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MONOTONICITY FAILURE UNDER STV
AND RELATED VOTING SYSTEMS
Nicholas R. Miller
Department of Political Science
University of Maryland Baltimore County
Baltimore, Maryland 21250
nmiller@umbc.edu
Preliminary and incomplete draft prepared for presentation at the
2002 Annual Meeting of the Public Choice Society,
San Diego, March 22-24, 2002
MONOTONICITY FAILURE UNDER STV AND RELATED VOTING SYSTEMS
It is now generally known that the single transferable vote (STV) and related voting systems
that entail runoffs of one sort or other are subject to the paradoxical feature that enhanced support
for a candidate can cost that candidate electoral victory (or, conversely, that reduced support can give
the candidate electoral victory) — that is, that they are subject to so-called “monotonicity failure.”
However, it remains unclear how extensive this problem is.
Is monotonicity failure merely a logical possibility that is highly unlikely to occur in practice,
or is it a problem that may occur with considerable frequency? Is the possibility of monotonicity
failure indicated by manifest characteristics of election results, or is it a problem that remains largely
hidden from view? In so far as there is a real problem, is it more substantial and/or more evident
under STV or under the other related voting systems that are (or might be) used in single-winner
elections. In particular, what is the extent of the monotonicity problem under so-called Instant
Runoff Voting [IRV] that now has a number of enthusiastic advocates in the United States?
This paper attempts to address questions along these lines by means of a (more or less)
complete analysis of single-winner elections with three candidates, in which case all voting systems
in this family are logically equivalent and, at the same time, are also distinct from simple plurality
voting (SPV), commonly called First Past The Post (FPTP) in the U.K.
1. Political and Theoretical Context
For well over a hundred years, the Electoral Reform Society of the United Kingdom has
enthusiastically — indeed, passionately — advocated the Single Transferable Vote (STV) as a
method of conducting parliamentary elections in Britain and elsewhere (as is already done in
Ireland). For many years, Australia (in addition to electing Senators from statewide multi-member
districts under STV), has filled seats in its (much more important) House of Representatives under
the so-called Alternative Vote (AV), which is simply STV applied to single-member districts. In the
United States, some non-partisan local elections and some party primary elections (especially in the
South) are conducted not on the basis of Simple Plurality Vote but with a runoff between the two
top candidates (SPV+R) if neither receives an absolute majority (or perhaps some lower threshold)
of the votes cast in the first round of voting. A similar system has been used over the past thirty
years in French Presidential elections. Advocates of abolition of the Electoral College for electing
U.S. Presidents and its replacement with a direct popular vote often recommend that such a vote be
conducted with a runoff in the same manner as French presidential elections (though some
recommend a lower threshold than 50% to trigger a runoff). Quite recently in the U. S., what has
come to be called Instant Runoff Voting (IRV) has been enthusiastically advocated for U.S.
Presidential and other executive elections (and sometimes also for legislative elections in single-
member districts). Finally, the newly created office of the Mayor of London is filled by means of
the so-called Supplementary Vote (SV), which is a variant of IRV.
Monotonicity Failure page 3
In practice, voters are likely to truncate their rankings. Different quotas and vote transferring algorithms
1
are used to deal with this and other complexities.
Since it is arbitrary which votes are surplus, votes are usually transferred in proportion to all the second
2
preference votes on the elected candidate’s ballots. This means that either fractional votes may be transferred or
transferred votes must be rounded to integer totals.
All these voting systems belong to a still more general family dubbed point-runoff voting
systems by Smith (1973), in which through two or more rounds of vote counting (or actual voting)
candidates are sequentially eliminated as possible winners on the basis of their having the lowest
support (by one measure or other) on the previous round. Within the STV family, support is
measured with respect to first preference [plus any transfer] support. Under ordinary runoff voting,
voters are called back for a second trip to the polls for the runoff election. Under the other systems,
voters come to the polls just once but are invited to indicate second and usually lower preferences
on the ballot. Runoffs are then “simulated” by transferring ballots for eliminated candidates to
surviving candidates based on the second and lower preferences indicated on the ballots. In the
simplest non-trivial case (which we focus on here) in which one candidate is to be elected from a
field of three candidates, all these systems are logically equivalent.
Single Transferable Vote. STV is an exceptionally complicated voting system for electing
m candidates in (usually small magnitude) multi-member districts whose basic characteristic will be
only sketched out here.
(1) Voter use an ordinal ballot to rank all the candidates in their order of preference.
1
(2) The total number of ballots cast is determined, on which basis of quota for election is
established. Normally this quota is the next integer above the quotient resulting from
dividing the total number of ballot cast by one more than the number of candidates to be
elected.
(3) The ballots are sorted according their first preferences. Any candidate who meets the quota
at this stage is elected. If m candidates achieve quota, vote counting stops.
