THE ALTERNATIVE VOTE (AND COOMBS) VERSUS FIRST-PAST-THE-POST: - PowerPoint Presentation

Slide1

THE ALTERNATIVE VOTE (AND COOMBS) VERSUS FIRST-PAST-THE-POST: A SOCIAL CHOICE ANALYSIS OF ENGLISH CONSTITUENCY ELECTIONS, 1992-2010

Nicholas R. Miller

University of Maryland Baltimore County (UMBC)

Second World Congress of the Public Choice Societies

Miami, March 8-11, 2012

http://userpages.umbc.edu/~nmiller/NRMILLER.AVvsFPTP.PCS12.pdfSlide2

Overview: An Opportunistic PaperPeter K-K’s invitation for paper on “empirical social choice”I initially expected to do something relating to the U.S. Electoral College.I was also working “Monotonicity Failure and IRV” and wanted to examine 2010 UK election data.I found

Pippa

Norris’s Shared Datasets website, with 1992-2010 UK data.

The UK held a May 2011 referendum on electoral systems: AV vs. FPTP

The paper has no particular thesis or research hypothesis.

Rather it examines a sample of 2642 “real” ballot profiles, in terms of

t

he plurality (and anti-plurality) status of candidates;

t

he Condorcet status of, and relationships among, the candidates ;

plurality-Condorcet interrelationships;

winners under the three alternative electoral systems: FPTP, AV [IRV

]

, Coombs;

the votes-seats proportionality of results under these electoral systems;

s

poiler effects under all three systems; and monotonicity failure under AV and Coombs.

I also examine several more hypothetical sets of ballot profiles that “bracket” the primary set.

I do not here come to grips with

“

tactical

voting” or a general assessments of the relative merits of the three systems.

FPTP won the

referendum

by a large margin.

The pre-referendum campaign did not stimulate

an especially enlightening discussion of the properties and relative advantages and disadvantages of the two electoral

systems

.Slide3
Slide4

Alternative Electoral SystemsOn an FTPT (“First-Past-The Post”) ballot, voters put an “X” beside the name of the candidate they wish to vote for, and

the

candidate with the most votes is elected.

On an

AV (“Alternative Vote”; known as Instant Runoff Voting [IRV] in US)

ballot, voters rank the candidates in order of preference.

If

one candidate has a majority of first preferences, that candidate is elected.

If no candidate is

supported by a majority of first preferences,

the

candidate with the fewest first preferences is eliminated and

his

or her ballots are transferred to other candidates on the basis of the second preferences expressed on the ballots for the eliminated candidate.

This

process is repeated until one candidate is supported by a majority of ballots and is elected.

“If

you like the Alternative Vote, you ought to know about the Coombs

rule” (

Grofman

and

Feld, 2004): similar AV but with a different elimination rule:

the candidate with the most last preferences is

eliminated.

Bernard

Grofman

and Scott Feld, “

If You

Like

the Alternative Vote

(a.k.a

. the Instant Runoff), Then You Ought to Know about

the Coombs Rule,”

Electoral Studies

, 2004.Slide5

Political ContextSince they were displaced by Labour as the main opposition to the Conservatives in the inter-war period, the Liberals have joined the Electoral Reform Society in advocating the Single Transferable Vote for general elections.STV isthe multi-member district generalization of AV and

(quasi-) proportional in nature.

More recently the Liberal Democrats have advocated AV as less radical but more acceptable alternative to FPTP.

Nick Clegg: “AV is a baby step in the right direction – only because nothing can be worse than the status quo

.”

The Liberals demanded the AV referendum a condition for joining the Conservatives in a coalition government after the May 2010 general election.Slide6

Political Context (cont.)As a generally centrist party, the Liberals have a lot of second preference support, though relatively little first preference support.Since AV asks voters to rank candidates and often counts second preferences, at first blush it seems that AV would help the Liberals.

