VOTING TO ELECT A
SINGLE CANDIDATE
Single-Winner Elections
Single-Winner Elections
• Election of a [unitary] executive office.
• Election of a representative from a singlemember district [SMD]
• Election of slate Presidential electors
running on a (“winner-take-all”) general
ticket/slate.
Binary Voting
• Just two options (so one winner and just
one loser)
– Single-winner elections with just two
candidates
• “straight fight” [British terminology]
– Yes/No voting
• referendums
• parliamentary voting
Kenneth May’s Theorem
• Choices for voters in straight fight between
A and B
– vote for A
– vote for B
– abstain (if indifferent)
• Possible voting outcomes
– A is the winner
– B is the winner
– deadlock (no winner, “hung electorate”)
May’s Conditions (Reformulated)
• Anonymity (of voters)
• Neutrality (between options A and B)
• Resoluteness (no deadlock)
• Preliminary Theorem: No voting system can
be simultaneously Anonymous, Neutral, and
Resolute.
• All voting systems considered here are
Anonymous and Neutral.
May’s Conditions (cont.)
• Almost Resoluteness: any deadlock is
broken by any voter changing his vote.
• Non-Negative Responsiveness: votes
don’t count negatively.
• May’s Theorem: Anonymity + Neutrality +
Almost Resoluteness + Non-Negative
Responsiveness <===> Simple (Relative)
Majority Rule
Strategyproofness
• Moreover, in a straight fight SMR is
strategyproof
– That is, no voter can ever improve the outcome with
respect to his or her true preferences by misreporting
those preferences on a ballot.
– Voters will never regret voting “sincerely” or
“honestly.”
– Sincere voting is Nash Equilibrium.
• However, in this respect SMR is not
unusual; in fact all non-negatively
responsive voting systems are strategyproof in a straight fight.
Three or More Candidates
• Once the number k of options/candidates
exceeds two, all sorts of problems arise.
– Different (“reasonable looking”) voting procedures, all
satisfying “May-like” conditions and equivalent to
SMR in a straight fight, may produce different winners
from the same “preference profile.”
– No voting procedure is strategyproof.
– No voting procedure is spoilerproof.
• This is essentially “Arrow’s Theorem.”
• One rationale for a two-party system is that it
typically produces (something very close to)
straight fights and therefore precludes these
problems.
Preference Profiles
With more than two candidates (k > 2), voters’ preferences
cannot be specified simply by listing their most preferred
(top-ranked) candidates; rather we must specify voters’
full preference ordering over all candidates (like the ice
cream flavors).
A collection of preference orderings, one for each voter is
called a preference profile.
We use British party labels to identify three candidates —
Labour, Liberal, and Conservative — one of whom is to
be elected. While there are six possible orderings of
three candidates, we first consider a simple profile in
which only three of these orderings are present and we
indicate the popularity of each.
We assume all have strict orderings (no indifference) and
there are no ties.
Preference Profile 1
# of voters
1st pref.
2nd pref.
3rd pref.
46
Labour
Liberal
Conserv.
20
Liberal
Conserv.
Labour
34
Conserv.
Liberal
Labour
Simple Plurality Voting (SPV) [“first past the post” or FPTP]
Plurality Ranking
Candidates
Votes Received (= First Preferences)
Labour
46 votes (winner)
Conservative
34 votes
Liberal
20 votes
Assuming “sincere” voting, SPV takes account of first
preferences only.
Plurality vs. Majority Winner
• Given a preference profile with k > 2:
– A plurality winner is a candidate who has
more first preferences than any other
candidate.
– A plurality winner always exists [in the
absence of ties].
– A majority winner is a candidate who is the
first preference of a majority of voters.
– Every majority winner is also a plurality
winner, but the reverse is not true.
– There may not be a majority winner.
Plurality Plus Runoff (or Instant
Runoff Voting [IRV])
• If there is no majority winner, there is a
runoff between the top two candidates in
the plurality ordering.
