Outline
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Collective choice
Arrow’s impossibility theorem
Direct democracyy
 Majority rule and cycles
 Single peaked preferences the median voter
 When does the median voter result in efficient provision
Representative democracy
Econ 2230: Public Economics
Lecture 10: Collective Choice
Collective Choice
Collective Choice

Examine a society with a finite set of individuals N and a finite set of
allocations denoted by A. Let Ri denote individual preference
ordering and RN = {Ri }Ni=1 preference profile for all N

Examine a society with a finite set of individuals N and a finite set of
allocations denoted by A. Let Ri denote individual preference
ordering and RN = {Ri }Ni=1 preference profile for all N

Collective choice question: Can we find a way to aggregate
individual preferences into a social choice function?
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Collective choice question: Can we find a way to aggregate
individual preferences into a social choice function?
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How do we turn {Ri }Ni=1 into a collective ordering: f: {Ri }Ni=1 → R
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How do we turn {Ri }Ni=1 into a collective ordering: f: {Ri }Ni=1 → R
X P1 y P1 z
x P2 z P2 y
xPzPy
y P3 z P3 x
{Ri }Ni=1
R
1
Collective Choice
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Arrow examined a series of different voting mechanisms
They all seemed to have weaknesses
Caused him to investigate
g
all voting
g mechanisms
Started by proposing criteria for the voting rule or social choice
function
What criteria must f satisfy?
What criteria do we want f to satisfy?

Pareto Principle (P):
 for any a,b ∈ A and {Ri }Ni=1 if a Pi b ∀i ∈ N → a P b

Independence
p
of Irrelevant alternatives ((I))
 ranking of two states does not depend on potential in other states
Transitivity (T)
 a P b and b P c → a P c
Unrestricted Domain (U):
 all orderings are permissible
Non-dictatorship (ND)
 there is no single individual j such that whenever a Pj b → a P c
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Arrow’s Impossibility Theorem
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Theorem: When N is finite and |A|≥3 then any f that satisfies (P), (U),
(I) and (T) will be dictatorial
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Implication:
 Any voting rule will be less than ideal
 Any decision-making mechanism will fail on one of the five criteria
described
Arrow’s Impossibility Theorem

Theorem: When N is finite and |A|≥3 then any f that satisfies (P), (U),
(I) and (T) will be dictatorial

Proof:
As we will need to show that f is dictatorial it will be useful to define
what it means to be decisive
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Definition: We say that a coalition S ⊂ N is decisive for a,b ∈ A if
[a P b] whenever [a Pi b] i ∈ S and [b Pj a] j ∈ N\S
2
Properties of decisive set J
Properties of decisive set J
(1) N ∈ J and ∅ ∉J
(P)
(2) If S ∈ J and T ∈ J then S∩T ∈ J
 S \ S∩T: c P b P a
S decisive over c P b
 T \ S∩T: b P a P c
T decisive over a P c
 S∩T: a P c P b
 N \S∪T: b P c P a
 How do we know S∩T ∈ J?
 Transitivity → a P b → S∩T decisive over a P b thus S∩T ∈ J
(3) For any S ⊂ N either S ∈ J or N \ S ∈ J
 i ∈ S:
a P i c Pi b
 i ∈ N \ S : b Pi a P i c
 Either
a P b or b P a
Proof of Arrow’s Impossibility theorem
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Plan for proof. (A) Show if a group is decisive over two alternatives then
decisive over all. (B) show that there is just one member of such a
decisive group
(A) if a group is decisive over {a
{a,b}
b} then decisive over all
If S is decisive over {a, b} then S is decisive over {a,c}, where a≠c
 Consider the profiles:
 i ∈ S: a Pi b Pi c
 i ∈ N \ S: b Pi c Pi a [need to disagree of a,b and a,c]
 S∈J→aPb
 (P) → b P c
 (T) → a P c
c, thus S is decisive over {a
{a,c}
c}
 (I) this holds independent of the ordering of {c,b} i.e would also hold if
i ∈ N \ S: c Pi b Pi a
If S is decisive of {a,b} then also decisive over {b,c} where b≠c
If S is decisive over {a,b} then also decisive over {x,y} ∈ A, ∀x,y
(4) If S ∈ J and S ⊂ T → T ∈ J
 Suppose not.
 N \ T ∈ J then by (2) (N \ T) ∩ S ∈ J
[why problematic?]
 but (N \ T) ∩ S = ∅ and we know from (1) that ∅ ∉J [so how do we
know T ∈ J?]
 Thus (N \ T) ∉J thus by (3) above we know that T ∈ J
Proof of Arrow’s Impossibility theorem
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(B) show that there is just one member of such a decisive group