But while second-preference support helps a candidate if and when he gets into a runoffs, second-preference support doesn’t help a candidate get into a runoff in the first place.

As Aldrich et

a

l. (2012) observe, if AV were applied at the national level (e.g., in a direct election of Prime Minister), the Liberals would still lose, since their candidate would not get into the runoff.

Aldrich et al. examine four alternatives to FPTP (Condorcet, AV, Coombs, and

Borda

) and conclude that the Liberals would have won such a national vote in 2010 under all four systems

except

the one they actually advocated.

A basic question is the extent to which individual constituencies are approximate replicas of the national electorate, implying that Liberal candidates would consistently fail to get into runoffs in which their second-preference support could elect them.

Coombs should be much more advantageous to Liberals because

second preferences not only help candidates win a runoff in the event they get in, but also

help them get into a runoff in the first place (by leaving them fewer last preferences).

John Aldrich et al., “Strategic Voting in the 2010 U.K. Election,” APSA Meeting, 2012. Slide7

Social Choice AnalysisThe analysis is based on three simplifying conditions:there are only three candidates,so there is at most only one “instant runoff.”v

oters ranks all three candidates.

In practice, AV may allow “truncated” ballots,

but Coombs really cannot do so.

In practice Coombs (even more than AV) may be vulnerable to “donkey voting.”

t

here are no plurality (or anti-plurality) or pairwise ties

.

A

ballot profile

is a set of

n

rankings of three candidates

X

,

Y

, and

Z

, where

n

is the number of voters. Slide8

Plurality StatusGiven a particular profile B: the candidate with the most first preferences is the

Plurality Winner

,

the

candidate with the second most first preferences is the

Plurality Runner-Up

, and

the

candidate with the fewest first preferences is the

Plurality Loser

.

Let

n

(PW),

n

(P2), and

n

(PL) be the number of ballots that rank the Plurality Winner, the Plurality Runner-Up, and Plurality Loser first

.

Given

three candidates,

n

(PW

) >

n

/3 >

n

(PL).

If

n

(PW) >

n

/2,

the

Plurality Winner is also a

Majority Winner

.

The

AV

winner is

the

Majority Winner if one exists and

otherwise either

the Plurality Winner or the Plurality Runner-Up, depending on the outcome of the instant runoff between

them.

Note that all these definitions

depend

on the distribution of first preferences only.Slide9

Condorcet RelationshipsCondorcet relationships take account of second (and lower) preferences.Given ballot profile B, candidates X, Y, and Z have

x

,

y

, and

z

first preferences respectively, where

x + y + z = n

.

Likewise

x

y

is the number of voters who have a first preference for

X

and second preference for

Y

(and therefore a third preference for

Z

),

x

z

is the number who have a first preference for

X

and a second preference for

Z

, so

x

y

+

x

z

=

x

; and likewise for other candidates.

If

under election profile

B

a majority of voters rank

X

over

Y

, i.e

., if

x +

z

x

> y +

z

y

, we say that ‘

X

beats

Y

’ in a “straight fight” (to use

an appropriately

British phrase).

This

is the basic

Condorcet relationship

.Slide10

Condorcet Relationships (cont.)Given three candidates and a particular profile

B

:

a

Condorcet Winner

is a candidate who beats both other

candidates

.

a

Condorcet Loser

is a candidate who is beaten by both other candidates; and

Condorcet

Carrier

is a candidate who beats one candidate and is beaten by the other

.

In the theory

of directed graphs, a

carrier is

a point with ‘in-degree’ and ‘out-degree’ both equal to

1.

If

X

beats

Y

,

Y

beats

Z

, and

Z

beats

X

or if

Y

beats

X

,

X

beats

Z

, and

Z

beats

Y

,

t

here is a

Condorcet cycle

, and

all

three candidates are Condorcet carriers. Slide11

Anti-Plurality StatusGiven ballot profile B:the candidate with the fewest last preferences is the

Anti-Plurality Winner

,

the

candidate with the second fewest last preferences is the

Anti-Plurality Runner-Up

, and

the

candidate with the most last preferences is the

Anti-Plurality Loser

.