• Under IRV, Profile 1 would produce a
runoff between Labour and Conservative
[first and second in the plurality ranking],
which Conservative would win.
Approval Voting
• Voters can vote for (“approve of”) more than one
candidate.
• The candidate with the most approval wins.
• Approval voting outcomes are indeterminate.
Presumably a voter casts an approval vote for A
and not for B only if the voter’s preference
ordering ranks A over B [“no skipping”].
• In Profile 1
– Labour wins if all voters cast only one vote.
– Conservative wins if only voters in the middle bloc
cast two votes
– Liberal wins if all voters in the right bloc cast two
votes or if all voters cast two votes.
• S. Brams and P. Fishburn, Approval Voting
Borda Point Voting
• Votes rank the candidates on the ballot.
• Candidates are awarded three points for each
ballot on which they are ranked first, two points
for each ballot on which they are ranked second,
and one point for each ballot on which they are
ranked third (if k = 3).
– In general, m points, m-1 points, etc., when there are
m candidates.
• Borda Ranking for Profile 1:
• Liberal
• Labour
• Conservative
220 points (winner)
192 points
188 points
Condorcet Voting
• Votes rank the candidates on the ballot.
• Examine all pairs of candidates and see who
wins each straight fight.
• For Profile 1:
– Liberal vs. Conservative:
– Conservative vs. Labour:
– Liberal vs. Labour:
Liberal wins by 66-34
Conservative wins by 54-46
Liberal wins by 54-46
• Assemble the Majority (or Condorcet) Ranking
– 1st pref.
– 2nd pref.
– 3rd pref.
Liberal (Condorcet Winner)
Conservative
Labour (Condorcet Loser)
• Precisely opposite of Plurality Ranking (based on 1st
prefs. only) and different from Borda Ranking (also
based on full rankings)
Condorcet Winners
• A Condorcet winner is a candidate who can beat
every other candidate in a straight fight.
• A Condorcet loser is a candidate who is beaten
by every other candidate in a straight fight.
• A majority winner is always a Condorcet winner,
but the reverse is not true.
• A plurality winner may not be a Condorcet
winner.
• A Condorcet winner may not be a plurality
winner.
– Indeed, a Condorcet winner may have the fewest first
preferences (e.g., Liberal in Profile 1).
Condorcet Voting (cont.)
• Moreover, Condorcet Voting appears to
be:
– Strategyproof, because it is built on Simple
Majority Rule, which is strategyproof; and
– Spoilerproof, because (for example), regardless of whether Nader is a candidate or not,
Bush is the Condorcet winner only if he beats
Gore in a straight fight.
• So what about “Arrow’s Theorem”?
Problem: There may be no
Condorcet Winner
• This assertion seems puzzling because
every ranking has a highest ranked
element.
• But there may be no Majority Ranking.
Preference Profile 2
# of voters
1st pref.
2nd pref.
3rd pref.
46
Labour
Liberal
Conserv.
20
Liberal
Conserv.
Labour
34
Conserv.
Labour
Liberal
• First preferences are unchanged from Profile 1,
so the plurality winner is unchanged.
• As before, there is no majority winner.
• Conservative remains the Plurality Runoff
winner.
• Labour becomes the Borda point winner.
• Approval voting remains indeterminate.
Cyclical Majorities
Re-examine the straight fights:
Liberal vs. Conservative
Conservative vs. Labour
Labour vs. Liberal
Liberal wins by 66-34
Conservative wins by 54-46
Labour wins by 80-20
• There is no majority ranking, no Condorcet Winner, and
no Condorcet Loser.
• Instead we have a cyclical majority that behaves like the
“even stranger” ice cream customer.
• In general, Condorcet Winners may not exist and
Condorcet Voting is not a proper voting system, so
Arrow’s Theorem stands.
Preference Profile 3
# of voters
1st
pref.
pref.
rd
3 pref.
4th pref.