(4) Otherwise some votes are “transferred” and the counting continues. First preference votes
received by elected candidate in excess of the quota are “surplus” and are transferred to
unelected candidates according to their second (or lower) preference votes. As a result of
2
the transfer of surplus votes, additional candidates may meet the quota and be elected. If
m candidates now meet the quota, counting stops.
(5) Otherwise more votes are transferred and counting continues. The candidate with the fewest
(first preference plus transferred) votes is eliminated and all of the eliminated candidate’s
votes are transferred according to their highest preferences for any remaining (non-eliminated
and non-elected) candidates. As a result of this transfer, additional candidates may meet the
quota and be elected. If m candidate now meet the quota, counting stops.
Monotonicity Failure page 4
As defined here, IRV simulates (and is logically equivalent to) ordinary runoff elections but with two
3
practical differences. First, in ordinary runoff elections the composition of the participating electorate may be
different in the first and second rounds of voting. (In U.S. runoff elections, turnout typically declines in the second
round.) Second, in ordinary runoff elections, voters who voted in the first round for eliminated candidates can
(sincerely or strategically) revise their preferences between the two leading candidates (which in any case they had
not yet been called upon to express) prior to the runoff (perhaps taking account of new information, including the
vote totals from the first round). It should be noted that some people who advocate what they call IRV are actually
advocating what is here called AV.
In contrast to workings of list proportional representation, these self-defined opinion groupings need not
4
be organized political parties. (Indeed, advocacy of STV has a distinctly anti-party favor.)
(6) The vote transfer and counting process continues until m candidates have met the quota and
are elected.
Alternative Vote. AV is simply STV applied to single winner elections. The quota is
therefore a simple majority of the total number of ballots cast. If one candidate receives more than
half of the first preference votes, that candidate elected. Otherwise, the candidate with the fewest
first preference ballots is eliminated and these ballots are transferred on the basis of their second
preferences, and so on until some candidate accumulates a majority of (first preference and
transferred) ballots and is elected.
Instant Runoff Voting. IRV proceeds in the same way as AV at the outset. But if no
candidate receives a majority of the first preference votes, all candidates except the leading candidate
and the runner-up are eliminated all at once and their first preference ballots are transferred to one
or other of the two surviving candidates according to which is higher ranked on each ballot.3
Supplementary Vote. Under SV each voter casts one “regular” vote for one candidate (as
under Simple Plurality Voting) but can also cast a second “supplementary” vote. (In effect, voters
are invited to express first and second but no lower preferences.) The ballots are tabulated with
respect to the regular (first preference) votes, and a candidate who receives a majority of all such
votes is elected. If no candidate receives a majority of regular votes, all candidates except the leader
and runner-up are eliminated and the ballots whose regular votes were cast for eliminated candidates
are transferred to one or other surviving candidate on the basis of the supplementary vote — or not
transferred at all in the event the supplementary vote is for another eliminated candidate. Whichever
of the two surviving candidates has the most votes after these transfers (which is likely to be fewer
than half of the total ballots) is elected.
Advocates have generally claimed these points in favor of STV and related voting systems.
1. STV allows self-defined opinion groupings in the electorate to secure the election of
representatives of their choice in rough proportion to their proportion of the electorate in a
multi-member district. (Clearly this characteristic of STV does not extend to its single-
4
winner variants.)
Monotonicity Failure page 5
For example, see the description of STV ballots in Niemi (1970, p. 92). (Niemi examined ordinal STV
5
ballots from university elections to try to track down instances of cyclical majorities.)
However, it has long been recognized that quorum requirements can induce a kind of monotonicity
6
failure.
2. The great sin of Simply Plurality Voting or FPTP is that many votes are “wasted,” in that
they don’t contribute to the election of any candidate, either because (i) they were cast for
candidates who lost anyway, or (ii) they were cast for the candidate who would have won
anyway. Under STV almost all votes will ultimately be counted for, and (given the transfer
of surplus votes) be necessary to the election of, one of the elected candidates. (Under the
single-winner variants, wasted votes of the first type, but not of the second, are minimized.)
3. Because in SPV elections votes cast for trailing candidates (expected to place third or lower)
are almost certainly going to be “wasted,” voters who sincerely prefer such candidates are
encouraged vote “tactically” for whichever of the two leading candidates they relatively
prefer. Thus the trailing candidates do not get even their (small) “fair share” of vote.
However, under STV (and its single-winner variants) such voters can rank their most
preferred candidate first knowing that, in the event this candidate is not elected, their votes
will transfer to other candidates in way that reflects their preferences.
4. In so far as supporters of trailing candidates who actually have a preference between the two
leading candidates resist the temptation to vote tactically, the trailing candidates are potential
“spoilers” — that is, while they cannot win in any event, their entry into, or withdrawal from,
the election can determine which of the leading candidates does win. Eliminating this
“spoiler problem” is one of the main claimed advantages of IRV in the U.S.
