The probability of casting a pivotal vote in an Instant Runoff Voting election Samuel Baltz

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The probability of casting a pivotal vote in an Instant

Runoﬀ Voting election

Samuel Baltz∗

August 1, 2023

Word count: 6,930

Abstract

I derive the probability that a vote cast in an Instant Runoﬀ Voting election will

change the election winner. I phrase that probability in terms of the candidates’ ex-

pected vote totals, and then I estimate its magnitude for diﬀerent distributions of voter

preferences. The result is very similar to the probability of casting a pivotal vote in

a Single-Member District Plurality election, which suggests that Instant Runoﬀ Vot-

ing does not actually increase or decrease voters’ incentives to vote strategically. The

derivation uncovers a counter-intuitive phenomenon that I call “indirect pivotality”,

in which a voter can cause one candidate to win by ranking some other candidate on

their ballot.

∗Massachusetts Institute of Technology, sbaltz@umich.edu. This paper beneﬁtted from

discussions, including several technical improvements and corrections, from Luka Buli´c

Braˇculj, Edward Foley, Joelle Gross, Christa Hawthorne, Walter R. Mebane, Jr., Scott E.

Page, Charles Stewart III, and Gillian Thompson. Fact-checking was performed on comput-

ers available through a fellowship from the Michigan Institute for Computational Discovery

and Engineering.

arXiv:2210.01657v2 [cs.GT] 31 Jul 2023

1 Introduction

After a wave of recent reforms, Instant Runoﬀ Voting (IRV) is suddenly a widely used elec-

toral system in major elections, but little is known about how voters behave in this ruleset.

Because IRV is a ranked-choice electoral system, some scholars and advocates reason that it

allows voters to report their sincere preference order on their ballot (Gehl and Porter, 2020,

ch. 5). Another possibility, though, is that the ranked options in IRV provide voters with

distinct reasons to act strategically, and diﬀerent tools for doing so (Eggers and Nowacki,

2021, Santucci, 2021). Strategic voters attempt to inﬂuence election results by supporting

more popular candidates, so whether voters have more or less opportunity to inﬂuence elec-

tion outcomes in IRV depends on the probability that a particular ballot changes the result

of an IRV election. However, that probability has only been derived in IRV contests of up to

four candidates (Bouton, 2013, Eggers and Nowacki, 2021). So, how big are voters’ actual

incentives to vote strategically in IRV?

The core component of strategic voting models is the expected utility of each vote choice,

but calculating the expected utility of a ballot in IRV requires knowing the probability that

a vote cast in an IRV election will change the election winner. This question of pivotal

probabilities was ﬁrst posed about the more traditional Single-Member District Plurality

(SMDP) electoral system more than half a century ago, and various frameworks for esti-

mating the quantity have been explored in detail since then (Cox, 1994, Eggers and Vivyan,

2020, Mebane et al., 2019, Riker and Ordeshook, 1968, Vasselai, 2022). Pivotal probabil-

ities have also been extended to proportional systems (Cox and Shugart, 1996), and even

the alternative ranked-choice system of Borda count (Baltz, 2022). Researchers have even

identiﬁed the set of all pivotal outcomes in a three- or four-candidate IRV race, and mod-

eled the probability of each outcome arising in a realistic election (Bouton, 2013, Eggers

and Nowacki, 2021). However, the general question of pivotal opportunities in IRV remains

open: it is not known whether or not voters have more or less opportunity to cast pivotal

votes in IRV compared to more conventional systems like SMDP.

I derive the probability that a vote cast in a single-winner IRV election changes the

outcome of that election, when any number of candidates contest the election, and voters

can rank any number of them on their ballots. This is not a straightforward generalization

of the 3- or 4-candidate case; those situations were solved by exhaustive enumeration, but

the case with any number of candidates requires deriving an expression for the probability of

a pivotal event (any situation in which a single ballot could change the election result) as a

function of the ballots that are expected to be cast. The resulting equation is exceptionally

complicated, which I argue reﬂects real complications in the electoral system. The derivation

also uncovers a property of IRV that I call “indirect pivotality”, in which a voter can cause

one candidate to win by ranking some other candidate, and I argue that this represents a

substantive motivation to keep ballot length short in IRV.1I then show that, under one well-

studied method for estimating numerically speciﬁc pivotal probabilities, the opportunity

to cast a pivotal vote is similar in IRV and SMDP, which suggests that neither system

fundamentally encourages strategic voting more than the other.