2nd
35
B
A
C
D
33
C
A
D
B
32
D
A
B
C
• Example of a socially divided society (e.g., Shiites,
Sunnis, and Kurds in Iraq); Candidate A is the “alliance”
(cross-community) candidate.
• Candidate A is the Condorcet Winner even though A has
no first preferences.
• Candidate A is the Condorcet Winner even though the is
a majority cycle among the other candidates.
• Side point: If A, B, C, and D are parties, the “Alliance
Party” A wins no seats under list-PR.
Condorcet Consistency
• A voting rule is Condorcet consistent if,
given sincere voting, it always selects the
Condorcet winner when one exists.
• Previous examples showed that, given
Profile 1, Liberal may fail to win under
each of the other voting rules discussed,
so none of them is Condorcet consistent.
• Condorcet voting is obviously Condorcet
consistent but, since it is does not always
select a winner, it cannot be deemed a fullfledged voting rule.
Parliamentary (Yes/No) Voting
• There are options A, B, C, D, ….
• Voting is by a sequence of binary votes taken in
some fixed order (say alphabetical):
– (1)
– (2)
– (3)
A vs. B
winner of (1) vs. C
and so forth
• If there is a Condorcet Winner, it can never be
knocked out, so CW wins regardless of the
voting order.
• If there is a majority cycle and no CW, winner is
determined by the voting order (later entry is
advantageous).
Single-Peaked Preferences
To say that majority cycles may exist is not to say
that they typically are present.
Indeed, if preferences are structured in a simple
way by ideology (or otherwise), cycles cannot
occur.
In British politics, the three major parties are
generally perceived to be ideologically ranked
from left to right in the following manner:
More leftwing:
Labour
Relatively centrist:
Liberal
More rightwing:
Conservative
Single-Peaked Preferences (cont.)
1st pref.
2nd pref.
3rd pref.
“Admissible” Orderings
Leftwingers Centrists
Rightwingers
Lab
Lib
Lib
Con
Lib
Lab
Con
Lib
Con
Con
Lab
Lab
“Inadmissible”
Orderings
Con
Lab
Lab
Con
Lib
Lib
Why the name “single-peaked”?
In diagrams below, L = leftist, C = centrist, and R = rightist
Single-Peaked Preferences (cont.)
•
Note the strength of the “centrist” (Liberal) candidate in the admissible
orderings.
– While it may be that few voters most prefer the centrist, no one likes the
centrist least.
– The centrist candidate must be the Condorcet Winner unless a majority of
voters have the leftwing or rightwing ordering.
– In other words, the centrist candidate fails to be the Condorcet Winner only
if one of the “extreme” candidates is a majority winner.
•
In the general case, if all voters can be ranked from most leftwing to most
rightwing with respect to their first preferences,
– no cyclical majority occurs;
– some position on the ideological spectrum is the Condorcet Winner; and
– that CW position corresponds to the first preference of the median voter,
such that no more than half the voters are more leftwing and no more than
half are more rightwing (Duncan Black, Theory of Committees and
Elections.
•
The Hotelling-Downs theory of electoral competition (to be discussed later)
states that two competing vote-seeking parties or candidates achieve
equilibrium only when both adopt the position that corresponds to the first
preference of the median voter.
Strategic Voting under Plurality Rule
• Any voting rule with three or more candidates may give
some voters incentives to vote otherwise than sincerely.
• Consider Profile 1 again.
# of voters
1st pref.
2nd pref.
3rd pref.
46
20
34
Labour Liberal Conserv.
Liberal Conserv. Liberal
Conserv. Labour Labour
– Labour wins under Plurality Voting if voters are sincere.
– But a majority of 54 voters prefer both other candidates to
Labour.
– If they all vote for the same other candidate, that candidate wins,
an outcome they all prefer to a Labour victory.
– But doing this requires some voters among the 54 to vote
“insincerely,” i.e., for their second preferences.
– Thus simple Plurality Voting can encourage what the British call
tactical voting and most political scientists call strategic voting,
i.e., non-sincere voting.
