Public Choice
•  We investigate how people can come
up with a group decision mechanism.
•  Several aspects of our economy can not
be handled by the competitive market.
–  Whenever there is market failure, there is a
rationale for government intervention. For
example, universal health care provides a
solution to the adverse selection problem.
–  the government also plays a role in
redistributing income more equitably.
Slide 1
On whose behalf should the
government act?
•  We will talk about desirable features of a
group decision mechanism (a voting
procedure).
•  We’ll first ask what we want the voting rule to
accomplish if everybody votes sincerely
(votes according to his or her preferences).
•  Next, we’ll talk about how different voting
procedures may be manipulated by strategic
voters.
•  We’ll find two central messages comprised in
Arrow’s Impossibility Theorem and the
Gibbard-Satterthwaite Theorem.
Slide 2
Voting Rules in Canada
•  Canada's electoral system is referred to as a "singlemember plurality" or "first-past-the-post" system.
•  In every electoral district, the candidate with the most
votes wins a seat in the House of Commons and
represents that electoral district as its member of
Parliament.
•  An absolute majority (more than 50 percent of the
votes in the electoral district) is not required for a
candidate to be elected.
Slide 3
Electoral reform in Canada
•  I have listed a few editorials from the
Globe and Mail on this topic.
•  Note that editorials are opinions voiced
by editors of the newspaper. I don’t
share the view of all that is said in these
pieces, but they contain a lot of food for
thought.
Slide 4
Electoral reform in Canada
•  http://www.theglobeandmail.com/globedebate/editorials/electoral-reform-howthe-system-partly-fixed-itself-in-2015/
article28028337/
•  http://www.theglobeandmail.com/globedebate/editorials/electoral-reform-wantto-ditch-first-past-the-post-meet-thealternatives/article28006897/
Slide 5
Electoral reform in Canada
•  http://www.theglobeandmail.com/globedebate/editorials/want-to-rewritecanadian-democracy-hold-areferendum/article27969331/
•  http://www.theglobeandmail.com/globedebate/who-wants-trudeaus-electoralreform/article27973254/
Slide 6
Voting Rules - Examples
•  Plurality Rule (first-past-the-post): Each voter
casts a vote for his or her preferred
candidate. Elect the candidate who is named
most often.
•  Pair-wise Majority Rule: two candidates are
put against each other in a vote. Whichever
candidate is preferred by more people than
the other, this candidate is preferred by
society to the other candidate. Candidate who
receives a majority of votes when put against
every other candidate (the Condorcet winner)
is elected.
Slide 7
Voting Rules - Examples
•  Scoring Method: Assigns numbers to ranks,
then sums up the numbers a candidate gets
from all individuals based on their ranking.
E.g. assign 3 to first rank, 2 to second, 1 to
third and 0 to fourth. Then we sum up the
ranks for each candidate and then order the
candidates according to their descending
scores. Elect the candidate ranked highest.
Slide 8
Modified Plurality Rules
•  Plurality rule with run-off (e.g. how French president
is elected): If in the first round using the plurality no
candidate gains a majority, the two candidates with
the highest vote count go into a second round.
Whoever wins then is the winner of the election.
•  Ranked Ballot (e.g. used in Australia): Voters rank all
candidates. In first stage, only first-ranked candidates
are considered. If nobody receives a majority, the
candidate with lowest vote count is eliminated and
the second-ranked candidate of those voting for the
eliminated candidate is moved up to first rank. Apply
plurality and see if now one of the remaining
candidates gains a majority. If not, proceed with
eliminating candidates until one candidate receives a
Slide 9
majority.
Let’s vote! – Table 1
Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
Slide 10
Questions
•  Which candidate is elected if we use the
plurality rule?
•  Which candidate is elected if we use the
plurality rule with run off?
•  Which candidate is elected if we use the
ranked ballot?
•  Which candidate is elected if we use the
pair-wise majority rule?
•  Which candidate is elected if we use a
Slide 11
scoring method?
Answers – Plurality Rule
Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
D gets 3 votes, A gets 5 votes, B gets 6 votes, C gets 7 votes,
thus C is elected.
Slide 12
Answers – Plurality w/ runoff Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
D gets 3 votes, A gets 5 votes, C gets 7 votes, B gets 6 votes,
thus B and C go into the run-off. Then B gets elected with
Slide 13
11 votes.
Answers – Ranked Ballot
Preference Rankings
# of
voters:
5
3
6
1st
A
D
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
1.
B
7
2.
C
D gets 3 votes, A gets 5 votes, B gets 6 votes, C gets 7 votes,
thus D is eliminated. Then B is eliminated and C wins with
Slide 14
13 votes.
