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![Page 1: A Fourier-Theoretic Perspective on the Condorcet Paradox and Arrow ’ s Theorem. By Gil Kalai, Institute of Mathematics, Hebrew University Presented by:](https://reader036.fdocuments.us/reader036/viewer/2022062309/56649dba5503460f94aab922/html5/thumbnails/1.jpg)
A Fourier-Theoretic Perspective on the Condorcet Paradox and Arrow’s Theorem.
By Gil Kalai, Institute of Mathematics, Hebrew UniversityPresented by: Ilan Nehama
![Page 2: A Fourier-Theoretic Perspective on the Condorcet Paradox and Arrow ’ s Theorem. By Gil Kalai, Institute of Mathematics, Hebrew University Presented by:](https://reader036.fdocuments.us/reader036/viewer/2022062309/56649dba5503460f94aab922/html5/thumbnails/2.jpg)
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Basic notations n players m alternatives Each player have a preference over the
alternatives Ri a >i b := Player i prefers a over b Linear order I.e.
Full and asymmetric: a, b : (a>b) XOR (a<b) Transitive
The vector of all preferences (R1, R2,…,Rn) is called a profile.
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Basic notations
The preferences are aggregated to the society preference. a > b := The society prefers a over b
Full and asymmetric: a, b : (a>b) XOR (a<b)
We do not require it to be transitive
The aggregation mechanism is called a social choice function
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4
Basic notations
Probability space For a social choice function F and a
property φ Pr[φ(RN)]:=#{Profiles
RN:φ(F)]}/#{Profiles}
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Social choice function’s properties Social choice function is a function
between profiles to relations. The social choice function is called
rational on a specific profile RN if f(RN) is an order.
The social choice function is called rational if it is rational on every profile.
An important property of a social choice function is Pr[F is non-rational].
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Social choice function’s properties
IIA–Independence of Irrelevant Alternatives. for any two alternatives a>b depends
only on the players preferences between a and b.
{i: a>ib} determines whether a>b
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Social choice function’s properties
Balanced-For any two alternatives x,y : Pr[x>y]=Pr[y>x]
Neutral-The function is invariant under permutations of the alternatives
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Social choice function’s properties Dictator
Profile-For a profile each player i that the social aggregation over the profile agrees with his opinion is called a dictator for that profile.
General-A player that is a dictator on a ‘big portion’ of the profiles is called a dictator.
Dictatorship-A social choice function that have one dictator player is called a dictatorship.
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Main results There exists an absolute constant K
s.t.: For every >0 and for any neutral social
choice function If the probability that the function is non-
rational on a random profile < Then there exists a dictator such that for every
pair of alternatives the probability that the social choice differs from the dictator’s choice < K
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Main results For the majority function the
probability of getting an order as result (avoiding the Condorcet Paradox) approaches (as n approaches to infinity) to G
0.9092<G<0.9192