Let

n

(APW),

n

(AP2), and

n

(APL) be the number of ballots that rank the Anti-Plurality Winner, the Anti-Plurality Runner-Up, and Anti-Plurality Loser last.

Given

three candidates, it follows that

n

(APW) <

n

/3 <

n

(APL).

The

Coombs winner

is

the Majority Winner if one exists and

otherwise

is either the Anti-Plurality Winner or the Anti-Plurality Runner-Up, depending on the outcome of the instant runoff

(based on the Condorcet relationship) between them.

Slide12

Plurality--Condorcet InterrelationshipsSome elementary social choice propositions for the case of three

candidates:

PROPOSITION 1. A

Majority Winner is

always

a Condorcet

Winner

.

PROPOSITION 2. The

Plurality Winner always wins under FPTP (by definition

).

PROPOSITION 3. Plurality and Condorcet status are completely independent, in particular,

the

Plurality Winner may be a Condorcet

Loser, and

the Plurality Loser may be a Condorcet Winner, so

a

Condorcet Winner may fail to be elected under FPTP; and

a

Condorcet Loser may be elected under

FPTP.

PROPOSITION 4. Either

the Plurality Winner or Plurality Runner-Up may be

elected under AV,

so

FPTP and AV may produce different

winners.

PROPOSITION 5. The

Plurality Loser cannot be elected under

AV,

because it

is eliminated from the

instant runoff.

PROPOSITION 6. A

Condorcet Loser cannot be elected under

AV,

because

,

even if it

gets into the runoff,

it

will be

beaten in the runoff.Slide13

Plurality---Anti-Plurality—Condorcet InterrelationshipsFor the case of three candidates:PROPOSITION 7. Plurality and anti-plurality status are completely independent; in particular,

the

Anti-Plurality Winner may be the Plurality

Loser, and

the

Anti-Plurality Loser may be the Plurality

Winner.

PROPOSITION 8. Condorcet and anti-plurality status are

in the general case

completely independent; in particular,

the

Anti-Plurality Winner may be a Condorcet

Loser, and

the

Anti-Plurality Loser may be a Condorcet Winner

.

PROPOSITION 9.

However

, in the special case of single-peaked preferences,

t

he Condorcet winner has no last preferences and therefore is always the Anti-Plurality winner, so

Coombs always elects the Condorcet winner.

In fact, this is true regardless of the number of candidates.Slide14

Empirical DataThe empirical analysis that follows is based on constituency-level data from U.K. general elections from 1992 through 2010. However, I use data

from English

constituencies only,

because

virtually all

constituency elections

in England are essentially three-party (

Labour

, Liberal Democrat, and Conservative) affairs,

while

those in Wales, Scotland, and Northern Ireland almost always include strong (and often winning) candidates of ‘nationalist’ parties as

well.

A

handful of English constituencies that do not fit the three-party pattern are also excluded

.

Over

the five general elections, this gives us a sample of

2642

three-candidate elections (527 to 531 per year

).

This

data

comes

from

Pippa

Norris’s Shared Datasets website (http://www.hks.harvard.edu/fs/pnorris/ Data/Data.htm). I am extremely grateful to Professor Norris for making this valuable data readily available.Slide15

The Problem of Second PreferencesAn obvious problem is that these elections were conducted under FPTP and therefore the election data provides us only (what we take to be) the first-preferences of voters, while analysis of AV (and Coombs) elections requires voters’

full preference rankings

of the three major-party candidates.

I

have addressed this problem in a way that

is

fairly standard among those

British

psephologists

,

which

is to

allocate second preferences (and

by

default third preferences) in each district in proportion to second preferences nationwide, as determined by surveys that produce individual level data about second

preferences.

