$ with the number of elections having multiple spoilers ranging
from about one-sixth to about one-half of all the elections containing
spoilers. Again, both Bloc and SNTV perform worse in this regard—with a higher
proportion of elections with spoilers having multiple spoilers. STV, in
contrast, almost never has multiple spoilers.

Weak Spoilers Overall, both SRCV and STV are very unlikely to experience weak
spoilers—particularly spoilers that are plurality losers. For most voting
methods, and under all models, the probability of the spoiler being a top-$k$
loser is greater than the probability of the spoiler being a plurality-loser.
(The exceptions are Cham Cour, greedy-CC and SNTV.) The differences are
particularly strong under Bloc, in part because in comparison to a top-$k$
loser, a plurality loser is much more likely to be in the winning set, and
hence cannot be a spoiler. The reverse is true of SNTV, where the plurality
loser is never a winner and hence has the potential to be a spoiler.

Complete versus Partial Rankings With a few exceptions, the probability that a
voter profile is susceptible to the spoiler effect is less when partial
ballots are allowed, although the differences are generally minor. The
exception is Bloc, where the probability of having a spoiler drops with
partial ballots under IC and IAC models. This is likely because if partial
ballots are allowed, a significant number of them may rank fewer than $k$
candidates. If a candidate not appearing on the ballot is eliminated, it will
not change how the ballot is counted. Thus, it is less likely that an
eliminated candidate changes the winning committee. This is not the case in
the 1D-spatial model, because of the limitations of the kinds of ballots
allowed. Note that unlike profiles containing only complete ballots, if
partial ballots are allowed, spoilers are possible for all choice of $m$ and
$k$. (Because of bullet voting, for instance, the winning committee may
consist of the left-most and right-most candidates. The elimination of a
middle candidate may thus cause another middle candidate to become a winner.)

Probabilities for the spoiler effect also increase under partial ballots under
the OM models for Borda, Cham-Cour, and greedy-CC, Note that under Borda (OM),
the probability of a spoiler increases slightly with partial ballots, while
under Borda (PM), the probability decreases slightly. This is to be expected;
under Borda (OM), candidates left unranked on voters’ ballots are assumed to
be tied for the next open spot on the ballot. Thus they are assigned a higher
weight than had the ballots been completed, and therefore are more likely to
affect the outcome of the election if they leave. The reverse is true under
Borda (PM), where unranked candidates are assigned minimal weight and thus are
unlikely to be spoilers. The same rationale applies to the likelihood of a
spoiler under complete and partial ballots for Cham-Cour (OM) and Cham-Cour
(PM), as well as greedy-CC (OM) and greedy-CC (PM). In each case, the OM
unranked candidates are assigned a relatively high weight under the OM version
in comparison to that under the PM version. Thus, candidates removed from the
election are likely to have a bigger impact on the overall society
satisfaction levels, and hence be spoilers, under the OM version than the PM
version.

The fact that the differences between spoiler susceptibility are so small for
complete and partial ballots under STV, is likely due to the fact that STV the
method as a whole is relatively immune to spoilers and because $m$ and $k$ are
relatively close in our simulations.

Due to limits of computation time, we did not perform simulations with more
than five candidates. If the difference between $m$ and $k$ were substantially
larger, we would expect to see a larger difference between the partial ballot
and complete ballot cases.

##  4\. Empirical Results

In this section we consider the likelihood of the spoiler effect using
empirical data collected from 999 elections held in Scotland between 2007 and
2022 to fill council seats for local government elections. We first briefly
describe the data and then present our results.

###  4.1. About the Database

For the purposes of local government, Scotland is partitioned into 32 council
areas, each of which is governed by a council. The councils provide a range of
public services that are typically associated with local governments, such as
waste management, education, and building and maintaining roads. The council
area is divided into wards, each of which elects a set number of councilors to
represent the ward on the council. The number of councilors representing each
ward is determined primarily by the ward’s population, although other factors
play a role. Every Scottish ward has used STV for local government elections
since 2007 and council elections are held once every five years. We have
access to the ballot data for 1100 Scottish elections, most of which come from
the 2012, 2017, and 2022 election cycles (most of the data from 2007 is
unavailable). The full dataset was collected by the first author for [26] and
is available at https://github.com/mggg/scot-elex.

In the vast majority of wards, $k=3$ or $k=4$: thirty elections satisfy $k=1$,
five satisfy $k=2$, 554 satisfy $k=3$, 508 satisfy $k=4$, and three satisfy
$k=5$. Because the spoiler effect cannot occur under any voting method if
$m=k+1$, not all 1100 elections are useful for our analysis. If for each
election we use the same $k$ value as was used in the actual election, there
are 999 elections which satisfy $m>k+1$ and $k>1$. These are the elections on
which the results of this section are based. Across these 999 elections,
partial ballots are extremely common. Approximately 57.6% of voters rank fewer
than $k$ candidates on their ballots. Only 12.0% of ballots provide a complete
ranking111By “complete ranking” we mean that a ballot ranks $m$ or $m-1$
candidates., which is lower than the percentage of ballots with only a single
candidate ranked, 13.6%. The median number of candidates for these elections
is seven (with a maximum of fourteen), and thus our ballot data is missing a
significant amount of ranking information.

###  4.2. Analysis

The results of the spoiler analysis are presented in Table 1. The columns are
organized in the same way as in Section 3: the first number is the estimated
probability (as a percentage) using (almost) complete ballots, and the second
is the estimated probability using partial ballots. In this case, the second
number represents the “actual” result—the result obtained from using the
actual ballots. The first number represents a spoiler probability obtained
from ballot data in which we have “filled in” in voter preferences so that the
ballots are “less partial;” we discuss these more hypothetical results, as
well as how the ballots were completed, at the end of the section.

Based on the results corresponding to the actual ballot data (the second
number in each column), the probability of a spoiler occurring under all
voting methods is distinctly lower than under any of the simulations,
suggesting that the simulations provide only an upper bound for the likelihood
of the spoiler effect occurring. SRCV behaves the best, with only a total of
28 elections susceptible to a single spoiler. Similarly effective are STV and
Borda (PM). Even Bloc and SNTV, which appeared highly susceptible to spoilers
under the IC and IAC models, appear moderately robust. As with the results in
Section 3, the OM versions of Borda, Cham-Cour and greedy-CC all behave worse
than the PM versions. This is to be expected since the OM versions assume
candidates not appearing on partial ballots are assumed to be higher on the
ballot and hence have greater impact on election results when eliminated.