If S ∈ J and includes more than one member, then there is a S’ ⊂ S,
S’≠S such that S’ ∈ J
 Take any h ∈ S
 If S\ {h} ∈ J done
 If not then {S\ {h}} ∉ J and N \ {S\ {h}} ∈ J by (3)
 Note that N \ {S\ {h}} = {N \ S } ∪ h ∈ J
 Then S ∩ [{N \ S } ∪ h ] = h ∈ J
 i.e., a subset S’ ∈ J
There is an h ∈ N s.t. h ∈ J (the dictator!)
 By (1) N ∈ J
 True by iteratively applying the step above (N finite)
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Implication
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No voting rule is ideal.
Problem is that we want it to always work. Want social rankings for
all preference orderings independent of how likely or unlikely they
may be
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E.g. majority rule – we know it must fail one of the conditions for f
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Definition: simple majority rule:
 a wins by majority over b (a M b) if a gets more than (N+1)/2
votes
Majority rule
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MR first analyzed by Condorcet (results later rediscovered by Arrow)
Examples (one dimensional issue)
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Example 2:
 Individual 1: a P1 b P1 c
 Individual 2: b P2 c P2 a
 Individual 3: c P3 a P3 b
 What will happen?
 a M b, b M c and c M a
 Condorcet paradox: voting cycle. Thus not transitive. Arrow’s
impossibility theorem makes clear that this is not an anomaly
 Implication: Outcome of the majority rule is sensitive to order. The
agenda setter has room for manipulation
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Consider a one dimensional issue: all propositions to be decided can be
linearly ordered, left to right. All voters agree on the left to right ordering,
but disagree on their choices.
Example 1:
 Individual 1: a P1 b P1 c
 Individual 2: b P2 c P2 a
 Individual 3: c P3 b P3 a
Assume :
Direct democracy. The citizens themselves vote on the policy
Open agenda. Citizens vote over pairs of policy alternatives, such that
the winning policy in one round is posed against a new alternative in the
next round and the set of alternatives includes all feasible policies.
Sincere voting – i.e., individuals vote “truthfully” rather than strategically.
What will happen?
 b M a and b M c → b wins
 b is the Condorcet winner (b beats all other alternatives in a pairwise
vote)
Restricting domain: Single peaked preferences
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Definitions:
 ai* is i’s ideal point iff Ui(ai*) > Ui(ai) ∀ai ≠ai*
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i’s preferences are single peaked iff
Ui(b) > Ui(c) ↔ ||b- ai*|| < ||c - ai*||
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Median voter: rank all individuals according to their ideal points ai*.
Suppose that N is odd. Then, the median voter is the individual
who has exactly (N-1) /2 ideal points to his left and (N-1) /2 ideal
points to his right
right. Denote this individual by m and his ideal point
by am*
When does majority rule work? When we restrict domain.
4
Black’s Median Voter Theorem
Black’s Median Voter Theorem
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If all voters have single peaked preferences over a one dimensional
issue then the median (m) voter’s ideal point am* cannot lose under
majority rule
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If all voters have single peaked preferences over a one dimensional
issue then the median (m) voter’s ideal point am* cannot lose under
majority rule
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Pf.
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Pf.
b
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a m*
c
b
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If all voters have single peaked preferences over a one dimensional
issue then the median (m) voter’s ideal point am* cannot lose under
majority rule
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Pf.
N/2
N/2
b
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c
a m*
Sincere voting?
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c
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a m*
How do we know that am* cannot lose under majority rule?
Black’s Median Voter Theorem