Strategic Voting under Plurality Rule (cont.)
• But how will the 54 voter majority coordinate their votes?
Will they all vote for Liberal or for Conservative?
– While all 54 voters prefer to see Labour defeated, they disagree as to
how to defeat him, i.e., by voting Conservative or by voting Liberal.
– It is generally believed that, in practice, tactical voting in Britain mostly
leads Liberal supporters to shift their votes “tactically” to their secondpreference (Labour or Conservative) candidate, because they typically
observe pre-election polls showing Liberal trailing well behind both other
candidates, and they therefore conclude that a Liberal vote is “wasted”
and that they should vote for the one of the two leading (non-Liberal)
candidates that they prefer.
– This can happen even though Liberal is the Condorcet winner, reflecting
the fact that polls (almost always) ask only about first preferences and
Liberal's great strength lies in second preferences.
– If Liberal supporters find Labour and Conservative to be equally
objectionable, they have no incentive to vote tactically.
– If pre-election polls show something close to a tie for second place (or a
three-way tie), tactical voting becomes far more conjectural.
Strategic Voting under Runoff [or IRV]
# of voters
1st pref.
2nd pref.
3rd pref.
Preference Profile 1
46
20
Labour
Liberal
Liberal
Conserv.
Conserv.
Labour
34
Conserv.
Liberal
Labour
Under Plurality Runoff [or IRV], the 46 voters who most
prefer Labour would do better by ranking Liberal first, as
this assures a Liberal victory (without a runoff), which
outcome they prefer to the Conservative victory that
otherwise results.
Moreover, other voters have no countermoves available.
No strategic vote is available to 34 voters whose first
preference is Conservative, and the 20 voters whose first
preference is Liberal are already getting their first preference.
Strategic Voting Under Borda Point
Voting
Given Profile 1, no voters can change their Borda score
ballots in a way that improves the outcome for them.
Profile 2
# of voters
1st pref.
2nd pref.
3rd pref.
46
Labour
Liberal
Conserv.
20
Liberal
Conserv.
Labour
34
Conserv.
Labour
Liberal
Given Profile 2, if the bloc of 20 ranks Conservative first
and the bloc of 34 ranks Labour third, then Conservative
gets the most Borda points (208 vs. 200 for Liberal and
192 for Labour), an outcome all 54 such voters prefer to
victory by the sincere Borda winner Labour.
Strategic Voting Under Borda (cont.)
Preference Profile 3 [4 in Handout]
46
1st pref.
2nd pref.
3rd pref.
Labour
Liberal
Conservative
54
Conservative
Labour
Liberal
Labour wins if voting is sincere (demonstrating that Borda Point Voting
can deny victory to a majority winner).
But the 54 Conservative-preferring voters can elect Conservative if they
(“insincerely”) push Labour down to third place on their ballots.
In turn, the 46 Labour-preferring voters can counteract this by pushing
Liberal to the top of their ballots (the resulting Liberal victory being
preferable to Conservative).
Note that if strategic manipulation stops at this point (though it need
not), Liberal is elected even though everyone prefers Labor to
Liberal.
And things can get worse.
Borda Strategy: “Turkey Raising”
There are three candidates: a more or less reasonable
Democrat D, a more or less reasonable Republican R,
and a real “turkey” T. Everyone one ranks T last, except
two deranged T supporters. The profile is:
50 voters
D
R
T
48 voters
R
D
T
1 voter
T
D
R
1 voter
T
R
D
If everyone votes sincerely, the Borda Point totals are
D=249, R=247, and T=104.
Anticipating this defeat, Republican voters caucus and
notice an interesting feature of Borda Point Voting — it
can pay voters to engage in “turkey raising,” i.e., to
strategically raise the “turkey” in their ballot rankings, so
as to push the rival “serious” candidate down in their
rankings and increase the point spread between the two.
“Turkey Raising” (cont.)