5. Until fairly recently, the general claim of STV advocates has been that voters can safely rank
the candidates on their ballot in order of their true preferences. Indeed, STV ballots often
not only instruct voters to rank candidates in order of preference but indicate that, regardless
of how many candidates a voter orders, ranking one candidate of above others may help that
candidate’s chance of being elected and certainly can never reduce it. The general claim
5
that it is never advantageous to depart from voting one’s true preferences is the conjunction
of two separate claims: given all other ballots, (i) whether candidate a is elected is or not is
independent of how the voter ranks other candidates (e.g., whether the voter ranks b above
c or c above b); and (ii) a voter can never bring about candidate a’s election by ranking a
lower or bring about candidate a’s defeat by ranking a higher.
Voting and social choice theorists have had reason to question, at least in principle, such
claims of “strategyproofness” for STV since Gibbard (1973) and Satterthwaite (1975) published
theorems to the effect that no non-dictatorial voting procedure can be strategyproof given three or
more alternatives. However, voting procedures most commonly fail to be strategyproof as a result
failure of condition (i) described above. The kind of “monotonicity failure” entailed by failure of
condition (ii) described above is more surprising and might be thought not to characterize any
halfway reasonable voting system in actual use. But, in a quite extraordinary paper, Smith (1973)
6
Monotonicity Failure page 6
The report cited the work of Doron and Kronick, Riker, and Brams and Fishburn. The British philo-
7
sopher Michael Dummett (who has published work on voting procedures [1984] and electoral reform [1997]) also
made a direct submission to the Plant Commission that criticized STV on monotonicity (as well as other) grounds
(Farrell, 2001, p. 150).
showed that so-called “point-runoff” systems were subject to monotonicity failure. Several years
later, Doron and Kronick (1977) in effect observed that Smith’s class of point-runoff systems
includes STV and its single-winner variants, which are therefore subject to monotonicity failure.
This finding attracted some attention within U.S. academic political science (especially in Riker,
1982, pp. 49-50; also see Brams and Fishburn, 1983, and Fishburn and Brams, 1983) but evidently
little elsewhere.
However, in the early 1990s, the British Labour Party established a commission under the
leadership of political scientist Raymond (now Lord) Plant to review alternative voting systems and
to make recommendations for possible changes in the British electoral system that might be
supported by the party (and that might facilitate cooperation with the Liberal Democrats). The “Plant
Report,” though it was generally balanced and well informed by academic research, attracted much
attention and hostility from British electoral reformers because it rejected STV (while, ironically,
speaking favorably of SV) as an option for British parliamentary elections, and it did so largely on
theoretical grounds, laying emphasis on the problem of monotonicity failure.7
The outraged electoral reformers did not — for they could — dispute the logical possibility
of monotonicity failure (which can be definitively established by a single example, such as that
provided by Doron and Kronick). They did, however, offer a number of counter-arguments that were
potentially sensible and to the point (for example, Editorial, 1993 and Hill, 1994).
1. No voting system is perfect; all have problems of some sort or other. Therefore, one should
not summarily reject one voting system on the basis of one type of problem without
considering the problems to which other systems are subject. (Invoking the impossibility
theorems of voting and social theory in this way is entirely appropriate — but also has a very
different flavor from the kinds of claims that previously had been made in behalf of STV.)
2. While the Plant Report rejected STV on the grounds of monotonicity failure, it was
“particularly galling” that the Report recommended “the very inferior SV, that contains the
very same fault as that for which they rejected” STV (Hill, 1994). (This point seems
unanswerable.)
3. The problems to which particular voting systems (including monotonicity failure with respect
to STV and related systems) are demonstrated to be subject on the basis of particular
examples are merely logical possibilities and, even if the problem is quite serious in the event
that it actually occurs (which seems to be true of monotonicity failure), the problem may in
fact occur so infrequently that it can be dismissed for all practical purposes.
It is this third counter-argument that we focus on here. STV advocates cite two claims in
particular to support this third counter-argument.
Monotonicity Failure page 7
1. In a “hands-on assessment of STV” based on his twenty two years of experience as the Chief
Electoral Officer for Northern Ireland since the introduction in 1973 of STV (for MEPs, the
Northern Ireland Assembly, and local government elections) in that province, Patrick Bradley
(1995) reported that “the experience of the use of STV in Northern Ireland over the past 22
years, involving a range of election types and sizes, reveals no evidence to support in
practice the lack of monotonicity.” It is difficult to give much credence to this claim, since
instances of monotonicity failure do immediately reveal themselves (for example, in the
manner the “reversal of winners” problem to which districted electoral system are subject,
e.g., the 2000 U.S. Presidential election or the 1951 U.K. general election). Monotonicity
failure is not directly apparent from aggregate election results, or even from more detailed
tabulations showing the sequence of transferred votes. Rather it is necessary to inspect full
“ballot profiles” (as defined in the next section) and then subject them to extensive and
tedious analysis and computation.
