Lecture 2: Strategy-proof voting
rules
Jingyi Xue
Singapore Management University
Econ 241
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Three or more candidates
What voting rule would properly extend majority
voting among pairs?
The plurality rule: Each voter casts a vote for his or
her favorite candidate. Choose the candidate(s)
who is (are) named most often.
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Condorcet’s challenge
Nicolas de Condorcet (1743-1794): a French
philosopher, mathematician, and political scientist.
Plurality may contradict the majority opinion.
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Condorcet’s argument
Consider the following problem with 21 voters and 4
candidates.
3
a
b
c
d
5
a
c
b
d
7
b
d
c
a
6
c
b
d
a
The plurality rule chooses a with 3+5=8 votes.
Condorcet: But a is the worst candidate for a clear
majority of voters (7+6=13). This majority prefers
any candidate to a.
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Condorcet winner
3
a
b
c
d
5
a
c
b
d
7
b
d
c
a
6
c
b
d
a
Condorcet: c should be chosen if we abide by the
majority opinion.
Reason: A majority (7+6=13) prefers c to a.
Another majority (5+6=11) prefers c to b. A third
majority (3+5+6=14) prefers c to d.
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Condorcet consistent voting rule
A Condorcet winner is a candidate who defeats
every other candidate in majority comparisons.
• Defeat another candidate: More voters prefers
a Condorcet winner to another candidate.
• By definition, there is a unique Condorcet
winner if exists.
A voting rule is Condorcet consistent if the voting
rule chooses a Condorcet winner whenever it exists.
• The plurality rule is not Condorcet consistent.
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Non-existence of a Condorcet winner
Class exercise: Argue that there is no Condorcet
winner in the following problem.
8
a
b
c
7
b
c
a
6
c
a
b
• Condorcet consistency property only specifies
whom to choose whenever a Condorcet winner
exists. To completely define a voting rule, we
also need to specify whom to choose when a
Condorcet winner does not exist.
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Borda’s challenge
Jean-Charles de Borda (1733-1799): a French
mathematician, physicist, and political scientist.
Plurality may contradict the majority opinion.
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Borda’s argument
3
a
b
c
d
5
a
c
b
d
7
b
d
c
a
6
c
b
d
a
Borda: a is a poor candidate, but b, instead of c,
should be chosen if we abide by the majority
opinion.
Reason: b is ranked first by 7 voters (against 6 for
c), first or second by 16 voters (against 11 for c),
and first, second, or third by all voters (as is c).
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The Borda rule
Borda’s idea: The ranking of each candidate in the
voters’ opinions should count — if I rank some
candidate first, this should help the candidate more
than if I rank him second.
The Borda rule: A candidate receives no points for
being ranked last, one point for being ranked next
to last, and so on, up to |A| − 1 points for being
ranked first. A candidate with the highest total
score, called a Borda winner, wins.
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Scoring voting rules
Definition
Fix a nondecreasing sequence of real numbers
s0 ≤ s1 ≤ · · · ≤ s|A|−1 with s0 < s|A|−1 . A candidate
receives s0 points for being ranked last, s1 points for
being ranked next to last, and so on, up to s|A|−1
points for being ranked first. A candidate with the
highest total score wins.
• The Borda rule: sn = n where
n = 0, · · · , |A| − 1.
• The plurality rule:
0 = s0 = s1 = · · · = s|A|−2 < s|A|−1 = 1.
• How about a Condorcet consistent voting rule?
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How about a Condorcet consistent voting rule?
Theorem (Fishburn (1973))
There are problems where the Condorcet winner is
never chosen by any scoring voting rule.
• Implication: No Condorcet consistent voting
rule can be a scoring voting rule.
• Proof: Homework question.
• Condorcet consistent voting rules and scoring
voting rules are different families of rules. No
comprise.
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Borda vs. Condorcet (scoring vs. Condorcet
consistent voting rules)
Condorcet supporters: The notion of majority
comparisons is easy to grasp and seems closer to
the people’s opinion than scores.
Used by a number of private organizations: the
Wikimedia Foundation, the Pirate Party of Sweden.
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Borda vs. Condorcet (scoring vs. Condorcet
consistent voting rules)
Borda supporters: There is one compelling property
supporting scoring voting rules — participation.
Used for political elections (Slovenia, the Green
Party of Ireland) and elections by some educational
institutions in the US (Harvard, UCLA)
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Participation
Starting from now, we will focus on voting rules
that choose only one candidate for each collective
decision problem.
Participation: Suppose that candidate a is chosen
from the set A by the electorate N. Next consider a
voter i outside N. Then the electorate N ∪ {i}
should select a or some candidate whom voter i
prefers to a.
If an additional vote succeeds in changing the
outcome of the election. It can only be to the
benefit of this “pivotal” voter.
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Example
If a voting rule chooses a in the LHS problem and c
RHS, it violates participation.
Bob Cheryl
a
a
c
b
→
d
c
b
d
a chosen
Ann Bob Cheryl
b
a
a
a
c
b
c
d
c
d
b
d
c chosen
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Scoring vs. Condorcet consistent voting rules
Theorem (Moulin (1986))
All scoring voting rules — where ties are broken
according to a fixed ordering of A — satisfy
participation.