N/2
N/2
am* M b
am* M c
Thus we have a Condorcet winner when voting over a one
dimensional issue and preferences are single peaked
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What implications do single peaked preferences have for sincere
voting?
Example 1 revisited:
 Individual 1: a P1 b P1 c
 Individual 2: b P2 c P2 a
 Individual 3: c P3 b P3 a
Suppose open agenda (vote over pairs of policy alternatives).
Incentive to vote truthfully?
Under sincere voting b M a then b M c → b wins
D
Does
3h
have an iincentive
ti tto vote
t sincerely?
i
l ?aMb
b, th
then c M a
Not an incentive to vote sincerely under open agenda.
However when preferences are single peaked then there exists a
social choice rule that is strategy proof.
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When are preferences likely to be single
peaked?
Sincere voting?
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When preferences are single peaked then there exists a social
choice rule that is strategy proof. It can be shown that in this case
strategy proofness implies that the social choice rule must be an
“
“augmented
t d median
di voter”
t ” rule,
l which
hi h essentially
ti ll selects
l t th
the median
di
from a list of the ideal points of all individuals
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Expenditures
E.g. vote on size of military
x
G*
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Does majority rule give rise to efficient provision?
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Bowen (QJE 1943)
Let
 xi – private good (price 1)
 G – public good (price q)
Assume N individuals split cost: individual cost = qG/N
Max Ui(wi – qG/N, G)
Ideal point?
 Gi* where MRSi = q/N
Single peaked?
 Preferences are single peaked with ideal point Gi*
Suppose identical preferences
 Gi* increases with wealth
 Individual with median wealth (i=m) wins
 Provision of public good MRSM = q/N
 Is it efficient?
U
U
G
G*
G
Gm*
G
When are single
g p
peaked p
preferences less likely?
y
 Expenditure with private alternative
 Public school with private school alternative
 Preferences in regards to expenditure may be:
High P Low P medium
Does majority rule give rise to efficient provision?

Provision of public good MRSm = q/N
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Efficiency:
PAS
∑i MRSi = q
Does majority
j y rule result in efficient p
provision?
i.e. is q/N = MRSm = ? ∑i MRSi / N
Provision is only efficient if distribution of MRS is symmetric i.e.,
median MRS = mean MRS
Example 1: xi + ai f (G)
 ai must be symmetrically distributed
Example 2: Ui (xi , G) = U(xi , G)
∀I
 Income must be symmetrically distributed
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What if mean income > median income?
Inefficiently low provision
6
Example
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U.S. presidential election of 2000.
Florida: six million voters, plurality rule to determine the winner.
Plurality
y rule,, according
g to which the winner is the candidate who is
more voters’ favorite candidate (i.e., the candidate more voters rank
first) than any other. (also the method used to elect Senators and
Representatives in the U.S. )
Bush ahead of Gore with less than 600 votes
Ralph Nader with 100,000 voters, while irrelevant in terms of chance
of winning, his presence determined the outcome of the election
(I) is there to rule out spoilers
Would the outcome have been different under open agenda majority
rule?
Which voting rule is best?
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Given that no voting rule satisfies the five axioms all the time, which
rule satisfies them most often?
Maskin: Take any voting rule that differs from majority rule, and
suppose that it works well for a particular class of rankings. Then,
majority rule must also work well for that class. Furthermore, there
must be some other class of rankings for which majority rule works
well and the voting method we started with does not. In other words,
majority rule dominates every other voter rule: whenever another
voting rule works well, majority rule must work well too, and there will
be cases where majority rule works well and the other voting rule
does not
not.
Maskin “Is Majority Rule the Best Election Method?”
Example
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U.S. presidential election of 2000.
Should we have expected majority rule to result in a voting cycle?
Were p
preferences single
g p
peaked?
Bush> Nader > Gore ?
Nader>Bush> Gore ?
Unlikely preferences – thus MR would have elected a Condorcet
winner
Direct vs. representative democracy

Direct democracy:
 Legislative referenda: state legislatures can send out pieces of
state law for a popular vote by citizens
 Popular referenda: in some states citizens can collect enough
petition signatures to place on the ballot a question of whether to
accept or reject a piece of state legislation.