2. Based on a combination of mathematical analysis, empirical data, and statistical assumptions,
Crispin Allard (1995, 1996) estimated that the probability of STV monotonicity failure is so
low that “if the U.K. is divided into 138 multi-member constituencies, and assuming
quadrennial general elections, we would expect in the whole country less than one instance
every century of monotonicity failure under STV” (1995, p. 49). (We will review Allard’s
arguments in the conclusion of this paper.)
In some degree at least, this conclusion and this evidence supporting it is entering the general
literature on voting systems. Thus, in his very useful new book on voting systems, Douglas Amy
(2000, p. 55 [with respect to IRV] and p. 105 [with respect to STV]) says: “It is undisputed that
nonmonotonicity can theoretically occur in a ... [IRV or STV] election, but most experts believe that
the conditions needed for this paradox to occur are so special that it would be an extremely rare
occurrence. One statistical study [i.e., Allard] found that if ... [IRV-like or STV] elections were to
be held throughout the United Kingdom, a nonmonotonic result would occur less than once a
century.” (It should be noted that [non-]monotonicity does not apply to an “election result” but to
how elections results may change as ballot profiles change.) And in a recent text on electoral
systems, David Farrell (2001, p. 150) cites both Bradley and Allard in claiming that “there is no
evidence that it [monotonicity failure] is a common occurrence.”
2. Analysis
From here on out we focus on the case of a single-winner election with exactly three
candidates, in which case the voting systems discussed here are logically equivalent (and, at the same
time, logically distinct from FPTP). In the concluding section, we shall speculate a bit about how
our conclusion might be extended to more general case.
First of all, it will be useful just to present a simple example of how monotonicity failure may
occur under voting systems of this type.
Monotonicity Failure page 8
Suppose that there are three candidates a, b, and, c and that a highly accurate poll on the eve
of the election shows that (i) no candidate commands the first preference support of a majority of
the electorate and that (ii) candidate c has the least first preference support. Thus the ballots that rank
c highest will transfer in some mix to a and b in a (simulated) runoff between them. The poll also
elicits information about these second preferences and reveals that (iii) the mix of transferred votes
will be sufficiently favorable to a that a will win the runoff and therefore will be elected.
This is indeed good news for candidate a and his supporters, but they want to spare no effort
in securing electoral victory and therefore undertake further campaign activities. These last-minute
efforts are unambiguously successful, in that on election day some voters who previously ranked a
lower now give first-preference support to a (though a still falls short of absolute majority first-
preference support) and, at the same time, no voters have either moved candidate a down in their
rankings or changed their ranking of b and c.
But a’s additional first-preference support must have come at the expense of one or other or
both of the other candidates. In fact, it came mostly at the expense of candidate b, whose first-
preference support now falls below c’s. As a result, the runoff is now between a and c, and its
outcome depends on the mix of support for a and c in the second preferences of the remaining b
voters. And in fact this mix is (as it had been all along) sufficiently favorable to c that c wins the
runoff is therefore elected. Thus we have monotonicity failure — candidate a’s added support has
cost a electoral victory.
We now aim to pin down more precisely the conditions under which such monotonicity
failure can occur.
2.1 Plurality and Ballot Profiles
There are three candidates a, b, and c. Voters mark ballots according to one of six strong
orderings, where P with a subscript designates the proportion of voters with preferences as shown
below.
abc
P P P
))))) ))))) )))))
ab ac ba bc ca cb
PP PP PP
))))) ))))) ))))) ))))) ))))) )))))
a a b b c c
b c a c a b
c b c a b a
a b c ab ac ba bc ca cb
Clearly 0 # P #1 for all subscripts and P + P + P = P + P + P + P + P + P
= 1. For notational and analytic convenience, we assume that all voters mark complete (or non-
truncated) ballots and that the number of voters is sufficiently large that we can rule out plurality or
a b ab ac
majority ties (i.e., that always P P, P P, etc.).
abc
A plurality profile P is a list (P, P, P) indicating the relative popularity of each candidate
abc
with respect to voters’ first preferences. We normally label the candidates so that P > P > P (at
least until we modify preferences so as to try to produce monotonicity failure). Thus we assume that
Monotonicity Failure page 9
candidate a is the plurality winner and also that candidate a is the majority winner if there is one
a
(i.e., in the event that P > ½)
ab ac ba bc ca cb
A ballot profile B is a list (P, P, P, P, P, P) indicating the relative popularity of
each preference ordering (or ballot). We use the notation a B b to mean that a majority of ballots
in B rank a over b (so that, in a runoff between a and b, a wins). If a B b and a B c, a is the
Condorcet winner at B; and if b B a and c B a, a is the Condorcet loser at B.
ab ca cb c
We define ( as (P ! P)'P and likewise for other arrangements of candidates. Thus
ab c
( × P is the net relative advantage that candidate a realizes when the c votes are redistributed
abc
between a and b on the basis of second preferences. For example, if P = .45, P = .4, P = .15 and
ba
( = a, candidate b gains a × .15 = .05 on candidate a in a runoff, just enough to close the first
ab ab
preference gap between b and a. Clearly !1 # ( # +1 for all pairs of candidates. If ( = !1, all
ab ab
c voters prefer b to a; if ( = 0, c voters split equally between a and b; if ( = +1, all c voters prefer
ab ab
a to b. In any event, ( = !( and likewise for other pairs of candidates.
