VOTING GAMES
Assoc. Prof. Yeşim Kuştepeli
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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
Assoc. Prof. Yeşim Kuştepeli
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•
•
•
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.
Assoc. Prof. Yeşim Kuştepeli
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• The acceptance or rejection by a
corporation’s broad of directors of merger
offer tendered by competing firm.
• The 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.
Assoc. Prof. Yeşim Kuştepeli
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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.
• If 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.
Assoc. Prof. Yeşim Kuştepeli
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Two different models of voter:
•
•
Naive voter: he always casts his
ballot honestly, even when it is
not in his best interest to do so.
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.
Assoc. Prof. Yeşim Kuştepeli
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Choosing Among Voting Mechanisms
• Plurality voting: ballots are
prepared which voters choose
only one of the candidate and
vote for him. 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.
Assoc. Prof. Yeşim Kuştepeli
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• 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.
Assoc. Prof. Yeşim Kuştepeli
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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.
Assoc. Prof. Yeşim Kuştepeli
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Arrow’s impossibility theorem:
• 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 nondictatorial.
Assoc. Prof. Yeşim Kuştepeli
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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.
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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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.
Assoc. Prof. Yeşim Kuştepeli
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Majority Rule with Two Choices
• 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.
• In an election between two
candidates decided by a majority
rule, all voters will rationally
abstain or vote honestly for
their most preferred candidate.
Assoc. Prof. Yeşim Kuştepeli
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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
Assoc. Prof. Yeşim Kuştepeli
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Candidate and voter locations
on left-to-right scale
Location of
candidate A
0
Location of voter
Location of
candidate B
0,30
0,39
1
0,35
Assoc. Prof. Yeşim Kuştepeli
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• 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.
Assoc. Prof. Yeşim Kuştepeli
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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.
Assoc. Prof. Yeşim Kuştepeli
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• The 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.
Assoc. Prof. Yeşim Kuştepeli
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Distribution of Voter
preferences
Voter
position
0.0
0.1
0.2
Number of
voters
36
15
10
0.3 0.4 0.5
8
5
3
0.6
0.7
0.8
0.9
1.0
1
2
3
5
13
Assoc. Prof. Yeşim Kuştepeli
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Plurality Rule and the
Condorcet Candidate
• 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 1743-1794).
Assoc. Prof. Yeşim Kuştepeli
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Hypothetical Voter Preference
Voter
type
A
B
C
Number of voters
preference
with that 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.
Assoc. Prof. Yeşim Kuştepeli
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Voter Preferences That Produce No
Condorcet Candidate
Voter
type
Number of voters
with that type
preference
A
B
C
1
1
1
x>y>z
y>z>x
z>x>y
Assoc. Prof. Yeşim Kuştepeli
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• 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.
Assoc. Prof. Yeşim Kuştepeli
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Shareholder Preferences
Voter
Number of
Shares/voters
Preference
A
B
C
D
8
7
6
3
z>y>x
x>z>y
y>x>z
y>z>x
Assoc. Prof. Yeşim Kuştepeli
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The Outcome of the Election As
B’s Vote Changes When the
Other Voters Are Honest
Voter B’s Ballot
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
Assoc. Prof. Yeşim Kuştepeli
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Strategic Voting with Plurality Rule
• 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.
Assoc. Prof. Yeşim Kuştepeli
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an example
• 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.
 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)
Assoc. Prof. Yeşim Kuştepeli
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Voter Preferences
voter
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.
Assoc. Prof. Yeşim Kuştepeli
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The Naïve Voting Model
If all three members of the committee vote
naively, then the outcome of vote is
straightforward and intuitive:
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.
Assoc. Prof. Yeşim Kuştepeli
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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.
Assoc. Prof. Yeşim Kuştepeli
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• Consider Ms. Ayşe’s best response to
every possible pair of votes by Ms. Fatma
and Mr. Ali.
• 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.
Assoc. Prof. Yeşim Kuştepeli
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Payoff Matrix assuming Ms. Ayşe
votes for L
Ms. Fatma’s vote
Mr.
Ali’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
Assoc. Prof. Yeşim Kuştepeli
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Payoff Matrix assuming Ms. Ayşe
votes for L and Ms. Fatma does
not vote for L
Ms. Fatma’s vote
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)
Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe
Assoc. Prof. Yeşim Kuştepeli
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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
Ms. Fatma’s vote
Mr. Ali’s
vote
H
H
M
(Best, Middle,
worst)
(Middle, worst best,)
Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe
Assoc. Prof. Yeşim Kuştepeli
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• 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.”
• 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.
Assoc. Prof. Yeşim Kuştepeli
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