Answers –Pairwise Majority
Preference Rankings
Rule
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
•  Majority rule: A:B = 15:6, A:C=8:13, A:D =
12:9; B:C=11:10, B:D = 11:10, C:D=12:9. No
Condorcet winner; D loses against every Slide 15
other candidate.
Answers – Scoring Method
Preference Rankings
# of
voters:
5 Score
3 Score
6 Score
7 Score
1st
A 3
D 3
B 3
C 3
2nd
B 2
A 2
D 2
A 2
3rd
C 1
C 1
C 1
D 1
4th
D 0
B 0
A 0
B 0
•  A: 5*3+(3+7)*2+6*0=35
B: 5*2 +6*3 + (3+7)*0= 28
C: 7*3 + 14*1= 35, D: 3*3 + 6*2 + 7*1+5*0 = 28,
A and C tie at first place.
Slide 16
Any lessons?
•  Different voting rules potentially elect
different candidates, so not just voters’
preferences matter in who gets elected
but also the procedure by which we
elect candidates.
•  Some voting rules are more “decisive”
than others. Should we avoid
ambiguity?
Slide 17
First-past-the post vs
Ranked Ballot
•  In previous example, both voting rules
elected the same candidate. Of course
this is not always the case.
•  Next example illustrates this.
•  Note that the next example has a
Condorcet winner; D wins against any
other candidate in a pair-wise election.
Slide 18
Let’s vote! –Table 2
Preference Rankings
# of
voters:
3
5
7
6
1st
A
D
B
C
2nd
D
A
D
D
3rd
B
C
A
B
4th
C
B
C
A
Slide 19
Plurality vs. Ranked Ballot
•  With plurality rule, B wins.
•  With ranked ballot, D wins. D is the
candidate most often ranked second. It
is also the Condorcet winner.
•  So first-past-the-post fails to elect the
Condorcet winner. In this example,
ranked ballot doesn’t.
•  Does ranked ballot always elect the
Slide 20
Condorcet winner?
Let’s vote (again)! – Table 3
Preference Rankings
# of
voters:
3
7
5
6
1st
A
B
C
D
2nd
B
A
D
C
3rd
C
C
B
B
4th
D
D
A
A
Slide 21
Condorcet winner not
elected
•  The Condorcet winner is C.
•  First-past-the post, plurality with run-off,
and ranked ballot all fail to elect the
Condorcet winner sometimes.
•  These rules result in B or D being
elected.
•  I’d argue that B is a better candidate
than D, so ranked ballot seems to do
worse here than first-past-the-post.
Slide 22
Electoral Reform
•  Is going from first-past-the-post to
ranked ballot an improvement?
•  If a candidate wins a majority of the
votes in the first round, all of these
voting rules would elect the Condorcet
winner.
•  If no candidate wins a majority, it’s not
clear which of the two rules is “better.”
Slide 23
In search of the ideal voting
rule
•  No voting cycles (see first example and
outcome under pair-wise majority rule)
•  Pareto Optimality
•  Every vote counts (unrestricted domain
and non-dictatorship)
•  Independence
Slide 24
Transitivity
•  One desirable feature of a voting mechanism
is to prevent voting cycles. This idea is
reflected in transitivity.
•  Transitivity means that if X is preferred to Y
and Y is preferred to Z, then X must be
preferred to Z.
•  For example, the relation “greater equal” is
transitive.
Slide 25
More Conditions for an Ideal
Voting Mechanism
•  Unrestricted Domain: no matter what
preference ordering people might have, they
should have an equal say in the voting
process. That is, we cannot exclude a person,
because we think it is weird to prefer Y to Z
and Z to X.
•  Nondictatorship: no individual in society
should be so powerful that the voting
mechanism reflects only his or her
preferences over every set of alternatives put
up for a vote.
Slide 26
More Conditions
•  Pareto Optimality: If there is one
alternative that everybody prefers to
another alternative, say everybody
prefers X to Y, then Y should not be
elected.
Slide 27
Question: Do the majority rule and the
scoring methods satisfy Pareto
optimality?
•  PO is satisfied by both pair-wise MR, and the scoring
method (SM). If an alternative is preferred by every
voter to another alternative this alternative has a clear
majority over the other, hence the dominated alternative
cannot be the Condorcet winner. For SM the dominated
alternative always gets less points than the other
alternative and hence is higher up in group ranking due
to a higher score.
Slide 28
More Conditions
•  Independence: the social ranking of two
alternatives X and Y should only
depend on these two alternatives.
Slide 29
Plurality Rules and
Independence
•  First-past-the-post, plurality with run-off,
and ranked ballot do not satisfy
independence.
•  To see this, check out Table 1 and then
see if removing one of the candidates
that didn’t get elected by the rules will
yield a different winner of the election.