Given two ballot profiles B and BN, we say that BN is more favorable to a than B is (or B is
less favorable to a than BN is) if the two profiles are identical except that some ballots rank a higher
a
in BN than in B. We write this as BN B.
a
OBSERVATION 1. If BN B, then a B b implies a BN b and b BN a implies b B a.
Given two plurality profiles P and PN, we say that PN is more favorable to a than P is if
aa
PN > P . Note that, while each ballot profile B entails a unique plurality profile P, a more favorable
ballot profile need not produce a more favorable plurality profile (because the more favorable ballots
may move a only from third to second place). Conversely, a more favorable plurality profile need
not derive from a more favorable ballot profile (because some ballots may move a into first place
while others move a down). But if the more favorable BN also results in a more favorable PN, we say
a
that BN is strictly more favorable to a than B is. We write this as BN B.
We say that ballot profile B entails forward monotonicity failure with respect to candidate
a in the event that a wins at profile B but loses at some other profile BN that is more favorable to a
than B is. Conversely, we say that ballot profile B entails backward monotonicity failure with
respect to candidate a in the event that a loses at profile B but wins at some other profile BN than is
less favorable to a than B is. Given the voting systems under consideration (under which candidates
survive or are eliminated on the basis of first-preference support), the fact that the pairing of
candidates in the runoff changes when the ballot profile changes from B to BN implies that the
resulting plurality profile changes as well, so monotonicity failure can occur only with respect to
ballot profiles that are strictly more (or less) favorable to the focal candidate.
abc
OBSERVATION 2. Given P > P > P, forward monotonicity failure can occur with respect to
either candidate a or candidate b. However backward monotonicity failure can occur with only
respect to candidate a.
Monotonicity Failure page 10
Note that we are demarcating the region of possible B profiles. To produce monotonicity failure, the
8
abc c b
corresponding BN profiles must lie outside of the P > P > P region, since c must get into the runoff (i.e., PN > PN).
Since candidate c must lose at B, monotonicity failure can occur only with respect to
candidates a or b. That forward failure can occur with respect to either a or b was illustrated by the
informal example that opened this analysis section (in which it was specified only that c had the least
first-preference support). However, backward failure cannot occur with respect to b. Suppose
aa bb
otherwise. Since BN is less favorable to b than B is, it must be that PN $ P and P > PN but, in
ab ab
conjunction with P > P , this implies that PN > P , so there is no way b can get into a runoff with
c (which is the only way b could win).
OBSERVATION 3. Suppose there is a forward (backward) monotonicity failure such that
candidate a wins (loses) at some profile B but loses (wins) at profile BO that is more (less) favorable
aa
to a than B is. Then either: (i) there is a profile BN “between” B and BO — that is, BO BN B
a a ca ca cb cb
(B BN BO ) — at which a loses (wins) such that PN = P and PN = P or (ii) BO is self is
ca ca cb cb
such a profile, i.e., PO = P and PO = P .
This follows because monotonicity failure with respect to a depends on the balance of first
preference support between b and c and not on the absolute level of either b’s or c’s first preference
support. We say that such a shift from B to BN “holds c harmless.”
2.2 The Space of Plurality Profiles
It is well known (Viviani’s Theorem) that the sum of the distances from any point in an
equilateral triangle such as that shown in Figure 1 (including points on the edges and vertices) is
equal to the height of the triangle and is therefore the same for every such point. Thus the sets of
points in such a triangle can conveniently represent all possible sets of three positive real numbers
that sum to a constant amount and, in particular, can represent the space of all plurality profiles with
three candidates.
The three vertices represent the three profiles in which one candidate gets all the first
preferences. The point in the center of the triangle represents the profile in which the three candi-
dates split first preferences equally. Lines parallel to an edge represent sets of profiles in which one
candidate receives a constant proportion of first preferences (i.e., the candidate is “held harmless”).
(The edge itself represents all profiles in which one candidate receives a constant zero proportion
a
of first preferences.) Corresponding to such lines are equations such as P = .4 (or, equivalently,
bc
P + P = .6). Lines not parallel to any edge represent sets of profiles in which the proportion of
first preferences received by one candidate is a linear function of the proportion received by another
ab c
such as P = P + .5 P (which can be rewritten in various ways to exclude one of the three Ps).
Selected points and lines are labeled in Figure 1.
