Introduction Proof Conclusion
A Difficulty in the Concept of Social Welfare
KENNETH J. ARROW
Oren Roth - Ben-Gurion University, Beer-Sheva, Israel
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
1
Introduction
The problem
Formalization
2
Proof
Theorem Proof
3
Conclusion
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We paased couple of weeks in our seminar, let say we want to
take all the lecturs we saw already...
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We paased couple of weeks in our seminar, let say we want to
take all the lecturs we saw already...
and give some order on them (prefernces)
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
So we will ask all the students in the seminar (and Paz as
well) to give us an order on all of the lectures.
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
So we will ask all the students in the seminar (and Paz as
well) to give us an order on all of the lectures.
We want some mechanism to take all those orders and
combine it to one aggregate order.
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
Unanimity
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
Unanimity
Independence of irrelevant alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
Unanimity
Independence of irrelevant alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
Unanimity
Independence of irrelevant alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
We will want to keep some basic rules:
Unanimity
Independence of irrelevant alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
So what Arrow THM shows as that the only why to keep this
simple rules is by having...
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
So what Arrow THM shows as that the only why to keep this
simple rules is by having...
Dictator!
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Definition of the problem
So what Arrow THM shows as that the only why to keep this
simple rules is by having...
Dictator!
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formalization
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formal definitions
Assumption: n voters give their complete ranking on set A of
alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formal definitions
Assumption: n voters give their complete ranking on set A of
alternatives
L the set of linear orders on A (permutation).
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formal definitions
Assumption: n voters give their complete ranking on set A of
alternatives
L the set of linear orders on A (permutation).
Each voter i provides <i in L
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formal definitions
Assumption: n voters give their complete ranking on set A of
alternatives
L the set of linear orders on A (permutation).
Each voter i provides <i in L
Input to the aggregator/voting rule is (<1 , <2 , . . . , <n )
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Formal definitions
Assumption: n voters give their complete ranking on set A of
alternatives
L the set of linear orders on A (permutation).
Each voter i provides <i in L
Input to the aggregator/voting rule is (<1 , <2 , . . . , <n )
Our Goal: find a function W : Ln → L, (called a social welfare
function) that aggregates voters preference into a common
order
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
if ∀i a <i b ⇐⇒ a <0i b therefore
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
if ∀i a <i b ⇐⇒ a <0i b therefore
a < b ⇐⇒ a0 < b
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
if ∀i a <i b ⇐⇒ a <0i b therefore
a < b ⇐⇒ a0 < b
Then it is dictatorial:
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
if ∀i a <i b ⇐⇒ a <0i b therefore
a < b ⇐⇒ a0 < b
Then it is dictatorial:
there exists a voter i where
W (<1 , <2 , . . . , <n ) =<i
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
The problem Formalization
Arrows Impossibility Theorem
Every Social Welfare Function W over a set A of at least 3
candidates, If it satisfies:
Unanimity
W (<, <, . . . , <) =<
for all < in L
Independence of irrelevant alternatives
Let (<1 , <2 , . . . , <n ) and (<01 , <02 , . . . , <0n ) s.t.
W (<1 , <2 , . . . , <n ) =< and W (<01 , <02 , . . . , <0n ) = W 0
if ∀i a <i b ⇐⇒ a <0i b therefore
a < b ⇐⇒ a0 < b
Then it is dictatorial:
there exists a voter i where
W (<1 , <2 , . . . , <n ) =<i
for all <1 , <2 , . . . , <n in L
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
Theorem Proof
Proof
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
Conclusion and open problems
Questions?
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare
Introduction Proof Conclusion
Conclusion and open problems
Thanks
KENNETH J. ARROW
A Difficulty in the Concept of Social Welfare