GOV 2005: Game Theory
Section 3: Social Choice
Alexis Diamond
adiamond@
Agenda
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
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?
To Consider the Preferences of a Society
Modus Operandi
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)
(1) encourage honest information-sharing, and
(2) use that information in a reasonable way?
Are there systems which
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
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
Additional “Desirable” Properties of a SWF
May’s Theorem: Step I
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
Define the Setting:
Consider the case of complete information, and
define basic rules for allowable preferences
May’s Theorem: Step II
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
Consider the rules that are still admissible
So, what is still admissible?
May’s Theorem: Step III
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
Now, decide whether these rules provide incentive for honest revelation of preferences
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
May’s Thm.
Arrow’s Theorem:
Enter the Third Alternative
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
Define the Setting:
Consider the case of complete information, and
define basic rules for admissible preferences
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
Arrow’s Theorem: Step II
Consider the rules that are still allowable
Condorcet Preferences
Alternative Preferences
Is social order in Condorcet world going to match the social order in the Alternative world?
B vs. C? A vs. B? A vs. C?
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
Borda Count allowed?
Majority rule allowed?
Arbitrary SWF allowed? Can we reject Condorcet preferences?
Is a dictatorship the only allowable SWF?
Condorcet Preferences
Alternative 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!
Arrow’s Theorem: Step III
Now, decide whether these rules provide incentive for honest revelation of preferences
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, ., a social choice matching the individual preferences of a particular person (the dictator) regardless of others’ preferences
Arrow’s Thm.
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
2
4
9
10
12
18
a
a
a
a
a
b
b
c
b
d
c
c
d
d
e
e
d
e
c
b
c
b
e
d
e
e
d
c
b
a
VI
V
IV
III
II
I
Category =
No. of voters =
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”
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.
Foreign Aid
Military
Welfare
Welfare
Foreign Aid
Military
Military
Welfare
Foreign Aid
Party 3
Party 2
Party 1
No Condorcet Winner
Most Preferred
Least Preferred
Single-Peaked Preferences ()
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
A > B > C C > B > A C > A > B
B > C > A B > A > C A > C > B
Can be ruled out by .
Assume a society: {(A,B,C), (A,B,C), (B,C,A), (C,B,A), (B,A,C)}
Is violated? Is there a Condorcet winner? Why?
A
B
C
Utility
Policy
Median Voter Theorem
With single-peaked preferences and an odd number of voters, the median of the voters’ ideal points is the Condorcet winner.
Who is the median voter? Why is the MV the CW?
m
a1
a2