We aim to specify regions in this space within which a profile may be subject to (forward or
abc
backward) monotonicity failure. In so far as we assume that P > P > P , we focus on the one sixth
of full triangle with that constitutes the right triangle with vertices (1,0,0), (½ ,½ ,0), and (a,a,a).8
Monotonicity Failure page 11
We first establish necessary conditions for monotonicity failure in terms of bounds on plurality
profiles.
It is immediate that in the event of (forward or backward) monotonicity failure, we must have
aa aa
½ > P . Suppose to the contrary that P > ½, so a wins at B without a runoff. If BN B, then PN
a
$ P > ½, so a also wins at BN without a runoff, and there can be no forward monotonicity failure.
And there can be no backward monotonicity failure at B as a already wins at B.
c b
We now consider bounds on P and P in the event of forward failure. Consider Table 1A,
which shows the relationship between plurality profiles associated with B and BN in the event of
forward monotonicity failure associated with candidate a (where BN is selected to hold c harmless).
aa b b
Let ) be the shift in relative pluralities as we move from B to BN, i.e., ) = PN !P = P !PN .
aaccb
Since a loses at BN , it must be that ½ > PN . Thus ½ ! P > ) . Since P = PN > PN , it must
bc a bc bc
also be that ) > P ! P . Thus know that ½ ! P > ) > P ! P. Substituting (1 !P ! P) for
a bc bc c ab
P , we get ½ !(1 !P ! P) > P ! P , which simplifies to P > ¼ or, equivalently, ¾ > P + P.
ab b
Since P > P , we must have d > P .
Table 1A: Forward Monotonicity Failure with Respect Candidate a
abc
P½> P > P > P>¼
¸º ç
abc
PN½> PN >PN < PN>¼
Now consider Table 1B, which shows the relationship between plurality profiles associated
with B and BN in the event of forward monotonicity failure associated with candidate b. By the same
bac c b
reasoning as above, ½ ! P > ) > P ! P, which likewise implies that P > ¼ and d > P .
Table 1B: Forward Monotonicity Failure with Respect Candidate b
abc
P½> P > P > P>¼
º¸ ç
abc
PN½> PN <PN > PN>¼
.))))))))))) < ))))))))))))-
bc
We now consider the bounds on P and P in the event of backward failure. Consider Table
1C, which shows the relationship between plurality profiles associated with B and BN (where BN is
selected to hold b harmless) in the event of backward monotonicity failure associated with candidate
aa bb a
a. Now ) = P !PN = PN !P . Since a loses at B, it must be that b B a, and since B BN, it
follows that b BN a. Since a wins at BN, a must be paired with (and beat) c in the runoff. Thus it
abb cbb ab bc
must be that PN > PN = P and also PN > PN = P . Thus we must have P ! P > ) > P !P.
ab c ab b ab b
Substituting (1 !P !P) for P, we get P ! P > P !(1 !P ! P) , which simplifies to a > P
ac a c 6ab
or, equivalently, P + P > b . Since always ½, > P we must have P > / . Since ½ > P , P
1
Monotonicity Failure page 12
cbcb
+ P > ½ and since P > P therefore P > ¼.
Table 1C: Backward Monotonicity Failure with Respect Candidate a
abc6
P½> P > P > P>/
1
º ç¸
abc
PN½> PN >PN < PN>¼
We can summarize these conclusions in the form of a proposition.
abc
PROPOSITION 1. Given that P > P > P :
abc
(a) forward monotonicity failure occurs only if ½ > P > P > P> ¼ (which also
b
implies that d > P) ; and
abc 6
(b) backward monotonicity failure occurs only if ½ > P > a > P > P > / (which
1
b
also implies that P > ¼).
Using Proposition 1, we can demarcate the regions of the plurality profile space in which
monotonicity failures may occur.
abc
Given that we are in the P > P > P region of Figure 1, forward monotonicity failure can
aa c c
occur only on the P < ½ side of the P = ½ line and only on the P > ¼ side of the P = ¼ line,
i.e., within the right triangle with vertices (½,¼,¼), (d,d,¼), and (a,a,a). Allowing for all
permutations of candidate labels, forward failure can occur only within horizontally marked regions
of Figure 2.
abc
Given that we are in the P > P > P region, backward monotonicity failure can occur only
aa b b
on the P < ½ side of the P = ½ line and only on the P > a side of the P = a line, i.e., within
6
the right triangle with vertices (½,¼,¼), (½,a, / ), and (a,a,a). Allowing for all permutations of
1
candidate labels, forward failure can occur only within the vertically marked regions in Figure 2.
b
A better understanding of Figure 2 can be provided by an example. The horizontal line P
= 0.3 is shown. Moving across this line from left to right indicates what happens as candidate c’s
first preference support increases at a’s expense, while candidate b is held harmless at 30% of the
vote). The story is told across the line at the bottom of Figure 2. From the profile (.7, .3, 0) to (.5,
.3, .2), a wins without a runoff; from (.5, .3, .2) to (.4, .3, .3) a goes into a runoff with b, the outcome
of which depends on the second preferences of the c voters. From (.4, .3, .3) to (.3, .3, .4) a goes into
a runoff with c, the outcome of which depends on the second preferences of the b voters. From (.3,
.3, .4) to (.2, .3, .5) c goes into a runoff with b, the outcome of which depends on the second
preferences of the a voters Finally, from (.2, .3, .5) to (0, .3, .7) c wins without a runoff.