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Agenda Defining the mathematical base –
The Discrete Cube The probability of irrational social
choice for three alternatives The probability of the Condorcet
paradox A Fourier-theoretic proof of Arrow’s
theorem
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Discrete Cube
Xn={0,1}n=P([n])=[2n] Uniform probability f,g:X->R
2
[ ]
2 0.5
[ ]
, : ( ) ( )2
, ( ( ))2
,
n
n
s n
s n
f g f S g S
f f f f f S
f g f g
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An orthonormal basis: us(T)=(-1)|ST|
[ ]
, : ( ) ( )2n
s n
f g f S g S
| | 2
[ ]
| | | |
[ ]
| | | | | ( { })| | ( { })|
[ ]\{ }
| | | | | | | |
[ ]\{ }
, (( 1) ) 12
. \
, ( 1) ( 1)2
( 1) ( 1) ( 1) ( 1)2
( 1) ( 1) ( ( 1) )( 1)2
2
n
n
n
n
n
S RS S
R n
S R T RS T
R n
S R T R S R x T R x
R n x
S R T R S R T R
R n x
u u
S T x S T
u u
[ ]\{ }
0 0R n x
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us(T)=(-1)|ST| form an orthonormal basis
2
[ ]
[ ]
2 2
[ ]
( )
( ) ,
, ( ) ( )
( , ) ( )
SS n
S
S n
S n
f f S u
f S f u
f g f S g S
f f f f S
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For f a boolean function f:X->{0,1}. F is a characteristic function
for some AX. A2(2[n])
P[A]:=|A|/2n
2
2
2
[ ]
0
, 2 ( ) 2 | | [ ]
: ( ) ( 1) 1
( ) , [ ]
n n
S n
f f f f S A P A
R u R
f f u P A
[ ]
, : ( ) ( )2n
s n
f g f S g S
Boolean functions over X
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Agenda Defining the mathematical base –
The Discrete Cube The probability of irrational social
choice for three alternatives The probability of the Condorcet
paradox A Fourier-theoretic proof of Arrow’s
theorem
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17
Domain definition
F is a social choice function < = F(<1, <2,…,<n) F is not necessarily rational Three alternatives – {a,b,c}
F is IIA {i: a>ib} determines whether a>b
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Each player preference can be described by 3 boolean variables xi=1 <=> a>ib yi=1 <=> b>ic zi=1 <=> c>ia
Domain definition
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19
F can be described by three boolean functions of 3n variables
f(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> a>b g(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> b>c h(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> c>a
Domain definition
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F is IIA {i: a>ib} determines whether
a>b
f,g,h are actually functions of n variables
f(x)=f(x1,..,xn) g(y)=g(y1,..,yn) h(z)=h(z1,..,zn)
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21
Define
1
2
3
p =P[{x| f(x)=1}] = f ( )
p =P[{y| g(y)=1}] = g( )
p =P[{z| h(z)=1}] = h( )
F will be called balanced when p1=p2=p3=½
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22
The domain of F is: Ψ = {all (x,y,z) that correspond to
rational profiles}= {(x,y,z) | i (xi,yi,zi) {(0,0,0),
(1,1,1)}
P[Ψ] = (6/8)n
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W=W(F)=W(f,g,h) is defined to be The probability of obtaining a non-rational
outcome (from rational profile) f(x)g(y)h(z)+(1-f(x))(1-g(y))(1-h(z))=1
<=> F(x,y,z) is non-rational
( , , )
( ( ) ( ) ( ) (1 ( ))(1 ( ))(1 ( ))
| |x y z
f x g y h z f x g y h z
W
W- Probability of a non-rational outcome
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Theorem 3.1
| | 1
[ ]
1
2
3
, : ( ) ( )( 1/ 3)
p =P[{x| f(x)=1}] = f ( )
p =P[{y| g(y)=1}] = g( )
p =P[{z| h(z)=1}] = h( )
S
s n
f g f S g S
1 2 3 1 2 3(1 )(1 )(1 ) ( , , , ) / 3W p p p p p p f g g h h g