Data

for 1992 through 2005 comes from

Curtice

(2009), which in turn comes from the British Election Study (post-election) for 1992 and 1997 and from ICM/BBC (pre-election) for 2001 and 2005. Data for 2010 comes from Ritchie and

Gardini

(2012), which in turn was taken a (pre-election) poll conducted for ITV News and The Independent newspaper.

Survey

respondents who gave a ‘nationalist’ or other fourth-party second preference, or who did not give a second-preference were excluded in these calculations, and proportions were calculated on the basis of

Labour

plus Liberal plus Conservative second preferences only.

John

Curtice

,

“

Recent History of Second Preferences” (http://news.bbc.co.uk/

nol

/shared/

spl

/hi/

uk_politics

/10/ alternative

_vote/alternative_vote_june_09_notes.pdf

)

Ken Ritchie

and Alessandro

Gardini

, “Putting Paradoxes into Perspective — In

Defence

of the Alternative Vote

,”

in Dan S.

Felsenthal

and

Moshé

Machover

, eds.,

Electoral Systems: Paradoxes, Assumptions, and Procedures

,

2012 Slide16

Survey-Based Second PreferencesAs

we would

expect

, that British voter preferences are “partially

single-peaked,”

that

is, most but not all

Labour

(presumptively ‘left-of-center

’) voters have the Liberals (the

presumptively ‘centrist

’ party) as their second

preference,

and

most

but not all Conservative

(presumptively ‘right-of-center

’) voters likewise have the Liberal as their second preference, while

Liberal

voters have second preference more equally divided between the two other parties (though generally leaning in the

Labour

direction but with the proportion varying considerably from election to election).Slide17

Other Second-Preference Assumptions Strictly Single-Peaked Preferences: all

Labour

and Conservative voters have Liberal as their second preferences.

No

ideological structuring of

preferences:

Random

second

preferences

: on average,

second preferences are equally

split, and

Impartial

second

preferences

: on

each ballot profile,

the second preference of each voter is

determined by a flip of a

coin.

SSP and Random/Impartial in effect “bracket” survey-based data.

In

addition, we occasionally compare the English data with

wholly random ballot profiles and

wholly impartial (culture) ballot profiles

.,

in

which first as well as second preferences are determined “randomly” or “impartially.” Slide18

Plurality Status of Party Candidates(All Second-Preference Assumptions)Slide19

Plurality Rankings of Party Candidates (All Second-Preference Assumptions)Slide20

Condorcet Relationships(All Second-Preference Assumptions Plus Random Profiles and Impartial Culture)Slide21

Plurality-Condorcet InterrelationshipsSlide22

AV vs. FTPT WinnersGiven a switch from FPTP to AV, an average 7.5% constituencies change party hands. Liberals are consistently helped by AV, losing not a single seat, gaining on average 4.3% of all seats (a 65% gain in seats). Conservatives are almost as consistently hurt by AV, gaining a handful of seats (from Labour

in 1992) but losing on average 6.3% of all seats (a 14.5% loss in seats).

Labour

is inconsistently affected by AV, gaining seats in good

Labour

years, losing seats in bad

Labour

years, and gaining on average 1.9% of all seats (a 3.8% gain in seats.Slide23

Coombs vs. FPTP Winners

Given

a switch

from FPTP to

Coombs,

an average

16%

constituencies change party hands.

Liberals are

greatly and consistently

helped by

Coombs,

losing not a single seat, gaining on average

11.2%

of all seats (a

170%

gain in seats

), and placing second in three years.

Conservatives are almost as consistently hurt by

Coombs,

gaining

seats from

Labour

in 1992

but losing on average

11.4%

of all seats (a

26%

loss in seats).

Labour

is

inconsistently affected by Coombs,

gaining seats in good

Labour

years, losing seats in bad

Labour

years, and

gaining/losing nothing on average.Slide24

Coombs vs. AV WinnersGiven switch from AV to Coombs, an average 9.5

% constituencies change party hands.

