Game Theory
Section 3: Social Choice
Agenda
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•
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Motivating questions and modus operandi
Key terms
May’s Theorem
Arrow’s Theorem
The Condorcet winner
– Condorcet Procedure: Voting Scheme
– Cycling
• Single-peaked preferences
• Median Voter Theorem
Motivating Questions
To Consider the Preferences of a Society
• What are reasonable/desirable characteristics of
the social (aggregate) preference?
– What are the reasonable/desirable characteristics of
individual preferences?
• How to aggregate individual preferences?
– Which schemes are possible, and which are impossible,
and what are the real-world implications?
• Voting, democracy, institutional design
• Under what circumstances will it be rational to
reveal information (or preferences) honestly?
– When won’t it be rational to reveal info in this way?
Modus Operandi
(1) encourage honest information-sharing, and
Are there
systems which (2) use that information in a reasonable way?
• First: Define the setting
– Consider a case of perfect information
– Define basic rules for allowable preferences
• Second: Now, consider rules that are still admissible
– Identify new allowable rules with desirable properties
• Third: Decide whether these rules provide incentive
for honest revelation of preferences
Remember: We work with ordinal preferences (including ties)
Key Terms
• Social Welfare Functional (SWF)
– Societal decision rule that creates a social ranking by
aggregating individual rank orders
• “Desireable” properties of the SWF
– Paretian Property: unanimity rules
– Symmetry among Agents: voting can be anonymous
without changing the result
– Neutrality among Alternatives: given 2 options, labels
given to the possible alternatives don’t matter
– Positive Responsiveness: if the social ranking is
indifferent between two alternatives, and one individual
changes preference in favor of one alternative, the
social preference must change to favor that alternative
Key Terms
Additional “Desirable” Properties of a SWF
• Transitivity:
– If x is preferred to y, AND
– If y is preferred to z,
– Then x must be preferred to z
• Independence of Irrelevant Alternatives (IIA)
– Social preference for A vs. B is independent of
individual’s preferences for all alternatives except A vs. B
– Changes in individual preferences that don’t change
preferences for A vs. B do not change social preference for
A vs. B
– The social preference for A vs. B is NOT affected by
changes in individual i’s preference for A vs. C or D vs. E
– IIA is also known as the pairwise independence property
May’s Theorem: Step I
Define the Setting:
Consider the case of complete information, and
define basic rules for allowable preferences
Given
• N individuals
• Revealing preferences over two outcomes
– i’s preferences:pi = (x>y) or (x>y) or (x~y)
– SWF(p1, p2 , …, pN) = (x>y) or (x>y) or (x~y)
• Allowing a social preference that complies with
– Symmetry among agents (anonymity)
– Neutrality among alternatives
– Positive responsiveness
May’s Theorem: Step II
Consider the rules that are still admissible
• Symmetry among agents
– The # of people with given preferences matters, not
who the people are
• Neutrality among alternatives (2 alternatives)
– None of the alternatives is favored by the SWF
– The support for alternative A required to get A >s B is
identical to the support for B necessary for B >s A
• Positive responsiveness
– Departures from a tie must break a tie
So, what is still admissible?
May’s Theorem: Step III
May’s
Thm.
A social welfare functional F(p1, p2, … , pN)
corresponds to majority voting iff it satisfies:
(1) symmetry among agents;
(2) neutrality among alternatives; and
(3) positive responsiveness
Now, decide whether these rules provide incentive
for honest revelation of preferences
In majority rule, you should vote for what you want.
Either (1) no effect (2) break a tie (3) tie
But a vote for what you don’t want can hurt you.
Everything thus far has been in a two-alternative world
Arrow’s Theorem:
Enter the Third Alternative
Define the Setting:
Consider the case of complete information, and
define basic rules for admissible preferences
Given
• N individuals
• Revealing preferences over three outcomes
– i’s preferences:pi = (x>y~z), for example
– SWF(p1, p2 , …, pN) = (x~z<y) for example
• Allowing a social preference that complies with
somewhat weaker properties than last time
– Transitivity
– Unanimity
– Independence of Irrelevant Alternatives
Arrow’s Theorem: Step II
Consider the rules that are still allowable
• Transitivity
– Social preferences must satisfy A > B, B > C, A > C
• Unanimity rules
• IIA (pairwise independence):
– Only my A vs. B is relevant to society’s A vs. B
– For its A vs. B, society only needs my A vs. B
Condorcet Preferences
Alternative Preferences
X
Y
Z
1
B
B
C
A
2
A
C
B
Is social order in
Condorcet world
going to match the
social order in the
Alternative world?