Table 1. Likelihood of a Spoiler in the Scottish Election Database Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 30.3, 7.0 | 18.1, 4.5 | 12.0, 4.2 | 16.2, 4.2
Borda (OM) | 19.0, 16.6 | 6.2, 4.1 | 7.0, 6.6 | 8.4, 5.7
Borda (PM) | 11.2, 4.4 | 3.9, 1.8 | 3.9, 1.8 | 7.1, 2.2
Cham-Cour (OM) | 12.2, 10.2 | 4.9, 4.5 | 6.4, 4.9 | 3.0, 3.8
Cham-Cour (PM) | 10.4, 6.4 | 4.1, 3.5 | 5.6, 4.8 | 3.2, 4.1
Greedy-CC (OM) | 14.0, 10.5 | 5.6, 4.5 | 6.7, 4.8 | 3.9, 3.7
Greedy-CC (PM) | 12.8, 7.0 | 5.0, 3.3 | 6.2, 4.9 | 4.3, 4.2
SRCV | 14.5, 2.8 | 0.2, 0 | 0, 0.1 | 2.7, 0.4
SNTV | 12.9, 11.0 | 5.4, 5.0 | 4.0, 3.3 | 3.6, 3.0
STV | 10.6, 4.9 | 2.0, 0.9 | 1.4, 0.9 | 2.4, 1.0

In general, the elections in which there are multiple spoilers were closely
contested. We illustrate this with an analysis of the 2022 election in Ward 5
of the City of Edinburgh Council Area in which 10 candidates ran to fill 4
seats and all 6 losing candidates were spoilers.

######  Example 1.

In the original election, the winning set consisted of Bandel, Mitchell,
Nicolsn and Osler. Bandel wins a seat in the last round because Nicolson
surpasses quota by a wide margin, allowing for a large transfer of surplus
votes to Bandel as shown in Table 2 (TOP). If Herring is eliminated, Nicolson
does not have nearly as many surplus votes to transfer, and Wood replaces
Bandel in the winner set as shown in Table 2 (MIDDLE). The result is the same
(with different numbers of surplus votes) if any other candidate besides Wood
is eliminated. If Wood is eliminated, Munro-Brian replaces Bandel in the
winner set as shown in Table 2 (BOTTOM).

Actual Election, Quota $=2684$
---
Candidate | Votes by Round
Bandel | 1714 | 1740.1 | 1740.1 | 1741.4 | 1751.5 | 1767.8 | 2221.6 | 2379.8 | 2383.7 | 2959.0
Herring | 853 | 863.1 | 863.1 | 867.1 | 889.3 |  |  |  |  |
Holden | 96 | 97.4 | 98.4 | 109.4 |  |  |  |  |  |
Laird | 53 | 53.6 | 53.6 |  |  |  |  |  |  |
McNamara | 17 | 17.3 |  |  |  |  |  |  |  |
Mitchell | 1836 | 1877.5 | 1878.7 | 1883.8 | 1896.8 | 2643.6 | 2767.6 |  |  |
Munro-Brian | 1684 | 1713.0 | 1715.0 | 1721.0 | 1736.2 | 1755.3 |  |  |  |
Nicolson | 2641 | 2657.3 | 2659.3 | 2663.3 | 2668.4 | 2683.4 | 2936.8 |  |  |
Osler | 3117 |  |  |  |  |  |  |  |  |
Wood | 1405 | 1700.6 | 1702.7 | 1711.7 | 1725.3 | 1765.1 | 2275.5 | 2303.0 | 2337.8 |

Herring Removed, Quota $=2677$
---
Candidate | Votes by Round
Bandel | 1728 | 1757.5 | 1757.5 | 1759.8 | 1760.4 | 1773.6 | 2331.8 | 2336.0 |
Holden | 108 | 110.0 | 111.0 | 124.0 | 124.0 |  |  |  |
Laird | 60 | 60.8 | 60.8 |  |  |  |  |  |
McNamara | 18 | 18.3 |  |  |  |  |  |  |
Mitchell | 2530 | 2585.3 | 2586.5 | 2595.6 | 2595.7 | 2635.8 | 2768.4 |  |
Munro-Brian | 1698 | 1730.7 | 1732.7 | 1741.7 | 1741.9 | 1760.0 |  |  |
Nicolson | 2654 | 2672.1 | 2674.1 | 2678.1 |  |  |  |  |
Osler | 3168 |  |  |  |  |  |  |  |
Wood | 1418 | 1754.0 | 1757.2 | 1767.2 | 1767.2 | 1783.9 | 2344.7 | 2382.5 | 3342.3

Wood Removed, Quota $=2681$
---
Candidate | Votes by Round
Bandel | 1726 | 1991.3 | 2036.3 | 2036.3 | 2041.6 | 2059.6 | 2082.9 | 2108.4 |
Herring | 886 | 1028.8 | 1029.3 | 1029.7 | 1035.5 | 1061.0 |  |  |
Holden | 98 | 113.9 | 114.6 | 115.6 | 129.2 |  |  |  |
Laird | 53 | 64.1 | 64.4 | 65.8 |  |  |  |  |
McNamara | 17 | 21.4 | 21.5 |  |  |  |  |  |
Mitchell | 1845 | 2227.4 | 2228.8 | 2230.6 | 2238.2 | 2257.4 | 3151.1 |  |
Munro-Brian | 1706 | 2054.3 | 2065.7 | 2068.7 | 2077.7 | 2101.9 | 2128.2 | 2218.0 | 3436.0
Nicolson | 2645 | 2756.1 |  |  |  |  |  |  |
Osler | 4447 |  |  |  |  |  |  |  |

Table 2. (Top) The votes in each round for each candidate in the 2022 election
in Ward 5 of the City of Edinburgh Council Area. (Middle) How the election
would unfold if Herring were removed. (Bottom) How the election would unfold
if Wood were removed.

Example 1 demonstrates that multiwinner elections can demonstrate different
spoiler dynamics under STV than single-winner elections. If $k=1$ then it is
not possible for all losing candidates to be spoiler candidates under STV,
since the plurality loser cannot be a spoiler in this case.

Weak Spoilers The results in Table 1 also indicate that many of the spoilers
are weak—being either plurality or top-$k$ losers. As with the simulation
results, both SRCV and STV are very unlikely to experience weak spoilers. With
the exception of Borda (PM), the probability of a spoiler being a plurality
loser is greater than the probability of a spoiler being a top-$k$-loser
spoiler. Among the other methods, the PM versions of Borda, Cham-Cour and
greedy CC are all more likely to have weak spoilers (either plurality-losers
or top-$k$-losers) than their OM counterparts. Since they are less likely to
have spoilers overall, this means that the proportion of weak spoilers among
all spoilers under the PM methods is much higher than the similar proportion
under OM models. This pattern differs from the simulation results and is
likely due the large number of partial ballots in the data. Hence the weak
candidates are likely to be very weak. Under the PM model, they have a minimal
chance to be a winner and hence have more potential to be a spoiler.

Stability under Spoilers Despite the large number of spoilers in Example 1,
the analysis suggests a certain degree of stability: the elimination of each
of these 6 candidates results in only two alternate winning committees.
Moreover Osler, Nicolson and Mitchell are in the winning sets in both the
original election and in the alternate winning sets. This is perhaps excepted
under STV, since candidates that easily surpass the quota will do so
regardless of who drops out. In fact, this stability occurs under several
voting methods.