Liberals are consistently helped by AV, losing not a single seat, gaining on average 6

.9%

of all seats (a

63%

gain in seats).

Conservatives are almost as consistently hurt by AV, gaining a

few

seats (from

Labour

in 1992)

but losing on average

5.1%

of all seats (a

13.8%

loss in seats).

Labour

is inconsistently

impacted

by AV, gaining seats in good

Labour

years, losing seats in bad

Labour

years, and

losing

on average

1.8%

of all seats (a

3.5% loss

in seats.Slide25

AV and FPTP Winners with DifferentSecond PreferencesThe Liberals gain slightly fewer seats under AV if preferences are strictly single peaked.With random second preferences, there is very little difference between AV and FPTP (random “erosion” of the FPTP distribution).

With impartial

second

preferences, t

here is essentially no difference between AV and FPTP.Slide26

Coombs and FPTP Winners with DifferentSecond Preferences

The Liberals gain

vastly more

seats under

Coombs

if preferences are strictly

single-peaked, winning more seats than either

Labour

or the Conservatives,

e

xactly paralleling Condorcet Winner status (from previous table).

With random second preferences, there is almost no difference between AV and FPTP (random “erosion” of the FPTP distribution).

With impartial second preferences, there is essentially no difference between AV and FPTP.Slide27

Coombs and AV Winners with DifferentSecond Preferences

The Liberals gain vastly more seats under Coombs if preferences are strictly single-peaked, winning more seats than either

Labour

or the Conservatives,

With

random second preferences, there is almost no difference between AV and FPTP (random “erosion” of the FPTP distribution).

With impartial second preferences, there is essentially no difference between AV and FPTP.Slide28

Disproportionality(Survey Second Preferences)Gallagher Index of Disproportionality =

Coombs delivers quite proportional results in 2010 but highly disproportional results in 1997 and 2001.

But overall

all

three system are

substantially and approximately

equally

disproportional.

Michael Gallagher

,

“Proportionality

,

Dispropor-tionality

and Electoral

Systems,”

Electoral

Studies

, 1991

 Slide29

Disproportionality(Strictly Single-Peaked)In this data, Coombs vastly overcorrects the penalty imposed on the Liberals by FPTP and generally is less proportional than FPTP and AV.Overall

all three system are substantially

disproportional, FPTP and AV being about equal disproportional as under the previous data.Slide30

Spoiler EffectsGiven a potential three-candidate (X, Y, and Z) election, the entry of candidate X into what would otherwise be a two-candidate contest (or X’s exit from a three-candidate contest) can have three effects:(1) No Spoiler Effect

I

: the same candidate (Y or Z) wins in either event;

(2)

No

Spoiler Effect

II

: X wins if X enters, Y or Z wins otherwise; or

(3)

Spoiler Effect

: X does cannot win even if X enters, but Y wins if X is out and Z wins if X is in, i.e., X’s entry “spoils” Y’s election (or X’s exit “spoils” Z’s election).

Two types of spoiler effects:

u

nilateral spoiler

: X’s entry spoils Y’s election but Y’s entry does not spoil X’s election;

m

utual spoilers

:

X’s entry spoils Y’s election

and

Y’s entry

spoils

X’s

election (e.g., X and Y are “clones”);

In the following discussion,

majority, plurality, anti-plurality, and Condorcet status

refers to

(potential) the

three-candidate

ballot profile.Slide31

Spoiler Effects (cont.)PROPOSITION 10. If there is a Majority Winner, no spoiler effects

occur

under FTPT, AV, or Coombs.

Otherwise we examine eight possible election configurations with respect to

the plurality status of the candidates and the Condorcet relationships among them (for FPTP and AV); and

t

he anti-plurality status of the candidates and the Condorcet relationships among them (for Coombs).

