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Prof. Yeşim Kuştepeli ECO 4413 Game Theory
VOTING GAMES
Prof. Yeşim Kuştepeli ECO 4413 Game Theory
OUTLINE
Choosing Among Voting Mechanisms
 Majority Rule with Two Choices
 Plurality Rule and the Condorcet Candidate
 A Modified Plurality Rule: The Single
Transferable Vote
 Strategic Voting with Plurality Rule

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

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

The individual is a consumer: individuals
demand a variety of consumption goods
and services from which they drive
welfare.
The individual provides productive
services: the most obvious resource
provided by the individual is labor.
The individual participates in the
political process: by voting and other
political activities, the individual
expresses his or her preferences
regarding the government’s provision of
goods and services.
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 The
acceptance or rejection by a
corporation’s broad of directors of merger
offer tendered by competing firm.
acceptance or rejection by a union’s
rank and file of the labor contract
negotiated by union’s officers.
 The
acceptance or rejection by a state’s
voters of proposal to issue new bonds with
which to build new roads and bridges.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
 The
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 The
outcome of any vote depends on the
voting rules and the ballots submitted by
voters. Voting is a type of game and
amenable to game theoretical analysis.
the secret ballots are used, then the
voters are playing a static game.
 Strategically
sophisticated voter realize
that it may not always be optimal to vote
sincerely.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
 If
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TWO DIFFERENT MODELS OF VOTER:

Strategic voter: she always casts her ballot so
as to obtain the best outcome possible given her
information, the voting rules, and her beliefs
about how the other voters will behave.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Naive voter: he always casts his ballot honestly,
even when it is not in his best interest to do so.
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CHOOSING AMONG VOTING MECHANISMS


After the election, these ballots are tallied
(counted) and the person receiving the largest
number of votes is declared the winner.
This voting mechanism most commonly used
to make public decision or to elect political
candidates.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Plurality voting: ballots are prepared which
voters choose only one of the candidate and
vote for him.
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
Social choice mechanism (SCM): voting is
just one procedure the members of a group could
use to from a “collective” ranking of a group of
candidates on the basis of their individual
rankings. We will refer to any such mechanism
as a Social choice mechanism
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Approval voting: voters could be allowed to
vote for as many candidates as they approve of.
The ballots are tallied and the candidate
receiving the highest number of approval votes
is declared the winner.
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ARROW’S IMPOSSIBILITY THEOREM:

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
There is no social choice mechanism that
simultaneously satisfies the Pareto condition ,
the independence of irrelevant alternatives
condition, and the transitivity with unlimited
domain condition, and is also non-dictatorial.
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ARROW’S REQUIREMENTS:
1) No candidate is ranked higher by every member of the group
than the candidate ranked highest by SCM.
3) The relative ranking of any two candidates provided by the
SCM depends on only the individual rankings. It does not
depend on how these two candidates are ranked against
any other candidates.
4) The SCM is not a dictatorship. There is no individual whose
ranking always matches the ranking of the SCM,
whatever the ranking of the other members of the group
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
2) The ranking provided by the SCM is always complete and
transitive whenever the ranking of every member of the
group is complete and transitive.
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MAJORITY RULE WITH TWO CHOICES

In an election between two candidates
decided by a majority rule, all voters will
rationally abstain or vote honestly for their
most preferred candidate.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Majority rule: the winner is the candidate
who receives more than half of the votes
cast. And the winner of election is that
candidate who receives the most votes.
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WE WILL STUDY THE BEHAVIOR OF
VOTING RULES ASSUMING THAT:
all voters are well informed
 their preferences are common knowledge
 they take voting mechanism
 the slate of candidates as given

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
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CANDIDATE AND VOTER LOCATIONS ON
LEFT-TO-RIGHT SCALE
0
Location of voter
Location of
candidate B
0,30
0,39
0,35
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Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Location of
candidate A
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

Positioning at 0.50 is a weakly dominant strategy
for both candidates, and so both candidates
positioning themselves at 0.50. This is also the
unique Nash equilibrium for this game.
0.50 is the position of the median voter. candidates’
positions converge to the position of the median
voter.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Candidate Position Game: the candidates are
assumed to choose their positions simultaneously
and cannot budge from that choice once it is made.
We will also suppose that voter preferences and
candidates objectives are common knowledge. We
will refer to the resulting game as the Candidate
Position Game.
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 The
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Median Voter Theorem: if there
are two candidates, the two candidates’
political positions can be represented by
their location on a continuous linear
scale, each voter’s preference over these
positions is singe-peaked, the
distribution of voter preferences is
common knowledge, and the election is
decided by majority rule; then the unique
Nash equilibrium strategy in the
Candidate position game is both
candidates to position themselves at the
median voter’s position.
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DISTRIBUTION OF VOTER
PREFERENCES
0.0
0.1
0.2
Number of
voters
36
15
10
0.3 0.4 0.5
8
5
3
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Voter
position
0.6
0.7
0.8
0.9
1.0
1
2
3
5
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PLURALITY RULE AND THE CONDORCET
CANDIDATE