Slide 30
Independence – Use Table 1
Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
Slide 31
Arrow’s Impossibility Theorem
•  No voting mechanism exists that
satisfies all conditions simultaneously.
•  That’s unfortunate, but it also makes our
lives more interesting J
Slide 32
Single Peaked Preferences
and the Median Voter
•  If we restrict preferences to be single peaked, the
pair-wise majority rule always generates a transitive
group preference, so there is a Condorcet winner.
•  Think of it as ranking of political parties from left to
right. Order political parties along the left-right scale
as follows: A, B, C, D.
•  Define the median voter as the voter whose
preferences lie in the middle of the set of all voters’
preferences; half the voters are located to the right of
the median voter and half of the voters are located to
the left.
•  Table 3 is an example of single-peaked preferences.
Tables 1 and 2 violate single-peakedness.
Slide 33
Not Single-peaked
Preferences – Table 1
Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
Slide 34
The Median Voter Theorem
•  As long as all preferences are singlepeaked, the outcome of pair-wise
majority voting reflects the preferences
of the median voter.
•  Moreover, it is only necessary for the
voting mechanism to know the peak of
each voter in order to compute the
Condorcet winner of the election.
Slide 35
Median Voter TheoremThe 11 voter
prefers C.
applied – Table 3
th
Preference Rankings
# of
voters:
3
7
5
6
123
4 5 6 7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
1st
A
B
C
D
2nd
B
A
D
C
3rd
C
C
B
B
4th
D
D
A
A
Slide 36
The Median Voter Theorem
•  implies that a party close to the center will
receive a majority of votes. Pierre Trudeau
once said: “We are in the extreme centre, the
radical middle. That is our position.”
•  Pair-wise MR is as easy to administer as
plurality rule, but has better properties.
•  Pair-wise MR and plurality rule are the same
when there are only two candidates.
Slide 37
Criticism of MVTh
•  Political beliefs may not always be ranked
along a single spectrum. The median voter on
the issue of subsidizing day care may not be
the same person as the median voter on the
issue of provincial versus federal rights.
•  If people have multi-peaked preferences over
political parties, the median voter theorem
does not hold either.
•  Implies that all politicians would adopt the
preferred policies of the median voter to get
elected.
•  However, even if politicians want to be
elected, ideology, personality and leadership
play a role in their decisions and political Slide 38
positions.
Vote Manipulation
•  Thus far we have assumed that people vote
in such a way that reflects their preferences.
If a voter thinks that her first choice has no
chance of being selected, she may decide to
vote for her second choice or even third
choice to prevent an alternative she considers
disastrous from being chosen. This process is
called strategic voting.
•  There are advocates of the ranked ballot who
claim voters wouldn’t vote strategically with
this method, but they do with first-past-thepost.
Slide 39
First-past-the-post and
strategic voting – Table 3
Preference Rankings
# of
voters:
3
7
5
6
1st
A
B
C
D
2nd
B
A
D
C
3rd
C
C
B
B
4th
D
D
A
A
Slide 40
Ranked Ballot and strategic
voting – Table 1
Preference Rankings
# of
voters:
5
3
6
7
1st
A
D
B
C
2nd
B
A
D
A
3rd
C
C
C
D
4th
D
B
A
B
Slide 41
The Gibbard-Satterthwaite
Theorem
•  When a single outcome is to be
chosen from more than two
alternatives, the only voting rule
that cannot be manipulated is a
dictatorial one.
Slide 42
Is there a way out of GSTh?
•  Again this theorem seems to be rather
pessimistic.
•  However, once again by restricting
individual preferences to be singlepeaked, we have a positive result.
–  Given single-peaked preferences, the pairwise majority rule cannot be manipulated.
(Moulin 1988).
Slide 43
Single-peakedness and
Median Voter
•  To see why the pair-wise majority rule cannot
be manipulated with single-peaked
preferences, recall that no voter would claim
a peak past the median voter’s peak,
because that would mean voting for a
candidate that is worse than the one based
on sincere voting. But since any claimed peak
between a voter’s actual peak and the
median voter’s peak would not change the
outcome of the election, nobody has an
Slide 44
incentive to vote strategically.
Median Voter TheoremThe 11 voter
prefers C.
applied – Table 3
th
Preference Rankings
# of
voters:
3
7
5
6
123
4 5 6 7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
1st
A
B
C
D
2nd
B
A
D
C
3rd
C
C
B
B
4th
D
D
A
A
Slide 45
Conclusion
•  Although the Arrow Impossibility Theorem
and the Gibbard-Satterthwaite Theorem are
rather pessimistic, the conclusion is that
society has to live with some imperfections in
the voting mechanism, and NOT that we
should have a dictatorship. And so I conclude
with Winston Churchill who once said that
“Democracy is the worst form of
government except for all those others
that have been tried.”
Slide 46