There are two configurations of each following type:

PW (or APW) is CW [(1a) and (1b)];

P2 (or AP2) is CW [(2a) and (2b)];

PL (or APL) is CW [(3a) and (3b)];

t

here is no CW (cyclical) [(4a) and (4b)].

The outcome of every straight fight is determined by the Condorcet (“beats”) relationship.Slide32

Spoilers under FPTPUnder FPTP, the winner of the three-candidate contest is (by definition) always PW

.

The number by each candidate shows the spoiler effect that results when that candidate enters/exists the election.Slide33

Spoiler Effects under FPTPPROPOSITION 11. The Plurality Winner can never be a spoiler under FPTP. PW wins whenever it enters, so always No Spoiler Effect II.PROPOSITION 12. FPTP is invulnerable to spoiler effects if and only if PW = CW [(1a) and (1b)].PROPOSITION 13. FPTP is vulnerable to mutual spoiler effects if and only if PW = CL [(2b) and (3b)].PROPOSITION 14. Otherwise FPTP is always vulnerable to unilateral spoiler effects.Slide34

Spoilers under AVUnder AV, the runoff (if any) is between PW and P2.So the winner of a three-candidate election isPW if PW  P2, and

P2 if P2

 PW.Slide35

Spoiler Effects Under AVPROPOSITION 15. The Plurality Loser can never be a spoiler under AV. PL never makes the runoff and, in the runoff, second preferences are distributed just as they would be if PL never entered.PROPOSITION 16. The Condorcet Winner is never a spoiler under AVPROPOSITION 17. AV is vulnerable to spoiler effects if and only if PL = CW or the profile is cyclical [(3a), (3b), (4a), (4b)].

PROPOSITION 18. AV is invulnerable to mutual spoiler effects.

PROPOSITION 19. AV does not [logically] “dominate” FPTP with respect to resisting spoiler effects; in particular, PW can be a spoiler under AV (if it is not the AV winner).Slide36

Spoilers under CoombsUnder Coombs, the runoff (if any) is between APW and AP2.So the winner of a three-candidate election is APW if APW  AP2, and

AP2 if AP2  APW

.Slide37

Spoiler Effects under CoombsPROPOSITION 20. The Anti-Plurality Loser can never be a spoiler under Coombs. [APL never makes the runoff, and second preferences are distributed just as they would be if APL did not enter

.]

PROPOSITION 21. The Condorcet Winner is never a spoiler under AV.

PROPOSITION 22. Coombs is vulnerable to spoiler effects if and only if APL = CW or in the event of a cycle

[(3a), (3b), (4a), (4b)].

PROPOSITION 23. Coombs is

invulnerable to mutual spoiler effects.

PROPOSITION 24. Coombs

does not

[logically] “dominate

” FPTP with respect to resisting spoiler

effects.

If preferences are single-peaked,

the Condorcet Winner has no last preferences, and

the Anti-Plurality Winner is therefore always the Condorcet Winner,

s

o only configurations (1a) and (1b) can occur.

PROPOSITION 25. If preferences are single-peaked, Coombs is invulnerable to spoiler effects.Slide38

Frequency of Spoilers in England under FPT, AV, and CoombsOverall, in this data AV and Coombs are considerably less vulnerable to spoilers effects than FPTP.The incidence of spoilers is slightly greater under Coombs than AV, even though the data is partially single-peaked, and

Coombs in invulnerable to spoilers given strictly single-peaked profiles.

It is notable than the 2010 election (close to a three-way tie with respect to first preferences) was highly vulnerable to spoiler effects under both FPTP and AV but not Coombs.Slide39

The “Nader Effect”A fundamental virtue of AV (and Coombs) is that it precludes the “Nader effect,” i.e.,the fact that a minor candidate (i.e., one who wins only a small percent of the vote) can be a spoiler

under

FPTP

(in the manner of Ralph

Nader in

FL [and therefore the US as a whole] in

2000).