Table 3 compares the stability of the winning sets of the different voting
methods. Columns 2 and 3 indicate the number of elections with a spoiler and
with more than one spoiler respectively. Column 4 identifies the maximum
number of spoilers in a single election. Column 5 indicates the number of
elections with multiple spoilers in which there was more than one alternate
winning set (depending on which candidate was eliminated). Column 6 indicates
the largest number of alternate different winning sets in a single election.
SRCV, of course, had no instances of multiple spoilers and hence the single
spoiler resulted in only one alternate winning set. SNTV stands out for the
large number of elections resulting in multiple alternate winning sets.
(Again, the OM versions of Borda, Cham-Cour and greeedy-CC behave worse in
this respect than their PM versions). Overall, the maximum number of alternate
winning committees in column 6 is not large. Even for Borda (OM), under which
one voter profile was susceptible to different 7 spoilers (the 2017 election
in the 4th Ward of the East Renfrewshire Council Area), the elimination of
these spoilers resulted in only 3 alternate winning sets. The fact that the
number of alternate winning committees is not generally correlated to the
number of spoilers under any voting method is likely due to the lack of
closeness in most of the elections in the database.

Omitted from Table 3 are three elections in which the result was a tie. In the
2017 election of the 8th Ward of the Moray Council Area, there were 7
candidates running for 4 seats. Under both Cham-Cour (OM) and greedy-CC (OM),
the result is a tie in which all candidates are potential spoilers. The same
is true of the 2017 election in the 7th Ward of the Glasgow City Council Area
and the 2022 election in the 4th Ward of the Clackmannanshire Council Area, in
which all 9 (resp. 7) candidates running for 4 seats resulted in a tie under
Cham-Cour (OM) (resp. Cham-Cour (PM)). Unsurprisingly, in these instances,
there were a larger number of alternate winning sets depending on which
candidate was eliminated.

Table 3. Numbers of Spoiler Elections and numbers of different winning sets in the Scottish Election Database Method | Spoilers. | Mult. | Greatest | Num. Mult. | Greatest Num.
---|---|---|---|---|---
| (Num.) | (Num.) | Num. Sp. | Winning sets | Winning Sets
Bloc | 70 | 45 | 5 | 13 | 2
Borda (OM) | 166 | 41 | 7 | 13 | 3
Borda (PM) | 44 | 18 | 3 | 4 | 2
Cham-Cour (OM) | 102 | 45 | 5 | 7 | 2
Cham-Cour (PM) | 64 | 35 | 5 | 3 | 2
Greedy-CC (OM) | 106 | 45 | 5 | 9 | 2
Greedy-CC (PM) | 70 | 33 | 3 | 5 | 2
SRCV | 28 | 0 | 1 | 1 | 1
SNTV | 110 | 50 | 5 | 32 | 3
STV | 49 | 9 | 6 | 2 | 2

A closer analysis of the impact of spoilers in the database yields one more
measure of stability. With only two exceptions, the alternate winning
committees under all voting methods differed from the actual winning
committees by only one candidate. These exceptions both involved Cham-Cour:
under Cham-Cour (PM), the 2012 election in the 3rd ward of the Aberdeenshire
Council Area demonstrated an alternate winner set which differed from the
actual winner set by more than one candidate; similarly for greedy-CC (OM) in
the 2022 election in the 13th ward of the Glasgow City Council Area. This
result is somewhat surprising, since the theoretical maximum number of
candidates that could be effected by a single spoiler is certainly greater
than 1. This result is likely again due to the fact that most of the elections
in the database were not close and suggests that in practice, spoilers are
less likely to have as big an impact as is theoretically possible.

(Almost) Complete versus Partial Rankings The results inTable 1 also include
the likelihood of a spoiler based on rankings that are almost complete. These
rankings were obtained through a process of extrapolation in which the partial
ballots were filled in proportionally based on the distribution of existing
similar but longer ballots. See Appendix B for a more detailed description of
this process. This resulted in rankings in which 71.5% of ballots were
complete and only 1.2% of ballots ranked fewer than $k$ candidates. None of
the ballots ranked only a single candidate. We note that we do not claim that
our methodology for extending ballots is “best” or “most natural” in any
sense. It would be a good avenue for future work to investigate and compare
other ways of approximating more complete preferences from actual data as a
way of better understanding the spoiler effect and other voting anomalies in
multiwinner voting methods

The frequency of spoilers under the different voting methods using these more
complete ballots is indicated by the first number in each entry of Table 1. As
with the simulations, the probability of a spoiler or a mutual spoiler is
greater with the more complete rankings. The differences under Bloc are
particularly large. With a couple of exceptions (SRCV and Cham-Cour (OM)), the
probability of a weak spoiler is also greater under the more complete
rankings.

Significant differences remain between the likelihood of spoilers under OM and
PM versions of Borda, Cham-Cour and greedy-CC under the more complete
rankings. This is a consequence of the fact that not all ballots were
completed, leading to different results in the OM and PM versions. Of course,
if we were able to extend each ballot completely then the differences between
OM and PM versions of the same method would disappear.

Comparison to the Single-Winner Case. The most comprehensive empirical
analyses of the spoiler effect for single-winner ranked-choice elections occur
in [18] and [27]. Both articles study single-winner elections in which no
candidate earns an initial majority of first-place votes (that is, the
elections considered all went to at least a second round). In [27], 2 out of
170 elections demonstrated the spoiler effect under STV, while 4 out of 185
exhibited this effect in [18]. Thus, estimated frequencies for the spoiler
effect when $k=1$ for STV are on the order of 1-2%, and this percentage would
decrease dramatically if those studies were to include elections with majority
candidates. None of these elections returned multiple spoiler candidates.

As we have shown, the situation is different for the multiwinner case. The
spoiler effect rate of 4.9% for STV is significantly larger than what we see
in the single-winner case, and the multiwinner setting produces several
elections which demonstrate multiple spoilers. _A priori_ , it is not clear
why the multiwinner case should produce such different results. There could be
a number of factors pushing the spoiler rate higher. We suspect that the main
reason it is easier to observe spoiler effects is that it is “easier” to
register a spoiler hit when there are multiple winners. In the single-winner
case we must replace the entire winner set, while in the multiwinner case we
need merely knock one of the current winners out of the committee. We note
that all elections analyzed [18] and [27] occurred in the United States, so it
is also possible that the spoiler effect might have different frequencies in
different jurisdictions. In particular, American elections do not have the
multi-party flavor of most Scottish local government elections. Future
research could more thoroughly explore if we should expect a much higher rate
of the spoiler effect when $k>1$.

###  4.3. The issue of “vote splitting”

In the popular discourse concerning ranked-choice voting and the spoiler
effect, the terms “spoiler effect” and “vote-splitting” are often used
interchangeably [27]. One reason for this is because of their frequent co-
occurance under plurality voting (the method of SNTV with $k=1$). Discussions
of the spoiler effect under plurality often involve examples such as that
shown in Table 4 where Candidate $A$ is the plurality winner, but only because
$W$ and $S$ “split the vote.” If $S$ were not in the election, $W$ would
easily win. Such examples motivate Tideman’s definition of “clone candidates”
[36] and its relationship to the spoiler effect. In this example, $W$ and $S$
are clones because they appear in adjacent rankings on every ballot.