Suppose the Republicans strategically modify all their ballots so as to
produce the following ballot profile:
50 voters
D
R
T
48 voters
R
T
D
1 voter
T
D
R
1 voter
T
R
D
The point totals would then be D=201, R=247, and T=152.
Democrats also notice this feature of Borda Voting and, concerned that
Republicans may engage in turkey raising, they engage in some
turkey raising of their own in order to counteract the anticipated
Republican stratagem. So the final ballot profile is:
50 voters
D
T
R
48 voters
R
T
D
1 voter
T
D
R
1 voter
T
R
D
The final point scores are D = 201, R = 197, and T = 202. May the
best turkey win!
Spoiler Effects
A sincere electorate using Plurality Voting may behave exactly the
confused ice cream customer.
# of voters
1st pref.
2nd pref.
3rd pref.
46
Labour
Liberal
Conserv.
Preference Profile 1
20
Liberal
Conserv.
Labour
34
Conserv.
Liberal
Labour
Conservative beats Labour in a straight fight, but Labour wins if Liberal
enters the election. So Liberal is a spoiler to Conservative.
Likewise, Liberal beats Labour in a straight fight, but Labour wins if
Conservative enters the election. So Conservative is also a spoiler
to Liberal.
Note that the effect of the spoiler’s entry is to elect the last preference
of the spoiler’s supporters, so the spoiler’s entry is self-defeating.
Note that, in this profile, Labour cannot be a spoiler, because if Labour
enters what had been a Liberal-Conservative straight fight, Labour
wins.
Spoiler Effects (cont.)
• Spoiler effects may be mitigated by strategic voting.
• Duverger’s Law: Single-Winner elections with Simple
Plurality Voting tend to produce two-party systems and
straight fights Strategic voting to mitigate spoiler effects
is one factor that drives Duverger’s Law.
– “Don’t waste your vote on a hopeless third candidate.”
• Candidate “entry deterrence” is the other, probably more
important, factor that drives Duverger’s Law.
– “Don’t run as a third candidate, you will spoil the chances of the
preferable of the two major candidates.”
• Multi-Winner elections [no elected executives] with any
variant of Proportional Representation [PR] sustains
multiparty systems. [Other half of Duverger’s Law]
Spoiler Effects under IRV
• Plurality Runoff/IRV is sometimes advocated on
the grounds that it precludes the spoiler effects
that characterize Simple Plurality.
• Plurality Runoff/IRV is an improvement over
Simple Plurality in this respect.
– Under IRV, a single third candidate (such as Nader)
with little first-preference support cannot act as a
spoiler in what is essentially a straight fight between
two major candidates, because the runoff will become
precisely that straight fight.
• However, if there are many candidates and/or
first preferences are dispersed, Plurality
Runoff/IRV is subject to spoiler effects.
Two Variants of IRV
• There are (at least) variants of IRV.
– They are equivalent if there are just three candidates
but are distinct (and may produce different winners) if
there are more than three candidates.
• The first variant mimics Plurality Runoff.
– If there is no majority winner, all candidates except
the leading and second-place candidate in the
plurality ranking are simultaneously eliminated.
– The ballots of all eliminated candidates are transferred to one or other surviving candidate on the basis
of lower preferences.
– The surviving candidate with the most (original plus
transferred) ballots is elected.
Two Variants of IRV (cont.)
• The second variant is distinct from plurality
runoff.
– If there is no majority winner, the candidate with the
fewest first preference ballots is eliminated.
– The ballots of the eliminated candidate are transferred to one of the surviving candidate on the basis
of second preferences.
– If there is still no majority winner, the surviving candidate with the fewest (original plus transferred) ballots
is eliminated and the ballots are transferred on the
basis of second or lower preferences.
– And so forth until there is a majority winner (which
necessarily occurs once all candidates but two have
been eliminated.
Two Variants of IRV (cont.)
• The second variant of IRV is also called the
Alternative Vote.
– It is the single-winner special case of a multi-winner
voting system called the Single Transferable Vote
(STV).