In the (.5, .3, .2) to (.4, .3, .3) interval, a goes into a runoff with b, which b may win
(especially where a’s first preference advantage is reduced toward the right end of the interval). But
in the (.4, .3, .3) to (.3, .3, .4) interval, a goes into a runoff with c, which a may win (especially where
Monotonicity Failure page 13
a retains some first preference advantage at the left end of the interval). Thus the shift in preferences
that moves the profile across the (0.4, 0.3, 0.3) threshold may entail backward monotonicity failure
as a’s decreasing support converts a win by b into a win by a. Likewise the shift in preferences
across the (0.3, 0.3, 0.4) mark may entail forward monotonicity failure with respect to c. And of
course, if such failures occur as c’s support increases at a’s expense, the reverse failures occur as a’s
support increases at c’s expense.
Looking at the full profile space in Figure 2, the triangles constituting the regions of possible
forward and possible backward monotonicity failure can be paired off. For example, a profile B in
the right triangle with vertices (½,¼,¼), (d,d,¼), and (a,a,a) may be subject to forward mono-
tonicity failure with respect to candidate a resulting from candidate a’s gaining first preference
support at b’s expense (while c is held harmless). This will entail a “southwest” shift from B along
c
a line parallel to the P = 0 edge to some profile BN located in the right triangle with vertices
6
(½,¼,¼), (½, / ,a), and (a,a,a), i.e., the region susceptible to backward monotonicity failure with
1
acb
respect to a given that P > P > P . In this event, a shift from BN back to B will entail backward
failure with respect to a. So these two triangular regions are paired off.
But a profile B in the triangle with vertices (½,¼,¼), (d,d,¼), and (a,a,a) may also be
subject to forward monotonicity failure with respect to (the second ranked) candidate b. This will
c
entail a “northeast” shift from B again along a line parallel to the P = 0 edge to some profile BN
6
located in the right triangle with vertices (¼,½,¼), ( / ,½,a), and (a,a,a), i.e., the region
1
bca
susceptible to backward monotonicity failure with respect to b given that P > P > P . (Note that
bac
the P > P > P region is skipped over.) In this event, a shift from BN back to B will entail
backward failure with respect to b. So these two triangular regions are also paired off.
2.3 The Underlying Ballot Profiles
The conditions on plurality profiles discussed in the previous section state only necessary
conditions for monotonicity failure. Such failure actually occurs or not depending on the second
preferences of the voters whose most preferred candidate does not make it into the runoff. Thus we
have to look beyond plurality profiles and look at (some aspects of) the full orderings given by ballot
profiles. However, monotonicity failure with respect to candidate a (for example) depends only the
ab ba ac ca
parameters ( (or () and ( (or () — the second preferences of the a voters play no role.
abc
Suppose as usual that P > P > P and that there is a ballot profile B that entails backward
monotonicity failure, necessarily with respect to candidate a. This means that B puts candidate a into
a runoff with candidate b, which a loses. Thus we have b B a or:
b cb a ca
P + P > P + P
b cb ca a
P + (P ! P) > P
bbaca
P + ( × P > P .
Candidates a and b tie in the event that:
abbac
P = P + ( × P (Equation 1)
Figure 3 shows these lines:
Monotonicity Failure page 14
abc b ca b b ac b cb
Since P = P + P '2 and P = P , P = 3 P '2 . Since P = 1 ! P ! P and P = P , P =
9
aaa a7
(1! P) ' 2. Thus P = 3 (1 ! P) ' 4, which simplifies to P = / .
3
bac
(i) P = ½ = P + P , which can be rewritten as
ac
2P + 2P = 1 or
ac ac
P + P = 1 ! P ! P
ac b
P + P = P
abc ba
P = P ! P , i.e., Equation 1 in the event ( = !1
ab ba
(ii) P = P , i.e., Equation 1 in the event that ( = 0 ; and
abc
(iii) P = ½ = P + P or [proceeding as in (i)]
abc ba
P = P + P , i.e., Equation 1 in the event ( = +1.
abbac
In general, the line representing the Equation 1 P = P + ( × P passes through the point
ab ab
(½,½,0) and falls between the (ii) P = ½ and (iii) P = ½ lines depending on the value of ( .
ab c ba
Given that P > P > P , we are concerned only with case (iii) in which ( is positive. In Figure
b a ba c ba
3, the line P = P + ( × P is drawn in for ( = ½. (By straightforward calculation, the line
bc 7ba
intersects the P = P line at a = / . ) Thus, given ( = ½ , it follows that b wins at all plurality
39
profiles “northeast” of this line.