Table 4. Example of a spoiler under plurality due to vote-splitting Num Voters | 100 | 90 | 40
---|---|---|---
1st choice | $A$ | $W$ | $S$
2nd choice | $W$ | $S$ | $W$
3rd choice | $S$ | $A$ | $A$

In this section we investigate the extent to which the spoiler effect can be
identified with the notions of clones and vote splitting. We use the 999
Scottish elections with the real $k$ values to examine if spoiler candidates
are “clone-like” with respect to the candidate they are preventing from being
a winner under different voting methods. To be precise, for each voting method
we identify all triples of the form $(A,W,S)$ where $A$ is a candidate who was
a member of the winning committee and $W$ is a “would-be winner” who would win
a seat if spoiler candidate $S$ were removed from the election. In all but two
cases, when a spoiler effect occurred under any voting method the winner set
changed by only one candidate; we ignore these two outlier cases. To measure
“clone-ness”, for each such triple, we count the number of ballots on which
$A$ (respectively $W$) and $S$ were ranked consecutively and both candidates
appear on the ballot; denote this count $B_{AS}$ (resp. $B_{WS}$). We say that
$S$ is more similar to $A$ if $B_{AS}>B_{WS}$; similarly, $S$ is more similar
to $W$ if $B_{AS}<B_{WS}$. In the classic vote-splitting scenario, $W$ is not
a winner because $S$ is more similar to $W$.

The results are summarized in Table 5. The entries under the $B_{AS}>B_{WS}$
column indicate the number of elections in which the spoiler $S$ was closer to
$A$ than to $B$; the entries under the $B_{AS}<B_{WS}$ are the reverse. To
make comparison between methods easier, we also include the values of the
ratio $\frac{B_{AS}<B_{WS}}{B_{AS}>B_{WS}}$. Methods where this ratio is less
than 1 can be thought of as slightly “clone-friendly.” In these cases, it is
more likely that the spoiler candidate $S$ is closer to the winning candidate
$A$ than the would-be winner $W$. So the presence of $S$ allows $A$ to win.
Methods where this ratio is great than 1 can be thought of as slightly “clone-
averse.” In these cases, the spoilers are more frequently closer to $W$ than
to $A$ and thus we see more evidence of a traditional notion of vote-
splitting.

As with previous spoiler results, we do not see any strong correlation between
a method’s categorization as proportional or excellence-based and the method’s
tendency to be clone-averse or clone-friendly. Proportional methods based on
Cham-Cour tend to produce clone-friendly spoiler results while STV tends to
produce results that are clone-averse. This is interesting in the case of STV
because some proponents of STV claim that it mitigates vote-splitting. In
terms of raw frequency this claim is largely borne out, but in the rare cases
that the spoiler effect occurs under STV this effect seems more akin to vote-
splitting than something like clone-friendliness.

Voting methods with ratio less than 1 include Borda (OM), Cham-Cour (OM) and
(PM) and greedy CC (OM) and (PM); for the more-complete ballots, the list also
includes Borda (PM). To better understand the Borda results, first consider
the data from the real ballots which contain many partial rankings. Suppose
that a voter ranked only $A$ and $S$. Under the OM model, all the remaining
candidate receive almost as many points as $S$; without $S$, all the
candidates aside from $A$ are treated equally. Thus the presence of $S$ on the
ballot creates a larger difference in point totals between $A$ and the other
candidates. Under Borda (PM), however, since unranked candidate receive 0
points, their point totals are not materially affected by the elimination of
$S$ in such a ballot. Under the more-complete ballots, the difference between
OM and PM is much smaller, since the ballots are longer. Hence their ratios
are more similar.

Table 5. Measure of similarity between $A$ and $S$ and between $W$ and $S$ in the Scottish database under different voting methods Method | More Complete Ballots | Actual Ballots
---|---|---
| $B_{AS}>B_{WS}$ | $B_{AS}<B_{WS}$ | $\frac{B_{AS}<B_{WS}}{B_{AS}>B_{WS}}$ | $B_{AS}>B_{WS}$ | $B_{AS}<B_{WS}$ | $\frac{B_{AS}<B_{WS}}{B_{AS}>B_{WS}}$
Bloc | 161 | 449 | 2.789 | 53 | 63 | 1.189
Borda (OM) | 241 | 24 | 0.1 | 214 | 10 | 0.047
Borda (PM) | 105 | 63 | 0.6 | 19 | 49 | 2.579
Cham-Cour (OM) | 159 | 38 | 0.239 | 151 | 20 | 0.132
Cham-Cour (PM) | 146 | 31 | 0.212 | 85 | 35 | 0.412
Greedy-CC (OM) | 180 | 50 | 0.278 | 157 | 23 | 0.146
Greedy-CC (PM) | 172 | 47 | 0.273 | 89 | 36 | 0.404
SRCV | 4 | 145 | 36.25 | 1 | 27 | 27
SNTV | 8 | 180 | 22.5 | 4 | 155 | 38.75
STV | 36 | 96 | 2.667 | 30 | 36 | 1.2

###  4.4. Using Candidate Subsets to Generate 4 and 5 candidate elections

Our simulations in Section 3 examined elections with $m\in\\{4,5\\}$. To
investigate how closely real-world data might compare to the simulations, we
use the Scottish database to generate tens of thousands of 4-and 5-candidate
elections and check them for the spoiler effect. To do this, for each
$t\in\\{4,5\\}$ and each Scottish election with $m\geq t$, we generated all
candidate subsets of size $t$. For each such subset, we used the preference
data from the original ballots and from the more-complete ballots to generate
a preference profile for only the candidates in the subset. This allowed us to
create tables similar to those we analyzed in Section 3 (see Tables 19, 20,
and 21 from Appendix C), essentially treating the real-world data as its own
model for $m=4$, $k=2$; $m=5$, $k=2$; and $m=5$, $k=3$. There may be limited
utility in generating such sub-elections and using a $k$-value that is often
different than what was used in the actual election from which the preferences
are pulled, but such an exercise can provide some direct comparison to the
theoretical results predicted by the models used in our simulations.

Unsurprisingly, the probabilities obtained from these sub-elections tend to be
much smaller than the probabilities predicted by the theoretical models. The
only exception are the probabilities for Bloc when compared to the
probabilities for the 1D-spatial model; these results strongly suggest that
the real-world sub-elections cannot be modeled by such a 1D model.

##  5\. Towards Spoiler-Proofness

None of the methods considered so far are immune to the spoiler effect. This
raises the question: is there a reasonable voting method which is immune? The
answer depends on one’s notion of “reasonable.” For example, a dictatorship is
spoiler-proof. For a more acceptable example, consider this method: first
truncate all ballots to length $k-1$ so that any preference data past the
$(k-1)$st ranking is discarded, and then use bloc voting. This method is
spoiler-proof and not as unreasonable as a dictatorship, but still probably
does not warrant serious consideration. In this section we consider two
reasonable (in our view) voting methods which are virtually spoiler-proof in
practice, meaning that the methods almost never exhibit the spoiler effect
when using the real-world data from Section 4. The first method is based on
the notion of a _Condorcet committee_ as defined in [22] and [32], and would
be used in a majoritarian setting. That is, the first method does not attempt
to achieve proportional representation. By contrast, the second method
arguably belongs in the family of proportional representation methods.