Representative democracy:
 Individuals rarely vote directly on the provision of public goods.
Elect politicians rather than policies
 Anthony Down (1957) extend the median voter theorem to
representative democracy
 Suppose two candidates and that voters vote for candidate closest
to ideal point. The candidate closest to the median voter wins the
election
7
Downsian Model of Political Competition
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1
1.
2.
3.
An odd number of voters i=1,..n.
A set of alternatives A (could be finite or infinite)
Every
y voter has a utilityy function ui((x),
), x∈ A. All voters are assumed
rational, i.e., maximize utility. No voter is indifferent between any two
alternatives
Two parties, A and B. Parties have no policy preferences. Each gets
a payoff R>0 from holding office. Aim to maximize the chance of
winning.
Sequence of moves:
Both parties simultaneously choose policies xA, xB∈ A
Voters simultaneously vote for A or B
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Assume single peaked preferences and median voter with ideal point
am*
N/2
N/2
xA
a m*
The winning party implements its policy xi chosen at stage 1
Downsian Model of Political Competition

Downsian Model of Political Competition
Assume single peaked preferences and median voter with ideal point
am*
Downsian Model of Political Competition

N/2
N/2
xAx
B
Assume single peaked preferences and median voter with ideal point
am*
N/2
N/2
xAx
a m*
B


a m*
Thus, the equilibrium outcome is that both parties set xA= xB=am*,
and wins with probability 1/2.
Both candidates adopt the same platforms, and both candidates
receive essentially the same number of votes.
8
Empirical Evidence

Do elected officials respond to the preferences of voters?
 Stratmann, Thomas, (2000) “Congressional Voting over Legislative
Careers: Shifting Positions and Changing Constraints,” The
American Political Science Review Vol. 94, No. 3, pp. 665-676
 LeVeaux, C. and Garand, J. C. (2003), Race-Based Redistricting,
Core Constituencies, and Legislative Responsiveness to
Constituency Change. Social Science Quarterly, 84: 32–51.
 Levitt, S, (1996), “How Do Senators Vote? Disentangling the Role
of Voter Preferences, Party Affiliation, and Senator Ideology,” The
American Economic Review Vol. 86, No. 3 (Jun., 1996), pp. 425441
 Washington, E. (2008), “Female Socialization: How Daughters
Affect Their Legislator Fathers' Voting on Women's Issues,” The
American Economic Review Volume 98, Number 1, pp. 311332(22)
Empirical Evidence
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Do elected officials respond to the preferences of voters?
 Miller, Grant “Women's Suffrage, Political Responsiveness, and Child
Survival in American History,” Quarterly Journal of Economics, August
2008 Vol.
2008,
Vol 123
123, No.
No 3
3, Pages 1287-1327
1287 1327
 Husted and Kenny, “The Effect of the Expansion of the Voting
Franchise on the Size of Government,” The Journal of Political
Economy, Vol. 105, No. 1 (Feb., 1997), pp. 54-82
 Lee, David, and Enrico Moretti, and Matthew J. Butler (2004). “Do
Voters Affect or Elect Policies? Evidence from the U.S. House,”
Quarterly Journal of Economics, 119(3), 807-859.
Do voters respond to the policy of the elected politician?
 Mullainathan,
Mullainathan S.
S and E.
E Washington (2009)
(2009). “Sticking
Sticking with Your Vote:
Cognitive Dissonance and Political Attitudes”, American Economic
Journal: Applied Economics, 1(1), 86-111.
 Wolfers, Justin (2007). “Are Voters Rational? Evidence from
Gubernatorial Elections”, unpublished working paper.
9