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25
Proof of Thm. 3.1 A,B are
boolean functions on 3n variables Subsets of 23n
A=ΧΨ
B=f(x)g(y)h(z)
3
3
( , , ) 2
2 ( ) ( ) ( ) , ( ) ( )n
n
x y z S
f x g y h z A B A S B S
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26
32
, ( ) ( )
A=
B=f(x)g(y)h(z)
nS
A B A S B S
3
3
3 | |
2
| || | | |3
( , , )
( )
2 : ( , , )
( ) , 2 ( )( 1)
| | | | | | | |
2 ( ) ( ) ( )( 1) ( 1) ( 1)
( ) ( ) ( )
n
y yx x z z
x y z
nx y z x k y k z k
n S RS
R
x x y y z z
S RS R S Rnx y z
R R R
x y z
B S
S S S S S S x S y S z
B S B u B R
S R S R S R S R
f R g R h R
f S g S h S
Proof of Thm. 3.1
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27
32
, ( ) ( )
A=
B=f(x)g(y)h(z)
nS
A B A S B S
3
1
3 | |
2
1
| |3
1 1( )
( )
: { , , }
1 S is a rational profile( ) {
0
( , , )
0 ' ' '( ', ', ') {
1
( ) , 2 ( )( 1)
| | | |
2 ( ) ( 1)
n
i i
i i
i i i i
n
i i ii
n S RS
R
n
i ii
n nS Rn
i ii iR
A S
F F x y z
A SOtherwise
A S S S
x y zA x y z
otherwise
A S A u A R
S R S R
A R
1
1
( )
n
n
i ii
A S
Proof of Thm. 3.1
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28
32
, ( ) ( )
A=
B=f(x)g(y)h(z)
nS
A B A S B S
1 2 31 2
1
| || 3|x y z
1
( )
( ) ( )
3 / 4
By direct computation: ( ) { 1/ 4 | | 2
0
( ) ( 1/ 4) (3 / 4) : belongs to two or none of S ,S ,and S( )
0
n
i ii
i
i i i
nn S S SS S S
i ii
A S
A S A S
S
A S S
otherwise
A S i iA S
Otherwise
Proof of Thm. 3.1
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29
| | | |x y z
1
x y z x y z
3
: belongs to two or none of S ,S ,and S( 1/ 4) (3 / 4)( ) ( ) {
0
We'll denote by Q the triplets (S ,S ,S ) for which (S ,S ,S ) 0
( ) ( ) ( ) ( )
2 ( ) (
x y z x y zS S S n S S Sn
i ii
x y z
n
i iA S A S
Otherwise
A
B S f S g S h S
f x g y
3( , , ) 2
| |
) ( ) , ( ) ( )
(1 )( ) 2 (1 )( ) ( )
2 ( ) 2 ( ) ( )
2 ( 1) ( )
( ){1 ( )
nx y z S
nS
R
n nS S
R R
n R S
R
h z A B A S B S
f S f R u R
u R f R u R
f S
f S S
Sf S
Proof of Thm. 3.1
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30
1
( , , )
3
( , , ) ( , , )
| | | |3 3
( , , )
| | ( ( ) ( ) ( ) (1 ( ))(1 ( ))(1 ( ))
6 2 ( ) ( ) ( ) (1 ( ))(1 ( ))(1 ( ))
2 ( ) ( ) ( )( 1/ 4) (3 / 4) 2 (1 )( )(1x y z x y z
x y z
x y z
n n
x y z x y z
S S S n S S Sn nx y z x
S S S Q
W f x g y h z f x g y h z
W f x g y h z f x g y h z
f S g S h S f S
| | | |
( , , )
| | | |3
( , , )
3
)( )(1 )( )( 1/ 4) (3 / 4)
2 ( 1/ 4) (3 / 4) [ ( ) ( ) ( ) (1 )( )(1 )( )(1 )( )]
2 (3 / 4) [ ( ) ( ) ( ) (1 )( )(1 )( )(1
x y z x y z
x y z
x y z x y z
x y z
S S S n S S Sy z
S S S Q
S S S n S S Snx y z x y z
S S S Q
n n
g S h S
f S g S h S f S g S h S
f g h f g h
3 | | | | | | | | | | | |
| |1 2 3 1 2 3
)( )] 2 ( 1/ 4) (3 / 4) ( ) ( ) ( 1/ 4) (3 / 4) ( ) ( ) ( 1/ 4) (3/ 4) ( ) ( )
[ (1 )(1 )(1 )] ( 1/ 3) [ ( ) ( ) ( ) ( ) ( ) ( )
x y x z y
yz x
n S n S S n S S n S
S S S S S S Sz S SSS S
S
f S g S f S h S h S g S
W p p p p p p f S g S g S h S f S h S
3 3 3 3
1 2 3 1 2 3
1 2 3
2 | | 1
]
, , ,(1 )(1 )(1 )
3
Note that if f=g=h then we get ( )
(1 ) , (1 ) ( )( 1/ 3)
S
S
S
f g g h h fp p p p p p
p p p p
W p p f f p p f S
Proof of Thm. 3.1
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31
Agenda Defining the mathematical base –
The Discrete Cube The probability of irrational social
choice for three alternatives The probability of the Condorcet
paradox A Fourier-theoretic prosof of
Arrow’s theorem
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32
The Condorcet Paradox
There are cases that the majority voting system (which seems natural) yields irrational results.