To find these two methods we investigated many ranked-choice methods beyond
those previously explored, and these performed the best with respect to the
spoiler effect with the Scottish data. It is an open question if there exists
a reasonable voting method for either the proportional or non-proportional
context that would return no instances of the spoiler effect in our real-world
data.

###  5.1. A Virtually Spoiler-Proof Majoritarian Method

The first method is based on the notion of a Condorcet committee, which is a
muitiwinner generalization of the single-winner Condorcet winner.

######  Definition 1.

([22] and [32]) A subset of candidates $\mathcal{C}$ is a Condorcet committee
if for each pair of candidates $(A,B)$ where $A\in\mathcal{C}$ and
$B\not\in\mathcal{C}$, more voters prefer $A$ to $B$ than prefer $B$ to $A$.

If an election contains a Condorcet committee of size $k$ then any voting
method which selects the Condorcet committee cannot demonstrate the spoiler
effect in the election, as eliminating candidates who are not in the Condorcet
committee does not affect the committee’s composition. Thus, to define a
voting method immune to spoilers, it is reasonable to select the Condorcet
committee of size $k$ if it exists; if one does not exist, we can start with a
Condorcet committee of size greater than $k$ and remove candidates in a
reasonable way until we reach a committee of size $k$. In theory, we could
modify Condorcet committees that are too small by adding candidates, but we
find it simpler to do the reverse because every profile has at least one
Condorcet committee of size at least $k$, the set of all candidates, but not
every profile has a Condorcet committee of size smaller than $k$.

######  Definition 2.

Let $k^{\prime}\geq k$ be the smallest number of seats for which there exists
a Condorcet committee of size $k^{\prime}$; denote this committee
$\mathcal{C}$. For each pair of candidates $(A,B)$ where $A,B\in\mathcal{C}$,
let PW$(A,B)$ denote the number of voters who rank $A$ above $B$ minus the
number of voters who rank $B$ above $A$. Let
score$(A)=\displaystyle\min_{B\neq A}$PW$(A,B)$. Under the Minimax Condorcet
Committee (MCC) method, eliminate the $k^{\prime}-k$ candidates from
$\mathcal{C}$ with the lowest scores; the resulting set of size $k$ is the
winner committee.

Observe that if an election contains a Condorcet committee of size $k$ or size
$k+1$ then the MCC method is not susceptible to the spoiler effect. In fact,
any voting method which selects the Condorcet committee if one exists and
otherwise eliminates candidates from the smallest Condorcet committee larger
than $k$ is immune to the spoiler effect when there is a Condorcet committee
of size $k$ or $k+1$.

Definition 2 does not take into account ties, leaving open the possibility of
more than one winning set. Such ties are rare in practice, and thus we ignore
this possibility in our empirical analysis of the MCC method.

To investigate how well the MCC method performs with respect to the spoiler
effect in practice, we analyzed its performance using the Scottish data in a
couple of ways. We first determined the number of instances the spoiler effect
occurred under MCC using the actual value of $k$ among the 999 elections with
$m>k+1$. Then we repeated the process using fixed a value of $k\in\\{2,3,4\\}$
across all the elections. As with all such hypothetical comparisons, we should
be careful not to assign too much weight to the results since voter and
candidate behavior might have been different for a different value of $k$.
However, given the large number of voter profiles in the database, we argue
that such results are still useful. In fact, for $k=2$, the size of the
database is larger since we can check for the spoiler effect among the 1068
elections with $m\geq 4$.

The results are shown in Table 6. When using the actual number of seats from
the real elections (row 1), only four elections (0.4% of the elections
processed) return a spoiler effect of any kind, and in only one election is
the plurality loser and the top-$k$ loser a spoiler candidate. The reason for
such a low number of potential spoilers is that all but four elections contain
a Condorcet committee of size $k$ or $k+1$. The election in which the
plurality loser (who is also the top-$k$ loser) is a spoiler candidate under
the MCC method is an outlier in the database in terms of “closeness”, which
seems to be the reason for its unique spoiler behavior. This election is the
2022 election of the Mid-Formartine Ward of the Aberdeenshire Council Area, in
which $n=6$ and $k=4$ (see Table 7). The election contains no Condorcet
committee of size 4 or 5. Table 7 indicates the number of first-place votes
and the number of top four rankings for each candidate. Note that the top and
bottom-ranked candidates under both measures are separated by a very small
margin; such small margins are rarely observed in real-world elections.

When we set $k=3$ or $k=4$ across the database we obtain similarly low rates
for the spoiler effect as when using the actual value of $k$. When we set
$k=2$, there is a jump in the number of elections affected by a spoiler,
although the overall rate of $14/1068=1.3\%$ is still quite low. The likely
cause of this increase is that very few of the elections actually corresponded
to $k=2$, and thus the voter profile is perhaps not indicative of how voters
and parties would behave in an election with fewer seats.

Overall, the results suggest the MCC method is virtually spoiler-proof. We
note that if we run the MCC method on the Scottish elections with the
preferences filled in, the spoiler effect rate of the method increases to
2.5%. This is much smaller than the rates for the other methods in this
setting, but this rate perhaps does not qualify as “spoiler-proof.”

Table 6. Spoiler effect results for the MCC method in the Scottish data. $k$ | Num. Elections | Spoiler Effect | Multiple Spoilers | Plurality Loser | Top-$k$ Loser
---|---|---|---|---|---
Actual | 999 | 4 | 0 | 1 | 1
2 | 1068 | 14 | 0 | 1 | 2
3 | 1032 | 7 | 0 | 0 | 0
4 | 922 | 3 | 0 | 1 | 1
Table 7. The first-place vote totals and the mentions in the top 4 rankings for the candidates in the 2022 election of the Mid-Formartine Ward of the Aberdeenshire Council Area Candidate | Hassan | Hutchison | Johnston | Nicol | Powell | Ritchie
---|---|---|---|---|---|---
First-Place Votes | 752 | 728 | 916 | 971 | 803 | 876
Top 4 Mentions | 2837 | 2147 | 2790 | 2238 | 2170 | 2246

Further analysis of the MCC method could be obtained through simulations.
However, given the large percentage of voter profiles from the database that
have Condorcet committees of size $k$ or $k+1$, it is likely that results
obtained using IC, IAC or the 1d-spatial model would be more indicative of the
lack of Condorcet committees of either of these sizes in the generated
profiles. This inference is supported by the large differences in the
probability of a spoiler under all voting methods using any of these models in
comparison to the empirical analyses, suggesting that these models do a poor
job of replicating actual voter behavior.

The MCC method, while virtually spoiler-proof in real-world elections,
achieves majoritarian-based outcomes and thus would not tend to produce winner
sets which achieve proportionality.

###  5.2. A Virtually Spoiler-Proof Proportional Method

The next method chooses the top $k$ ranked candidates by the single-winner
version of STV, referred to as instant-runoff voting, the plurality
elimination rule, the alternative vote, etc., in the single-winner literature.

######  Definition 3.

Under the top-$\mathbf{k}$ instant runoff voting (top-$k$ IRV) method the
election proceeds in rounds. In a given round we eliminate the candidate with
the fewest first-place votes and transfer those votes to the next candidate on
the ballot who has not been previously eliminated. Continue until there are
$k$ candidates remaining, and declare these candidates the winning committee.