B
3
C
A
A
B vs. C? A vs. B? A vs. C?
X
Y
Z
1
A
B
C
2
B
C
3
C
A
Arrow’s Theorem: Step II
Consider the rules that are still admissible
• Transitivity
– Social preferences must satisfy A > B, B > C, A > C
• Unanimity rules
• IIA (pairwise independence):
– Only my A vs. B is relevant to society’s A vs. B
– For its A vs. B, society only needs my A vs. B
Condorcet Preferences
X
Y
Z
1
A
B
C
2
B
C
3
C
A
Alternative Preferences
X
Y
Z
1
B
B
C
A
2
A
C
B
B
3
C
A
A
Borda Count allowed?
Majority rule allowed?
Arbitrary SWF allowed?
Can we reject Condorcet
preferences?
Is a dictatorship the only
allowable SWF?
Arrow’s Theorem: Step III
Arrow’s
Thm.
With 3 or more alternatives, the only social
welfare functional F(α1, α2, … , αN) satisfying
(1) unanimity;
(2) transitivity; and
(3) independence of irrelevant alternatives
and no restriction on the domain of preferences
is a dictatorship, i.e., a social choice matching
the individual preferences of a particular person
(the dictator) regardless of others’ preferences
Now, decide whether these rules provide incentive
for honest revelation of preferences
If you’re the dictator, you should vote your preference.
If you’re not, you don’t matter…Else, 1, 2, and/or 3 don’t apply.
If it’s IIA, be strategic! Consider the “irrelevant” alternatives!
The Condorcet Winner
• So, three or more alternatives spells trouble
• Run-off rules are a coping mechanism
– Because with 2 alternatives, you can use majority rule
and get honest voting & stable outcomes
– But Arrow’s Theorem throws a wrench in it
• Can’t always be honest voting in elimination stages
– One voting scheme: Condorcet procedure, which asks
• Can one option win a majority against each of the others?
• If so, that option is known as the Condorcet Winner
Condorcet Winner Example
Category =
No. of voters =
I
II
III
IV
V
VI
18
12
10
9
4
2
a
b
c
d
e
e
d
e
b
c
b
c
e
d
e
e
d
d
c
c
d
b
c
b
b
a
a
a
a
a
Most
Preferred
Least
Preferred
What constitutes a majority?
Is ‘a’ the Condorcet Winner? Is ‘e’ the Condorcet Winner?
Do > than ½ voters actually have the CW as their 1st choice?
This example was taken from Shepsle, Bonchek “Analyzing Politics”, which took it from Malkevitch 1990.
Cycling: The “Money Pump”
No Condorcet Winner
Party 1
Most
Preferred
Least
Preferred
Party 2
Foreign Aid Welfare
Party 3
Military
Military
Foreign Aid Welfare
Welfare
Military
Foreign Aid
Raiffa called these “money pump” preferences; if we’re at any one policy, we could
conceivably hold a new vote to change to any other policy, and just go round and round.
Single-Peaked Preferences (S.P.P)
A restriction on preferences that they are
monotonically decreasing away from an ideal point
Obviates Condorcet’s cycling problem
Illustration: 3 options & strict preferences WLOG
Can be
A>B>C
C>B>A C>A>B
ruled out
B>C>A
B>A>C A>C>B
by S.P.P.
Assume a society: {(A,B,C), (A,B,C), (B,C,A), (C,B,A), (B,A,C)}
Utility
Is S.P.P violated?
there a Condorcet winner?
Why?
Policy
A
B
C
Is
Median Voter Theorem
With single-peaked preferences and an odd
number of voters, the median of the voters’
ideal points is the Condorcet winner.
a1
m
a2
Who is the median voter? Why is the MV the CW?