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
A candidate who is preferred by a majority of
voters to any other alternative in a series of
pairwise comparisons is called condorcet
candidate after the Marquis de Condorcet
(French mathematician and philosopher 17431794).
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HYPOTHETICAL VOTER PREFERENCE
Voter
type
95
95
110
b>d>e
d>b>e
e>b>d
Cyclical majority: no one
candidate can muster a majority of
votes against all alternatives,
hence there is no Condorcet
candidate.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
A
B
C
Number of voters
preference
with that type
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Voter Preferences That Produce No
Condorcet Candidate
Number of voters
with that type
preference
A
B
C
1
1
1
x>y>z
y>z>x
z>x>y
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Voter
type
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
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Single transferable vote: voters are given a
slate of candidates and are asked to vote for one.
if no one candidate receives a majority of the
votes cast, a second ballot is taken in which the
candidate receiving the smallest number of votes
from the first ballot dropped from the list of
eligible candidates.
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SHAREHOLDER PREFERENCES
Preference
A
B
C
D
8
7
6
3
z>y>x
x>z>y
y>x>z
y>z>x
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Voter
Number of
Shares/voters
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THE OUTCOME OF THE ELECTION AS
B’S VOTE CHANGES WHEN THE
OTHER VOTERS ARE HONEST
x>z>y (honest)
x>y>z
z>x>y
z>y>x
y>x>z
y>z>x
Winner
Z
Y
Z
Z
Y
Y
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Voter B’s Ballot
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STRATEGIC VOTING WITH PLURALITY
RULE

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Chairman’s paradox: one member of a
committee -the chairman – is given the
power to break ties. the resulting
plurality-rule voting game has a unique
iterated dominant strategy equilibrium
whose outcome is the one considered the
worst by the “powerful” chairman.
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AN EXAMPLE



require no math courses for students majoring in
economics. L (low requirement)
require one term of Univariate calculus. M
(medium requirement)
require two terms of calculus: one of Univariate
calculus and one of multivariate. H (high
requirement)
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Three professors at economics department, Ms.
Ayşe, Ms. Fatma , and Mr. Ali , have been asked
by the economics department to choose the
mathematics requirement for economics major.
There are considering three options.
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VOTER PREFERENCES
preferences
Ms. Ayşe
Mr. Ali
Ms. Fatma
L > M >H
H > L >M
M > H >L
One of the
committee members,
Ms. Ayşe, has been
designated the Chair
of committee.
The chair carries the power to cast a tiebreaking vote. It is more likely that Ms. Ayşe’s
preferred outcome will be chosen. This
institution is confirmed by the naïve model.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
voter
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THE NAÏVE VOTING MODEL
Ms. Ayşe  L
Ms. Fatma  M
Mr. Ali  H
L is chosen by the
committee and
economics majors
will not be required
to take any math
course.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
If all three members of the committee vote
naively, then the outcome of vote is
straightforward and intuitive:
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THE STRATEGIC VOTING MODEL
Strategic voting behavior: the game essentially
reduces to a static game in which the set of
strategies and the set of moves coincide. The
concept we will employ to make predictions about
Strategic voting behavior is Nash equilibrium.
 But, if we can find dominant strategies and
iterated dominant strategies, we will use them as
our predictors. Because of Ms. Ayşe’s voting power,
it is likely that he has dominated strategies that
can be removed from the game.

Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
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

If Ms. Fatma and Mr. Ali vote for the same
candidate, then Ms. Ayşe s vote is irrelevant and
all three of Ms. Ayşe’s strategies are equally good.
If Ms. Fatma and Mr. Ali split their vote, then Ms.
Ayşe’s vote determines the outcome. Hence Ms.
Ayşe has a weakly dominant strategy of voting for
L. we would predict, and so presumably would Ms.
Fatma and Mr. Ali, that he will vote for L.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

Consider Ms. Ayşe’s best response to every possible
pair of votes by Ms. Fatma and Mr. Ali.
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PAYOFF MATRIX ASSUMING MS. AYŞE
VOTES FOR L
Mr.
Ali’s
vote
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Ms. Fatma’s vote
H
M
L
H
(Best, Middle, worst)
(Middle, worst best,)
(Middle, worst, best)
M
(middle, worst, best)
(worst, Best, middle)
(Middle, worst, best)
L
(middle, Worst, best)
(Middle, Worst, best)
(Middle, Worst, best)
Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe
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PAYOFF MATRIX ASSUMING MS. AYŞE
VOTES FOR L AND MS. FATMA DOES NOT
VOTE FOR L
Mr. Ali’s
vote
H
M
H
(Best, Middle, worst)
(Middle, worst best,)
M
(middle, worst, best)
(worst, Best, middle)
L
(middle, Worst, best)
(Middle, Worst, best)
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Ms. Fatma’s vote
Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe
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PAYOFF MATRIX ASSUMING MS. AYŞE
VOTES FOR L, MS. FATMA DOES NOT VOTE
FOR L, AND MR. ALI VOTES FOR H
Mr. Ali’s
vote
H
H
M
(Best, Middle,
worst)
(Middle, worst best,)
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory
Ms. Fatma’s vote
Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe
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
Of course, what drives the result is that Ms. Fatma
does not vote “naively”. she recognizes that if she
were to do so, then she would end up with what
she considers to be the worst outcome. Therefore,
she votes strategically.
Prof. Yeşim Kuştepeli ECO 4413 Game
Theory

H wins, even though the Chair of the committee,
Ms. Ayşe, considers this his worst outcome. The
“power” to break ties turns out to be a bad thing
for the person who is unlucky enough to be saddled
with it. Oddly enough, once the committee is
assigned, there should be a fight among the
members not to be chosen as Chair, even though
the Chair brings with it extra voting “privileges.”
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