As with MCC, this definition does not take into account ties. For the purposes
of analysis we break a tie as follows: if in some round there is a tie for the
candidate with the fewest first-place votes, we eliminate the candidate who
comes first alphabetically. (Other ways of breaking such a tie are probably
more sensible.) Since ties occur very rarely in the data, the choice of tie-
breaking mechanism almost never affects the eventual winner committee.

It is not clear from the description of top-$k$ IRV that the method should be
classified as one which achieves proportional representation. In particular,
top-$k$ IRV does not satisfy classical proportional representation axioms such
as Dummett’s proportionality for solid coalitions [11]. However, we argue that
this method generally achieves proportional representation in practice for
three reasons. First, no wining committees selected under top-$k$ IRV in the
database violates the proportionality for solid coalitions axiom. (This is not
as strong a statement as it may seem because the ballots are so highly
truncated [26], but it is noteworthy nonetheless.) Second, of the 1070
multiwinner elections in our Scottish database, top-$k$ IRV chooses the same
winner set as STV in 863 of them. Thus, in approximately 81% of real elections
top-$k$ IRV produces the same outcome as the method that is most associated
with proportional representation. (We note that other methods previously
discussed do not agree with STV at nearly as high a rate. For example, SRCV
produces the same winner set as STV in 55% of the Scottish elections.) Second,
of the 207 elections in which top-$k$ IRV and STV disagree, the number of
parties represented in the winning committee selected by top-$k$ IRV is at
least as great as the number of parties represented in the winner committee
selected by STV: the number of parties under both methods are equal in 91
elections and the number of parties under top-$k$ IRV is greater in 98
elections. (By contrast, when a majoritarian method like SRCV disagrees with
STV, the number of parties represented in the winning committee under SRCV is
generally less than the number of parties represented in the winning committee
under STV.) Thus, even though top-$k$ IRV fails to be proportional from an
axiomatic standpoint, in practice it belongs in the family of methods designed
for proportional representation.

Table 8. Spoiler effect results for the top-$k$ IRV method in the Scottish data. $k$ | Num. Elections | Spoiler Effect | Multiple Spoilers | Plurality Loser | Top-$k$ Loser
---|---|---|---|---|---
Actual | 999 | 4 | 0 | 0 | 0
2 | 1068 | 30 | 2 | 0 | 0
3 | 1032 | 8 | 0 | 0 | 0
4 | 922 | 4 | 0 | 0 | 0

We analyze the behavior of top-$k$ IRV with respect to spoilers in the same
way as for MCC. The results are indicated in Table 8. When using the actual
value of $k$, there are only four instances of the spoiler effect and no
instances of multiple spoilers. When we set $k=2$ across all elections then we
observe a sizable increase, just as we did for the MCC method. This large jump
disappears when we increase to $k=3$ or $k=4$ for all elections. By
construction, the plurality loser cannot be a spoiler candidate under top-$k$
IRV, and thus the zeroes in that column are to be expected. In addition,
however, top-$k$ IRV never results in a spoiler who is a top-$k$ loser. As
with MCC, we omit further analysis using simulations because the method is
perhaps not worth formal study.

##  6\. Conclusion

Our results suggest that STV and SRCV perform the best with respect to
susceptibility to spoilers overall while bloc and SNTV perform the worst. This
result is unsurprising: previous work has shown that a voting method like
plurality produces the spoiler effect at a relatively high rate. Furthermore,
because the definition of a spoiler candidate involves the removal of a
candidate from an election, it is unsurprising that methods such as STV and
SRCV (which, by design, remove candidates as an election unfolds) are more
resilient to spoilers. Our empirical results, in particular, demonstrate that
STV is not especially susceptible to the spoiler effect in real-world
elections.

It is worth repeating that the empirical must be interpreted cautiously since
the elections from which the data was collected were run under STV and not any
other method. It is not clear what the conclusions would be had it been
possible to collect a similar number of voter profiles from elections using
these other methods. The simulations, however, do confirm these overall
conclusions. While the differences in spoiler frequencies between simulated
and real voter profiles suggest that the IAC, IC and 1D-spatial assumptions
are poor models for actual voter behavior, the simulation results do indicate
a kind of upper bound for the likelihood of spoilers under the different
methods. The real data, being so much less close than the simulated data,
suggest that overall, spoiler frequencies are lower than predicted: good news
for STV given its prevalence in actual elections around the world. More
importantly, these results largely hold up using both partial and more
complete ballots.

This analysis also raises some interesting questions to explore further. The
two methods suggested in Section 5 are meant to be taken as suggestive rather
than proscriptive and arise from consideration of the actual database. Neither
one has been systematically studied or used (as far as we are aware). In
theory, basing a voting method on the existence of a Condorcet committee of
size $k$ is intuitively appealing, but theoretical models tend to produce
elections without such committees at fairly high rates. In practice, it is
interesting that Condorcet committees of correct size occur so frequently. The
fact that both MCC and top$-k$ IRV are so spoiler-proof in practice, suggests
that both methods are worth further study. Indeed it may be, as more ranked
data becomes available, that the relative desirability of different
multiwinner voting methods shifts as researchers gain a better understanding
of voter behavior using these methods.

Lastly, the results about stability are also interesting. Despite the relative
frequency of spoilers (or mutual spoiler) under several different voting
methods, it remains the case that the number of alternative winning committees
is relatively small as is the number of winning candidate affected in those
committees. It would be useful to determine theoretical bounds on these
numbers under the different voting methods.

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## Appendix A: Description of technique to complete ballots from Scottish
elections

In this appendix, we describe the process for extending the partial ballots in
the Scottish data to more complete rankings.

The process involves a series of steps, each time extending partial ballots of
length $k^{\prime}$ to length $k^{\prime}+1$ based on the probability
distribution of ballots which agree on the first $k^{\prime}$ entries. To be
more precise, suppose there are 10 ballots of the form $ABC$. To increase
these ballots by length 1, we consider all ballots of the form $ABC*$ which
have length at least 4. Suppose there are 38 such ballots as shown in Table 9.
We extend the ballots of the form $ABC$ proportionally, (column 3) and round
to a whole number using Hamilton’s apportionment method (column 4).

Table 9. Extending ballots of length 3 to length 4 Ballot | Number | Prop. | Num. ballots
---|---|---|---
ABCD | 9 | $(9/38)\cdot 10=2.368$ | 2
ABCE | 12 | $(12/38)\cdot 10=3.158$ | 3
ABCF | 17 | $(17/38)\cdot 10=4.474$ | 5
Total | 38 | 10 | 10

In theory, this process can be iterated to create complete ballots, assuming a
sufficient set of complete ballots. In practice, however, it does not make
sense to do so if the number of ballots of length at least $k^{\prime}+1$ is
not sufficiently large in comparison to the number of ballots of length
$k^{\prime}$. Thus, each ballot was extended until either the ballots were
complete or the number of ballots of length $k^{\prime}+1$ was less than 10%
of the number of ballots of length $k^{\prime}.$ This provided a fuller set of
voter profiles that, while not complete, allowed for a better comparison with
the simulated data.