For backward monotonicity failure, we must find some BN such that:
a
(i) B B N (the new profile is strictly less favorable to a);
a b c b
(ii) PN > PN and PN > PN (the runoff is between a and c); and
(iii) a BN c (a wins the runoff).
In order for a to win the runoff against c, we must have:
a ba c bc
PN + PN > PN + PN
a c bc ba
PN > PN + (PN ! PN)
accab
PN > PN + ( × PN .
We now rewrite this condition in terms of the original plurality profile, where as before )
is the shift in first preferences from candidate a to candidate b that characterizes the shift from B to
BN:
a ba c bc
P ! ) + P > P + ) + P
aac bc
P + ( × P > P + 2) .
We know that ) must be large enough to tip the balance of first preferences between candidates b
b c
and c, i.e., that ) > P! P . Thus we have:
a ac b cbc
P +( × P> P + 2P ! 2P .
caa
Replacing P with 1 ! P ! P and combining we get:
Monotonicity Failure page 15
ac b b
( × P > 3P ! 1
1
b
> P .
ac
(3+ ()
The equation
1
b
P = (Equation 2)
ac
(3+ ()
b
is represented by a horizontal line that lies below or above the P = a line depending on whether
ac
( is positive or negative. We have a BN c at all profiles that lie above this line.
Putting everything together, we can state the following:
abc
PROPOSITION 2. Given that P > P > P , backward monotonicity failure occurs (with respect
to a) at profile B if and only if these conditions hold:
abc 6
(1) ½ > P > a > P > P > / ;
1
bbaca
(2) P + ( × P > P ; and
1 b
(3) > P
ac
(3+ ()
ba
Suppose that ( = ½ (i.e., with respect to their second preferences, 75% of the c supporters
prefer b over a, so that the net relative increment in b’s support over a resulting from votes trans-
ac
ferred from c is equal to one-half of those votes) and ( = ¼ (i.e., with respect to their second
ba ac
preferences, 62.5% of b supporters prefer a over c). Given these parameters ( and ( , backward
monotonicity failure with respect to a at any profiles in the area shown in Figure 3a.
If second preferences are distributed “favorably” enough, these bounds enclose the entire
6
triangle with vertices (½,¼,¼), (½,a, / ), and (a,a,a), in which case monotonicity failure will
1
actually occur at all profiles that are potentially subject to it.
abc ba ac
COROLLARY 2.1. Given that P > P > P and also that ( = 1 and ( # 0, backward mono-
tonicity failure occurs (with respect to a) at every profile B satisfying the conditions specified in
Proposition 1(b).
ba a b ba c a b c
If ( = 1, the P = P + ( × P line becomes simply the horizontal P = P + P (or
a ac b ac
P = ½) line and, if ( = 0, P = 1 ' (3 + () = a, so the entire triangle with vertices (½,¼,¼),
6
(½,a, / ), and (a,a,a) lies between these lines. Put more substantively, in the event that every
1
c supporter prefers b to a and no more than half of the b supporters prefer a to c, every profile B such
abc 6
that ½ > P > a > P > P > / is subject to backward monotonicity failure with respect to
1
candidate a.
On the other hand, if second preferences are distributed “unfavorably” enough, these bounds
6
enclose the entire triangle with vertices (½,¼,¼), (½,a, / ), and (a,a,a) in which profiles are
1
potentially subject to monotonicity failure and such failure therefore in fact cannot occur at any
Monotonicity Failure page 16
profile.
abc ba ac
COROLLARY 2.2. Given that P > P > P and also that ( = 0 or ( = 1, backward mono-
tonicity failure cannot occur at any profile B.
ab ba
The region of monotonicity failure shrinks to zero area just as the three lines P = P + (
c b ac b c
× P , P = 1 ' (3+ (), and P = P intersect at a common point, as shown in Figure 3b.
ab bac
We can readily find a formula for the point at which the P = P + ( × P line intersects
b c b c b a
the P = P by substituting P for P and 1 ! 2 P for P . Thus we have:
bbbab
1 ! 2 P = P + ( × P
bbab
3P + ( × P = 1
ba b
(3 + () P = 1
1
b
P = ba
(3 + ()
b c
The region of monotonicity failure thus shrinks to a point of zero area on the P = P line
ac ba b c
when ( = ( , and its location on the P = P line depends on the common ( parameter value.
Combining these considerations with previous conclusions, we have the following:
abc
PROPOSITION 2N. Given that P > P > P , backward monotonicity failure occurs (with respect
aacbbac 6
to a) at profile B if and only if ½ > P > 1'(3+ () > P > 1'(3 + () > P > / .
1
abc
COROLLARY 2.3. Given that P > P > P , backward monotonicity failure occurs (with respect
ac ba
to a) only if ( > (..
Monotonicity Failure page 17
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