– STV is used in small multi-member districts (MMDs).
– It produces approximately proportional representation
of groups among the winners, even if those groups
are not political parties.
• contrast with List-PR
– STV is used to elect:
• the Irish Dail (Parliament)
• the Australian House of Representative
• the Cambridge MA City Council
Spoiler effects under Plurality
Runoff (and 1st IRV variant)
• When there are many candidates, the
Plurality Runoff and the first variant of IRV
are subject to spoiler effects in precisely
the same manner as Simple Plurality
Voting, with respect to the question of
which two candidates will survive into the
runoff.
• This is illustrated by the 2002 French
Presidential election (“Two Round” or
Plurality Runoff).
• Left splinter candidates “spoiled” Jospin’s chance to get
into the second round runoff against Chirac.
• However, if the second [AV] variant of IRV had been in
used:
– ballots for left splinter candidates would have (presumably)
transferred to Jospin;
– few ballot would have transferred to Le Pen; and
– Jospin would have gotten into the runoff and might have won.
Would Runoff/IRV Lead to a Proliferation of
Presidential Candidates?
Spoiler Effects under 2nd IRV Variant (AV)
• However, spoiler effects also exist under the second IRV
variant. This is illustrated by Preference Profile 1 with
just three candidates (in which case the two variants are
equivalent):
# of voters
1st pref.
2nd pref.
3rd pref.
46
Labour
Liberal
Conserv.
20
Liberal
Conserv.
Labour
34
Conserv.
Liberal
Labour
• Liberal wins a straight fight with Conservative, but Liberal
does not even make it into the runoff if Labour enters the
field. So Labour is a spoiler to Liberal under IRV.
• This is not a distinctive flaw in Plurality Runoff (or IRV)
because, as previously noted, the problem is unavoidable with three or more candidates.
Negative Responsiveness under Plurality
Runoff (or IRV)
• Plurality Runoff (or IRV) has another flaw that is
distinctive (and avoidable).
• Plurality Runoff (or IRV) can respond negatively
when a candidate’s position in a preference (or
ballot) profile becomes more favorable — put
otherwise, it can punish a candidate for gaining
more support.
– This sometimes called “monotonicity failure.”
– It is a violation of May’s Non-Negative Responsiveness condition
Negative Responsiveness under Plurality
Runoff or IRV (cont.)
Original Preference Profile 4
35
10
25
30
A
B
B
C
B
A
C
A
C
C
A
B
Revised Preference Profile 4
35
10
25
30
A
A
B
C
B
B
C
A
C
C
A
B
• Prior to the election, a poll indicates that voting intentions those
shown in Original Preference Profile 4. Candidate A is pleased with
this information, because it projects that A and B will go into the
runoff, which A will win.
• However, A doesn’t want to take any chances and urges his
supporters to go out and drum up still more support for A. A’s
supporters are successful, transforming the original profile into the
revised profile, in which 10 voters who previously preferred B to A
now prefer A to B. (No other preferences have changed.)
• As a result of this successful campaign, A loses the election.
IRV and the “No Show Paradox”
•
Here is a related peculiarity of Plurality Runoff [or IRV].
Preference Profile 5
5
6
4
B
C
A
C
B
B
A
A
C
[2]
[A]
[B]
[C]
• The preference profile is as shown above, but the two
individuals with the bracketed preference orderings fail to
vote.
– The election outcome is determined by the remaining 15 voters.
Candidates B and C are paired in a runoff, which B wins.
– This is somewhat disappointing for the two individuals who failed to
vote, in that their second preference won.
– They regret their failure to get to the polls and wonder whether their first
preference A might have won if they had not failed to vote.
– It can be checked that, if they had gotten to the polls and voted
according to their preferences, C would have won (so the outcome
would have been worse, not better, for them).