## Appendix B: Tables of Simulation Results

Table 10. Likelihood of a spoiler for IC when $m=4$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 49.7, 40.0 | 11.6, 7.3 | 24.7, 20.3 | 38.9, 32.2
Borda (OM) | 21.2, 21.1 | 0.1, 0.4 | 11.0, 10.4 | 13.2, 13.7
Borda (PM) | 21.2, 18.4 | 0.1, 0.3 | 11.0, 10.4 | 13.2, 12.8
Cham-Cour (OM) | 25.5, 25.4 | 0.3, 0.4 | 17.1, 17.2 | 14.1, 14.4
Cham-Cour (PM) | 25.5, 18.5 | 0.3, 0.2 | 17.1, 11.7 | 14.1, 10.8
Greedy-CC (OM) | 23.5, 23.8 | 0.5, 0.6 | 14.7, 15.5 | 13.5, 13.6
Greedy-CC (PM) | 23.5, 17.1 | 0.5, 0.4 | 14.7, 10.1 | 13.5, 9.8
SRCV | 18.4, 16.3 | 0.4, 0.3 | 2.9, 2.6 | 6.9, 6.6
SNTV | 37.9, 36.4 | 4.9, 4.7 | 29.8, 28.8 | 18.4, 18.3
STV | 15.6, 14.4 | 0.3, 0.4 | 0.3, 0.5 | 6.4, 5.8
Table 11. Likelihood of a spoiler for IC when $m=5$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 65.5, 63.1 | 33.2, 31.1 | 33.1, 31.7 | 42.9, 41.2
Borda (OM) | 32.3, 33.9 | 6.5, 7.0 | 13.8, 14.5 | 15.8, 16.7
Borda (PM) | 32.3, 29.8 | 6.5, 6.0 | 13.8, 12.6 | 15.8, 14.8
Greedy-CC(OM) | 36.0, 36.6 | 7.7, 8.3 | 16.1, 17.0 | 17.2, 18.1
Greedy-CC (PM) | 36.0, 30.2 | 7.7, 5.1 | 16.1, 13.0 | 17.2, 13.7
SRCV | 33.6, 31.7 | 6.6, 6.3 | 3.1, 2.8 | 7.9, 7.7
SNTV | 59.7, 59.1 | 23.9, 23.4 | 35.7, 35.4 | 28.0, 27.9
STV | 29.4, 28.0 | 5.3, 4.9 | 0.1, 0.1 | 7.9, 7.4
Table 12. Likelihood of a spoiler for IC when $m=5$ and $k=3$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 59.9, 47.1 | 17.6, 10.5 | 23.6, 18.2 | 47.1, 38.4
Borda (OM) | 19.9, 20.7 | 0.1, 0.4 | 9.2, 10.0 | 11.8, 12.1
Borda (PM) | 19.9, 18.1 | 0.1, 0.3 | 9.2, 8.4 | 11.8, 11.5
Greedy-CC (OM) | 27.9, 28.4 | 1.2, 1.3 | 17.0, 17.0 | 11.1, 11.9
Greedy-CC (PM) | 27.9, 23.9 | 1.2, 0.9 | 17.0, 13.7 | 11.1, 9.7
SRCV | 22.7, 21.1 | 0.8, 0.7 | 4.2, 4.0 | 9.1, 8.8
SNTV | 40.7, 40.4 | 5.6, 5.6 | 32.3, 32.1 | 14.9, 15.0
STV | 16.6, 16.2 | 0.5, 0.5 | 0.5, 0.7 | 5.0, 5.0
Table 13. Likelihood of a spoiler for IAC when $m=4$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 39.4, 33.0 | 8.5, 5.7 | 18.4, 16.1 | 29.9, 26.4
Borda (OM) | 19.6, 19.7 | 0.1, 0.3 | 10.7, 11.4 | 13.3, 13.4
Borda (PM) | 19.6, 18.0 | 0.1,0.3 | 10.7, 10.0 | 13.3, 12.8
Cham-Cour (OM) | 22.2, 22.8 | 0.3, 0.3 | 16.7, 17.1 | 11.2, 12.1
Cham-Cour (PM) | 22.2, 17.5 | 0.3, 0.3 | 16.7, 12.1 | 11.2, 9.8
Greedy-CC (OM) | 23.4, 25.1 | 0.6, 0.7 | 16.8, 17.7 | 12.6, 14.0
Greedy-CC (PM) | 23.4, 18.6 | 0.6, 0.3 | 16.8, 12.3 | 12.6, 10.6
SRCV | 16.9, 14.45 | 0.2, 0.1 | 3.3, 2.6 | 5.6, 5.6
SNTV | 27.1, 28.3 | 3.7, 3.7 | 20.5, 22.0 | 12.8, 14.0
STV | 17.4, 12.0 | 1.1, 0.0 | 4.9, 0.0 | 7.8, 4.8
Table 14. Likelihood of a Spoiler for IAC when $m=5$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 61.8, 60.2 | 29.7, 28.3 | 29.7, 29.5 | 38.6, 38.0
Borda (OM) | 31.8, 33.2 | 6.5, 6.9 | 13.7, 14.8 | 15.9, 16.8
Borda (PM) | 31.8, 29.1 | 6.5, 5.7 | 13.7, 12.4 | 15.9, 14.6
Greedy-CC (OM) | 37.8, 39.3 | 8.7, 9.1 | 18.4, 19.6 | 18.1, 19.1
Greedy-CC (PM) | 37.8, 33.6 | 8.7, 6.8 | 18.4, 15.2 | 18.1, 15.8
SRCV | 33.0, 30.7 | 6.1, 5.6 | 3.5, 3.2 | 7.0 , 7.1
SNTV | 53.0, 54.3 | 20.7, 21.1 | 30.0, 31.6 | 24.2, 25.3
STV | 28.7, 26.7 | 4.5, 4.1 | 0.5, 0.0 | 7.6, 7.0
Table 15. Likelihood of a Spoiler for IAC when $m=5$ and $k=3$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 55.1, 44.3 | 15.3, 9.4 | 21.4, 17.3 | 43.0, 35.8
Borda (OM) | 19.7 , 20.4 | 0.2, 0.4 | 9.0, 9.8 | 12.0, 12.1
Borda (PM) | 19.7, 17.9 | 0.2, 0.3 | 9.0, 8.5 | 12.0, 11.8
Greedy-CC (OM) | 28.1, 29.5 | 1.3, 1.5 | 18.4, 19.2 | 10.6, 11.9
Greedy-CC (PM) | 28.1, 25.5 | 1.3, 1.1 | 18.4, 15.5 | 10.6, 10.6
SRCV | 22.0, 19.7 | 0.6, 0.3 | 4.3, 3.8 | 8.7, 8.2
SNTV | 34.5, 35.6 | 4.7, 4.8 | 27.2, 38.6 | 12.3, 13.3
STV | 19.7, 14.4 | 1.3, 0.0 | 5.7, 0.0 | 7.1, 4.4
Table 16. Likelihood of a Spoiler for the 1D-spatial model for $m=4$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 0, 1.4 | 0, 1.0 | 0, 1.1 | 0, 1.2
Borda (OM) | 12.7, 11.6 | 0.0, 0.4 | 6.1, 5.9 | 12.7, 10.9
Borda (PM) | 12.7, 7.4 | 0.0, 0.6 | 6.1, 2.6 | 12.7, 6.9
Cham-Cour (OM) | 18.4, 16.7 | 0.0, 0.1 | 15.2, 13.8 | 6.1, 5.9
Cham-Cour (PM) | 18.4, 6.5 | 0.0, 0.0 | 15.2, 5.4 | 6.1, 2.2
Greedy-CC (OM) | 32.0, 22.2 | 1.6, 0.8 | 21.0, 17.5 | 11.1, 8.9
Greedy-CC (PM) | 32.0, 9.2 | 1.6, 0.1 | 21.0, 7.5 | 11.1, 4.3
SRCV | 9.5, 6.8 | 0.0, 0.0 | 0.0, 0.1 | 7.0, 5.7
SNTV | 27.9, 19.3 | 11.1, 6.0 | 21.1, 16.0 | 21.1, 14.3
STV | 20.6, 9.7 | 0.3, 0.5 | 1.7, 2.0 | 12.1, 6.2
Table 17. Likelihood of a Spoiler for the 1D-spatial model for $m=5$ and $k=2$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 36.5, 24.1 | 36.5, 22.7 | 13.7, 11.0 | 30.0, 20.4
Borda (OM) | 21.0, 23.7 | 4.1, 5.2 | 4.8, 8.4 | 8.5, 8.1
Borda (PM) | 21.0, 11.6 | 4.1, 3.3 | 4.8, 3.6 | 8.5, 5.8
Greedy-CC (OM) | 56.4, 45.0 | 12.8, 11.8 | 25.6, 22.5 | 14.9, 14.6
Greedy-CC (PM) | 56.4, 16.5 | 12.8, 4.1 | 25.6, 9.0 | 14.9, 7.3
SRCV | 22.5, 15.4 | 1.5, 0.7 | 0.6, 0.4 | 5.3, 2.8
SNTV | 43.4, 33.4 | 24.1, 14.8 | 20.9, 16.1 | 24.4, 18.3
STV | 37.3, 21.4 | 2.2, 2.2 | 1.1, 1.7 | 12.0, 8.0
Table 18. Likelihood of a Spoiler for the 1D-spatial model for $m=5$ and $k=3$ Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 0, 0.5 | 0, 0.2 | 0, 0.3 | 0, 0.4
Borda (OM) | 12.6, 12.1 | 0.0, 0.4 | 4.7, 4.5 | 12.6, 11.3
Borda (PM) | 12.6, 7.0 | 0.0, 0.6 | 4.7, 1.6 | 12.6, 6.4
Greedy-CC (OM) | 19.4, 20.2 | 0.5, 1.1 | 14.0, 15.3 | 6.7, 6.2
Greedy-CC (PM) | 19.4, 9.7 | 0.5, 0.3 | 14.0, 7.1 | 6.7, 3.3
SRCV | 6.1, 7.0 | 0.03, 0.04 | 0.01, 0.05 | 4.6, 6.2
SNTV | 26.9, 20.9 | 13.5, 8.7 | 21.5, 17.7 | 15.0, 11.8
STV | 23.7,11.9 | 0.6, 0.7 | 2.6, 2.5 | 13.4, 7.0