Three voters, three alternatives 1) a>1b>1c 2) b>2c>2a 3) c>3a>3b
Result: a>b>c>a
Marie Jean Antoine Nicolas Caritat, marquis de Condorcet
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33
Computing the probability of the Condorcet Paradox
3 alternatives n=2m+1 voters f=g=h are the majority function
G(n,3):=The probability of a rational outcome.
G(3):=limn→∞G(n,3)
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34
Computing the probability of the Condorcet Paradox It is known that
3 3 1(3) arcsin 0.91226
4 2 3G
We will prove 1 1 2 8 1
0.9092 1 ( ) (3) 1 ( ) 0.91924 2 9 9 2
G
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35
2 1 2
1
2 1 2
(2 1) |{ } |
[2 1] [2 1]\{ } [2 1]\{ }| | 1 | { }| 1 | | 1
2 2
1
: ({ })
2( 2 ) (2 1)
2 ({ }) ( 1) ( 1) 1
2 2 2
m
mk
mm
m k S
S m S m k S m kS m S k m S m
m m
i m i m
d f k
md m
m
f k
m m m
i i m
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36
2 1 2
1
2 1 2
({ })
2( 2 ) (2 1)
m
mk
m
d f k
mm
m
m
2 2 1 2 2 2 1 22
m+1
2 1 22 1 2m2
2 24
4
d is a decreasing sequence
2 2 (2 2)!( 2 ) (2 3) ( 2 ) (2 3)1d ( 1)!
(2 )!2d ( 2 ) (2 1)( 2 ) (2 1)( !)
(2 1) (2 2) (2 3) (2 1)(2 3)2
( 1) (2 1) (2 2)
m m
mm
m mm mm m
mm mmmm
m m m m m
m m m
2
2 2
2 1 2 2 1 2m 2
2 22 1 2
2
22 1 2
4 8 31
4 8 4
1lim
2
2 (2 )!d ( 2 ) (2 1) ( 2 ) (2 1)
( !)
Striling's approximation : ( !) 2
(2 ) 2 2( 2 ) (2 1)
( 2 )
2 2 1( 2 ) (2 1)
4
mm
m m
k k
m mm
m m
mm
m
m m
m m
d
m mm m
m m
k k e k
m e mm
m e m
mm
mm
1
2
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1 2 n 1 2 n
|R S| |R S| |R S|+1
R [n] R [n]\{x}
|R S|
R [n]\{x}
f is the majority function
f(1-x ,1-x ,...,1-x )=1-f(x ,x ,...,x )
f (S)=0 S , |S| is even
x S
f (S)= f(R)(-1) f(R)(-1) f(R {x})(-1)
(-1) [f(R)-f(R {x})]=
|R S|
R [n]\{x}|R|=m
| | 1i
0(|R S|=i)
| | 1
2i | | 1
0(|R S|=i)
(-1) [0-1]=
| | 1 2 1 | |(-1)
| | 1 2 1 | | | | 1 2 1 | |(-1) (-1)
| | 1 (| | 1 )
|
S
i
S
S i
i
S m S
i m i
S m S S m S
i m i S i m S i
S
| | 1
2i 1
0(|R S|=i)
| 1 2 1 | | | | 1 2 1 | |(-1) (-1) 0
)
S
i
i
m S S m S
i m i i m i
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38
3 32
| | 1
2 2| | 1 | | 1
|S| is odd |S| is odd
2 | | 1
|S| is odd
2
m
|S| =1
1 ( ,3) (1 ) ( )( 1/ 3)
¼- ( )( 1/ 3) ¼- ( )(1/ 3)
1 11 ( ) (3)
4 2
1 ( ,3) ¼- ( )(1/ 3)
1¼- ( ) ¼-d ¼
2
S
S
S S
S S
S
S
S
G n W p p f S
f S f S
G
G n f S
f S
1 1 2 8 1Proving 1 ( ) (3) 1 ( )