“Clone” Candidates
When I was 12 years old I was nominated to be treasurer of my class at
school. A girl named Michelle was also nominated. I relished the
prospect of being treasurer, so I made a quick calculation and nominated Michelle’s best friend, Charlotte. In the ensuing election, I
received 13 votes, Michelle received 12, and Charlotte received 11,
so I became treasurer.” T.N. Tideman
Consider the following preference profile, in which a Republican
minority is united behind a single candidate R but the Democratic
majority is split between the two “clone” candidates D1 and D2.
Preference Profile 6
Democrats
35%
D1
D2
R
25%
D2
D1
R
Republicans
25%
15%
R
R
D1
D2
D2
D1
Clone Candidates (cont.)
• Clones are candidates who very similar in terms of their
ideology, issue positions, etc. and are therefore adjacent
in all voters’ preference rankings.
• Simple Plurality is notorious for penalizing clone candidates.
– In this case, the Republican candidate wins due to the Democratic split, even though R is the Condorcet loser and would be
beaten by both D1 and D2 in straight fights.
– D1 and D2 are spoilers against each other.
• It is precisely the expectation of such outcomes under
Simple Plurality voting that leads to party formation and
party discipline.
– The Democrats [and the girls] have a huge incentive to hold a
prior nominating convention or primary to choose between D1
and D2 and then send just one of the two clones forward against
the Republican.
– Given the preference profile above, D1 would win the nomination
and then the general election.
Plurality Runoff [or IRV] and Clones
• Are there voting rules that can reduce, eliminate,
or even reverse the self-defeating effect of running clone candidates?
• Given Profile 6, Plurality Runoff (instant or otherwise) solves the clone problem.
– The first-round election functions as the (Democratic)
“primary” (except that Republicans also vote in this
primary).
– The runoff functions as the general election in which
the Democratic majority gets its way.
• If there are four or more candidates, Plurality
Runoff [or IRV] does not treat clones so well
and, as we have seen, it is subject to other
problems in addition.
Approval Voting and Clones
• Brams and Fishburn advocate Approval Voting
as a desirable a voting rule that (among other
things) does not punish clones.
– In Profile 6, presumably (almost all) Democrats would
vote for both D1 and D2, one of whom would be
elected.
– By not penalizing clones, AV does not encourage
party formation or party unity.
– For this reason, many political scientists are more
inclined to support AV for primary elections and nonpartisan elections than for partisan general elections.
Double-Vote List-PR and Clones
• A variation of one type of party-list PR (Proportional Representation) system is another voting
method that does not penalize clones who have
the same party affiliation.
– In a district that elects m candidates, each voter votes
for m candidate, but these vote counts in two ways:
• first, as a party vote to determine which party wins the
election; and
• second, as a candidate vote to determine which candidate(s)
of the winning party is elected. In the profile above (with m =
1), D1 would be elected.
Borda Point Voting and Clones
Perhaps surprisingly, Borda Point Voting actually
rewards the running of clones.
Suppose that there are two candidates and Republicans
are again in the minority.
Preference Profile 7A
60 voters
D
R1
40 voters
R1
D
With just two candidates, the Borda point rule is identical to
Plurality Voting (and SMR), so the Republican candidate
R1 loses.
Borda Point Voting and Clones (cont.)
• But, if Borda Voting is in use, the Republicans can reverse this
outcome by nominating an additional clone candidate R2 whom
everyone sees as identical to R1 with respect to issues and ideology
but inferior with respect to (let’s say) personal qualities.
Preference Profile 8A
60 voters
40 voters
D
R1
R1
R2
R2
D
• Democrats can counteract this Republican stratagem by strategically ranking R2 above R1.
• Alternatively, they can counteract it by running their own clone.
• Borda voting is highly susceptible to strategic maneuvers of this sort
(which, moreover, have the effect of expanding the candidate field
rather than winnowing it down in the manner of Plurality Rule).
• Also recall “turkey raising” under Borda
Overall Conclusion
• Two-candidate elections (“straight fights”)
are clean and simple.
• Elections with three or more candidates
are complex and inevitably quirky.