## Appendix C: Tables of Results From Scottish Sub-Elections

Table 19. Likelihood of a Spoiler for $m=4$ and $k=2$ using sub-elections from the Scottish data. Num elections 75405 Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 14.4, 5.6 | 2.8, 0.7 | 8.0, 4.2 | 12.6, 5.3
Borda (OM) | 7.2, 5.3 | 0.1, 0.3 | 5.1, 4.2 | 6.7, 5.1
Borda (PM) | 6.1, 3.9 | 0.4, 0.4 | 4.4, 3.2 | 5.8, 3.8
Cham Cour (OM) | 5.7, 3.9 | 0.1, 0.1 | 5.2, 3.7 | 4.0, 3.3
Cham Cour (PM) | 4.7, 2.4 | 0.1, 0.1 | 4.2, 2.2 | 3.3, 2.0
Cham Cour greedy (OM) | 7.5, 4.5 | 0.2, 0.1 | 6.6, 4.2 | 5.4, 3.7
Cham Cour greedy (PM) | 5.9, 2.5 | 0.1, 0.1 | 5.1, 2.3 | 4.4, 2.1
SRCV | 3.9, 0.8 | 0.0, 0.0 | 0.4, 0.2 | 2.9, 0.6
SNTV | 7.2, 4.5 | 2.3, 1.3 | 6.7, 4.3 | 5.0, 3.3
STV | 2.8, 1.2 | 0.1, 0.1 | 1.2, 0.7 | 1.8, 1.0
Table 20. Likelihood of a Spoiler for $m=5$ and $k=2$ using sub-elections from the Scottish data. Num elections 79428 Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 16.5, 6.3 | 9.1, 2.8 | 9.6, 3.3 | 9.5, 3.2
Borda (OM) | 13.3, 10.4 | 1.9, 2.1 | 6.7, 5.0 | 6.4, 4.4
Borda (PM) | 7.8, 4.4 | 2.6, 2.1 | 4.9, 2.8 | 5.5, 2.8
Cham Cour greedy (OM) | 11.2, 8.2 | 2.1, 1.6 | 6.1, 4.4 | 5.4, 4.0
Cham Cour greedy (PM) | 7.7, 2.7 | 1.8, 1.2 | 4.7, 1.9 | 4.4, 1.9
SRCV | 5.8, 1.4 | 0.1, 0.0 | 0.2, 0.1 | 0.4, 0.1
SNTV | 11.6, 7.6 | 5.1, 3.4 | 4.4, 2.7 | 4.1, 2.4
STV | 4.0, 1.9 | 0.6, 0.5 | 0.6, 0.5 | 1.1, 0.7
Table 21. Likelihood of a Spoiler for $m=5$ and $k=3$ using sub-elections from the Scottish data. Num elections 79428 Method | Spoiler Effect | Multiple Spoiler Effect | Plurality Loser | Top-$k$ loser
---|---|---|---|---
Bloc | 10.2, 3.8 | 1.2, 0.4 | 4.7, 2.8 | 9.1, 3.6
Borda (OM) | 6.2, 4.7 | 0.1, 0.2 | 3.9, 3.6 | 5.0, 4.3
Borda (PM) | 5.1, 2.9 | 0.2, 0.2 | 3.3, 2.3 | 4.4, 2.7
Cham Cour greedy (OM) | 6.8, 4.3 | 0.2, 0.2 | 6.3, 4.1 | 3.6, 3.1
Cham Cour greedy (PM) | 6.1, 3.2 | 0.2, 0.1 | 5.3, 2.9 | 3.2, 2.1
SRCV | 3.8, 0.9 | 0.0, 0.0 | 0.5, 0.2 | 3.1, 0.7
SNTV | 5.4, 3.6 | 1.2, 0.7 | 5.4, 3.5 | 3.2, 2.6
STV | 3.2, 1.2 | 0.1, 0.0 | 1.5, 0.8 | 1.8, 0.8