Under AV (and Coombs), (most) Nader voters could have (and presumably would have) ranked Gore second, so

their votes would have transferred to Gore in the runoff (as if Nader were not on the ballot or all Nader voters had voted “tactically” under FPTP).

However, if Nader and been a third “major” candidate, rather than a “minor” candidate, he might still have been a spoiler under AV.

Suppose that Nader had been more “centrist,”

so Gore had been “squeezed” between Bush and Nader,

rendering Gore the Plurality Loser, while still the Condorcet winner.

The runoff then would have been Bush and Nader, rather than Bush and Gore, and Bush would (as the near Majority Winner, even if most [but not quite all] Gore voters ranked Nader over Bush) have won the runoff.Slide40

The “Nader Effect” (cont.)We have seen that under AV, only the Plurality Runner-Up or Plurality Winner can be a spoiler

.

Since spoiler effects can occur only in the absence of a Majority Winner, the smallest vote the Plurality Runner-Up can receive and possibly be a spoiler is slightly more than 25%.

Thus a candidate who wins less than 25% of the (first-preference) vote cannot be a spoiler under AV.

Note that this does not imply that that AV is invulnerable to spoiler effects if the Plurality Loser in a prospective three-way contest wins less than 25% of the vote.

Such a prospective Plurality Loser may prevent the Plurality Winner from being a Majority Winner, and thereby allow the Plurality Runner-Up to be a spoiler, i.e.,

t

he Plurality Loser (e.g., Gore “squeezed between Bush and Nader) wins a straight fight with the Plurality Winner (e.g., Bush) but Plurality Winner wins if the Plurality Runner-Up (e.g., Nader) enters

and displace the Plurality Loser from the runoff. Slide41

Propensity to Spoiler Effects by Support for the (Prospective) Plurality Loser and Plurality Runner-UpSlide42

Are The Same Elections Vulnerable to Spoiler Effects Under the Different Electoral Systems?In this data, elections subject to spoiler effects under AV are a subset of those likewise subject under FPTP.In contract, elections subject to spoiler effects under Coombs and FPTP and under Coombs and AV are almost disjoint.Slide43

Frequency of Spoilers by PartyLiberal candidate are never spoilers under AV,evidently because they are typically Plurality Losers and/or Condorcet Winners.As we would expect, mutual spoilers are always ideologically adjacent, i.e., “near clones.”Slide44

Monotonicity FailureA disconcerting feature of AV (and Coombs) is that getting more (first preference) votes can cause a candidate to lose an election, and

getting

fewer votes can cause a candidate to win.

While FPTP is

monotonic

,

electoral systems that incorporate

(actual or ‘instant’)

runoffs are subject

to

(

upward

and/or

downward

)

monotonicity

failure

.

Though AV (and Coombs) are unarguably non-monotonic, the question arises how often instances of monotonicity failure arise in practice.

The English data provide a good opportunity to address this question.Slide45

Conditions for Monotonicity Failure under AVGiven that Z is the Plurality Loser, two separate conditions must hold for an AV ballot profile to be vulnerable to (Upward or Downward) Monotonicity Failure in the event that candidate

X

is moved up or down in some ballot orderings.

Condition

1 pertains to

X’s

runoff

opponent;

i

t requires

that the ballot

changes

deprive

Y

of enough first preferences (for Upward

Monotoncity

Failure),

or

give

Z

enough additional first preferences (for Downward

Monotoncity

Failure),

to

convert

Z

into the Plurality Runner-Up,

so

the runoff that had been between

X

and

Y

is now between X and Z.Condition 2 pertains to the runoff outcome; it requires that X must lose (for Upward Monotoncity Failure) or win (for Downward Monotonicity Failure) the new runoff with Z.

The conjunction of

the two conditions

is necessary and sufficient to make profile

B

vulnerable to (Upward or Downward) Monotonicity Failure.