If A contains four or more candidates, there is no
Condorcet consistent voting rule satisfying
participation.
• Homework question
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Voting: aggregating individual opinions???
If a voting rule does not satisfy participation, a
voter can influence the outcome of election in
his/her favor by not participating in the election.
Such a voting rule fails to aggregate individual
opinions!
What can a society do to ensure participation?
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Ensure participation
Punish those not participating by a fine!
Moral punishment: social opprobrium!
Change to a voting rule that incentives
participation: scoring voting rules.
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Ensure participation
Legal punishment: compulsory voting
Those not participating could face prosecution, face
difficulties getting a job in the public sector, and
lose the right to vote for 10 years.
Compulsory voting is enforced in Argentina,
Australia, Belgium, Brazil, North Korea, Singapore,
...
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Voting: aggregating individual opinions???
Strategic participation suggests that a voter, once
realizing that his/her own vote may influence the
outcome of election, will think twice and act like a
player in the game of election, trying to maximize
the returns from his/her vote.
No participation can be prevented, but how about
lying about one’s favorite candidate?
We vote and we lie!
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Real-life strategic (tactical) voting
2017 French presidential
election
“votez utile” (tactical
voting)
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Real-life strategic (tactical) voting
In the 2004 Canada federal election, the governing
Liberal Party was able to convince many New
Democratic voters to vote Liberal in order to avoid
a Conservative government.
For the 2015 UK general election, voteswap.org was
set up to help prevent the Conservative Party
staying in government, by encouraging Green Party
supporters to tactically vote for the Labour Party in
listed marginal seats.
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Strategic (tactical) voting under the Borda rule
The Borda rule (satisfying participation)
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
LHS problem (true preferences): a : 3, b : 6, c : 7,
d : 2 with c winning
RHS problem (misreported preference of Ann):
a : 2, b : 7, c : 6, d : 3 with b winning in favor of
Ann!
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Strategic (tactical) voting
The strategic aspect of voting is already noticed in
1876 by Charles Dodgson:
This principle of voting makes an election more of a
game of skill than a real test of the wishes of the
electors.
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Strategic (tactical) voting
A voting rule fails to aggregate individual opinions if
it is manipulable by strategic voting (lying), like by
not participating.
What can a society do to prevent lying?
Punish those who lie? Legally? Morally?
Change to a voting rule that voters cannot benefit
by lying?
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How to prevent strategic voting?
It is impossible to distinguish a strategically biased
report of a voter’s preference from a truthful one.
One’s opinion is private information (before a
reliable lie detector is invented).
Hence, an openly untruthful report is a perfectly
legal move.
Moral punishment?
Find a voting rule that is immune to strategic
voting!
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Strategy-proofness
A voting rule is strategy-proof if when any voter, no
matter what the other voters report, misreport
his/her own preference, the voting rule will choose a
candidate that is less preferred, according to his/her
true preference ordering, compared with the
candidate chosen if he/she has reported his/her true
preference.
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Recall: the Borda rule violates strategy-proofness.
The Borda rule (satisfying participation)
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
LHS problem (true preferences): a : 3, b : 6, c : 7,
d : 2 with c winning
RHS problem (misreported preference of Ann):
a : 2, b : 7, c : 6, d : 3 with b winning in favor of
Ann!
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Implication of Strategy-proofness
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
Imagine that LHS preference of Ann is her true
preference. Then
a chosen LHS =⇒ no restriction RHS
b chosen LHS =⇒ b or c or d chosen RHS
c chosen LHS =⇒ c or d chosen RHS
d chosen LHS =⇒ d chosen RHS
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Implication of Strategy-proofness
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
Imagine that LHS preference of Ann is her true
preference. Then
a chosen RHS =⇒ a chosen LHS
b chosen RHS =⇒ a or b chosen LHS
c chosen RHS =⇒ a or b or c chosen LHS
d chosen RHS =⇒ no restriction LHS
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Class exercise
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
Imagine that RHS preference of Ann is her true
preference. Then
a chosen RHS =⇒
b chosen RHS =⇒
c chosen RHS =⇒
d chosen RHS =⇒
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Class exercise
Ann Bob Cheryl
a
c
c
b
b
b
c
d
d
d
a
a
Ann Bob Cheryl
b
c
c
a
b
b
d
d
d
c
a
a
Imagine that RHS preference of Ann is her true
preference. Then
a chosen LHS =⇒
b chosen LHS =⇒
c chosen LHS =⇒
d chosen LHS =⇒
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Class discussion
How to prevent strategic voting?
Find a voting rule that is strategy-proof.
Is the majority rule strategy-proof ? Why?
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Two candidates: good news!
The majority rule: desirable in the sense of being
anonymous, neutral, and strictly monotonic.
It is also immune to strategic voting: No voter can
benefit by lying.
Reason: Any voter, no matter what other voters do,
by voting for his/her less favorite candidate, either
has no influence on the outcome of election, or
making his/her less favorite candidate elected.
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How about three or more candidates?
The Farquharson-Dummett conjecture: voting rules
with at least three issues face endemic tactical
voting.
Dummett and Farquharson (1961). “Stability in
voting”. Econometrica. 29 (1): 33-43.
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