Of course, the literal expected utility of each vote choice may or may not inﬂuence real

voting behaviour; exactly how strategic voters are in IRV, and what factors inﬂuence their

vote choice, is an open question that has been the subject of substantial recent research

(Atsusaka, 2023, Buisseret and Prato, 2022, Reilly, 2021, Simmons et al., 2022). However,

the pivotal probability of an IRV ballot is an important quantity even apart from its poten-

tial impact on voting behaviour. The foremost reason is that this probability shapes real

outcomes in IRV. If ties tend to be more common in IRV, then voters may ﬁnd themselves

acting pivotally more often, whether they plan to or not (this is naturally a larger concern

in smaller electorates). Another reason is that models of voting behaviour often require

knowing pivotal probabilities, even when the intention is not just to model completely ratio-

nal choice voting behaviour (Bendor et al., 2011, Eggers and Nowacki, 2021, Mebane et al.,

1By “ballot length” I mean the number of candidates that each voter is allowed to rank to determine the

single winner of an IRV election. I will also use “ballot” to mean just the section of the ballot relating to a

speciﬁc IRV election (recognizing that a ballot may have multiple oﬃces on it, but lacking a simple term for

“just the part of a ballot that corresponds to the IRV race under consideration”).

2019). Importantly, though, this exercise is valuable even for a reader who does not grant

that people might vote strategically, so long as they see value in knowing how exactly an

IRV ballot could change the results of an election: the pivotal probability derivation requires

writing down the set of all pivotal events in IRV, and even if nobody uses this set to vote

strategically, it is still an underlying and informative truth about that electoral system.

In that light, this paper makes three advances in our understanding of IRV. The core

contribution is to extend the classic calculus of voting to IRV. I derive the probability that

a ballot cast in an IRV election will change the outcome of the election, for the ﬁrst time

covering elections with any number of candidates and any length of ballot. This is the

central ingredient in any rational choice study of IRV voting. I also show how to estimate

pivotal probabilities in IRV using pseudocode, and in the appendix I illustrate and explain

the abstract derivation by providing simple numerical examples. After obtaining the full

expression for pivotal probability, I pause to consider how complicated it turns out to be. I

argue that this is not just a reﬂection of strategic voting being complicated, but rather that

the equation brings into relief something deeply complicated in the idea of instant runoﬀs.

That derivation also highlights a counter-intuitive property of IRV that I call “indirect

pivotality”. In IRV, one way for a voter to change the outcome of the election is to rank a

candidate somewhere on their ballot, and thereby causes that candidate to win the election.

However, in IRV, there is another possibility: by ranking some candidate on their ballot,

a voter can thereby cause a diﬀerent candidate to win the election. This phenomenon is

closely related to some well-studied phenomena (especially IRV’s failure of the monotonicity

criterion, and also the no-show paradox), but it is not the same as either one. Greater

opportunities for indirect pivotality in many-candidate elections may provide a motivation

for limiting the number of candidates that voters can rank in IRV elections.

Finally, I compare the probability of casting a pivotal vote in IRV to the probability of

casting a pivotal vote in SMDP, and I ﬁnd that pivotal probabilities in these two systems are

very similar. This suggests that voters do not have a larger incentive to behave strategically

in one system than the other.

2 Pivotal probabilities in Instant Runoﬀ Voting

For about a century IRV was almost uniquely used in Australian legislative elections, but now

it is suddenly being rapidly considered and even implemented in large democracies. A 2011

British referendum proposed adopting IRV countrywide, and IRV is reportedly the preferred

system of Canada’s prime minister (Kohut, 2016). The United States is experiencing a

surge of IRV adoptions, and this sytem now selects state and federal representatives in

Alaska, congressional representatives in Maine, and municipal oﬃcials in a number of cities,

including the country’s largest. One major stated motivation for adopting IRV is that it

reduces strategic behaviour, and “liberates citizens to vote for the candidates they actually

favor” (Gehl and Porter, 2020, ch. 5). But attempts to measure strategic voting in IRV do

not clearly show that the system reduces or eliminates strategic behaviour, and an even more

basic question has gone un-answered: do voters actually have less reason to vote strategically

in IRV than in other systems, or are their strategic opportunities similar or even larger?

Single-winner IRV elections work as follows: in an IRV race with κcandidates, voters

are allowed to rank some number of those candidates. The election administrator counts the

number of times that each candidate was ranked ﬁrst, and the candidate with the fewest

ﬁrst-place votes is eliminated. Then, any remaining candidate that was ranked on a ballot

immediately after the candidate that was eliminated has one vote added to their vote total.

Again, whichever candidate has the fewest votes is eliminated. This procedure is repeated

until κ−1 candidates have been eliminated, and the remaining candidate wins the election.

In the following section, I will derive the probability that a ballot cast in an IRV election

will change the outcome of the election. We will take the perspective of a voter who has

only three pieces of information: they know how many candidates are running, they know

how many candidates voters are allowed to rank on their ballots, and they know the number

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TheprobabilityofcastingapivotalvoteinanInstantRunoffVotingelectionSamuelBaltz∗August1,2023Wordcount:6,930AbstractIderivetheprobabilitythatavotecastinanInstantRunoffVotingelectionwillchangetheelectionwinner.Iphrasethatprobabilityintermsofthecandidates’ex-pectedvotetotals,andthenIestimateitsmagnitudef...

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