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Economics 113
UCSD
Winter 2010
Prof. Ross Starr, Mr. Troy Kravitz
February 23, 25, 2010
1
Lecture Notes, February 23, 25, 2010
Social Choice Theory, Arrow Possibility Theorem
BergsonSamuelson social welfare function W(u
1
(x
1
), u
2
(x
2
), …, u
#H
(x
#H
))
with
0
i
W
u
∂
>
∂
all i
.
Let the allocation x*
∈
R
N(#H)
+
maximize W subject to the usual technology
constraints.
Then x* is a Pareto efficient allocation.
Further, suppose x**
∈
R
N(#H)
+
is a Pareto efficient allocation.
Then
there is a specification of W so that x** maximizes W subject to constraint.
Paradox of Voting
(Condorcet)
Cyclic majority:
Voter preferences:
1
2
3
A
B
C
B
C
A
C
A
B
Majority
votes A > B,
B >
C.
Transitivity requires A > C but majority
votes
C > A.
Conclusion:
Majority voting on pairwise alternatives
by rational (transitive)
agents can give rise to intransitive group preferences.
Is this an anomaly?
Or systemic.
Arrow Possibility Theorem says systemic.
Arrow (Im) Possibility Theorem:
We'll follow Sen's treatment.
For simplicity we'll deal in strong orderings
(strict preference) only
X
=
Space of alternative choices
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View Full Document Economics 113
UCSD
Winter 2010
Prof. Ross Starr, Mr. Troy Kravitz
February 23, 25, 2010
2
Π =
Space of transitive strict orderings on X
H
=
Set of voters, numbered #H
Π
#H
= #H  fold Cartesian product of
Π ,
space of preference profiles
f:
Π
#H
→ Π ,
f is an Arrow Social Welfare Function.
P
i
represents the preference ordering of typical household i.
{P
i
} represents
a preference profile, {P
i
}
∈
Π
#H
.
P represents the resulting group (social)
ordering.
" x P
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This note was uploaded on 09/30/2011 for the course ECON 113 taught by Professor Starr,r during the Fall '08 term at UCSD.
 Fall '08
 Starr,R
 Economics

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