Note

that Condition 1 depends on the distribution of first preferences only, while Condition 2 depends on second preferences as well.Slide46

Upward Monotonicity Failure under AVIf ballot profile B is vulnerable to Upward Monotonicity failure,

Condition

1 requires that

X

can gain enough first preference ballots at

Y

’s expense that two things are simultaneously true in the resulting

profile

B

ʹ:

X

is still not a Majority Winner, and

Y

becomes the Plurality Loser instead of

Z

.

This

requires that

n

/2 −

x > y − z

.

Condition

2 requires that

Z

beat

X

under

B

ʹ, i.e., that

zʹ + yzʹ > xʹ + yxʹ. It turns out that Condition 1 can be simplified and Condition 2 can be restated in terms of the original ballot profile B, as follows:

PROPOSITION 26.

A ballot profile

B

in which

X

is the IRV winner and

Z

is

the Plurality

Loser is vulnerable to Upward Monotonicity Failure if and

only

if:

(1)

z > n

/4; and

(2)

z

 +

y

z

> x +

y

x

.

Nicholas

R.

Miller,

“

Montonicity

Failure in IRV Elections with Three Candidates,”

PCS, 2012

Dominique

Lapelley

et al.,

“The Likelihood of Monotonicity Paradoxes in Run-Off Elections,”

Mathematical Social Sciences

,

1996Slide47

Downward Monotonicity Failure under AVIf ballot profile B is vulnerable to Downward Monotonicity failure, Condition

1 requires that it is possible for

X

to lose enough first preference ballots in favor of

Z

that two things are simultaneously true in the resulting companion profile

B

ʹ:

Z

is no longer the Plurality Loser, and

Y

,

rather than

X,

becomes the Plurality Loser.

Thus

it must be that

x −  y > y − z

.

Furthermore

, in order for

Z

to gain these first preferences rather than

Y,

these (

y − z

) new first preference ballots for

Z

must all come from the

x

z

ballots that initially ranked

Z

rather than Y second. Condition 2 stipulates that X beats Z under Bʹ, i.e., xʹ + yxʹ >

z

ʹ +

y

z

ʹ.

Again

these conditions can be simplified and restated in terms of the original ballot profile

B

only:

PROPOSITION 27.

A ballot profile

B

in which

Y

is the IRV winner and

Z

is the Plurality Loser is vulnerable to Downwards Monotonicity Failure if and only if:

(1)

y

< n

/3 and

x

z

> y − z

; and

(2)

y +

y

z

<

n

/2

.

Nicholas R. Miller, “

Montonicity

Failure in IRV Elections with Three Candidates,” PCS, 2012

Dominique

Lapelley

et al., “The Likelihood of Monotonicity Paradoxes in Run-Off Elections,”

Mathematical Social Sciences

, 1996Slide48

Monotonicity Failure in English Data vs. Random ProfilesSlide49

Contrast Between English and Random DataThe English data contains considerably fewer profiles vulnerable to monotonicity failure (

1.7%)

than

the random data (14.1%).

This

might suggest that

simulated

data

is

irrelevant and

misleading:

once we look at

“

real” electoral data, the problem of monotonicity failure under

AV

almost

disappears.

However

, this low incidence reflects particular features of the English election data, and does not demonstrate that IRV’s

non-monotonicity

problem is practically irrelevant.

The

primary determinant of vulnerability to monotonicity failure is election closeness

, but

very few of these English elections represented closely contested three-candidate contests (in

part

because they were actually conducted under

FPTP, not AV).

In

all English ballot

profiles

60%

had a Majority

Winner, and

in only 4.2% did the Plurality Loser get as much as 25% of first-preference support (and 39.2% of these profile were vulnerable to Monotonicity Failure).Controlling for elections closeness, vulnerability to Monotonicity Failure looks very similar in the two data sets.Slide50

Monotonicity Failure by Closeness in English vs. Random DataSlide51

Monotonicity Failure by Election Closeness in English, Random, and SPWC Data