4 2 9 9 2G
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39
2 2 2 22
| | 3 |S|=1 | | 2
2
2| | 1
|S| is odd
2 2| | 1
| | 3 | | 3
2 8 1(3) 1 ( ) 0.9192
9 9 2
( ) ( ) ( ) ( )
0 ¼-
1 ( ,3) ¼- ( )(1/ 3)
¼- ( )(1/ 3) ¼- 1/ 9 ( )
¼- 1/ 9(¼- ) 2 / 9 8 / 9
1
S S
m m
S
S
Sm m
S S
m m m
G
f S f f f S f S
p p d d
G n f S
d f S d f S
d d d
G
( ,3) 2 / 9 8 / 9
2 8 11 ( )
9 9 2
m
m
n d
G n
1 1 2 8 1Proving 1 ( ) (3) 1 ( )
4 2 9 9 2G
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40
Agenda Defining the mathematical base –
The Discrete Cube The probability of irrational social
choice for three alternatives The probability of the Condorcet
paradox A Fourier-theoretic proof of Arrow’s
theorem
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41
Arrow’s Theorem At least three alternatives Let f be a social choice function which
is: unanimity respecting / Pareto optimal independent of irrelevant alternatives
Then f is a dictatorship.
Kenneth Arrow
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42
Lemma 6.1: For f a boolean function:If <f,uS>=0 S: |S|>1
Then exactly one of the following holds f is constant
f=1 or f=0 f depends on one variable (xi)
f(x1, x2,…,x1)=xi or f(x1, x2,…,x1)=1-xi
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43
<f,uS>=0 S: |S|>1f is not constant=> f depends on one variable
S S
2
S [n]
2 2 2
1 S [n] S
p [ ( ) 1]. With no loss of generality p ½
(otherwise we will prove for (1-f) , <1-f,u >= -<f,u >)
p= f ( ) f ( )
Assume that f is not constant and hence p [½,1)
f ({ }) f ( ) f ( )n
i
P f x
S
i S S
22
[n]|S|>1
2 2
1 1
f ( )
|f ({ }) | |f ({ }) | !i f ({ }) 0n n
i i
p p
i p p i p p i
Proof of Lemma 6.1
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44
i
{ }
2{ }
S [n] 1 1
2
1
1 f ({ }) 0Define x as: x {
0
01( ) {
11
1 ( ) f ( ) ( ) f ({ }) ( ) |f ({ }) | 1
½ |f ({ }) | ! f ({ }) 0, | f ({ }) | ½
ii
i
n n
s ii i
n
i
i
Otherwise
xu x
x
f x S u x p i u x p i p p p
p p i p p i i
<f,uS>=0 S: |S|>1f is not constant=> f depends on one variable
Proof of Lemma 6.1
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45
|{ } |{ }
[ ] [ ] [ ]|
i i
i i
i
f ({ }) , 2 ( )( 1) 2 ( )(1) ( )( 1)
½Pr[x =0 f(x)=1] - ½Pr[x =1 f(x)=1]
| Pr[x =0 f(x)=1] - Pr[x =1 f(x)=1] | = 1
Then one of the two cases:
x =0Pr[
n k S ni
S n S n S ni S i S
i f u f S f S f S
i
i i
x =1 ]=0 , Pr[ ]=1 ( )
f(x)=1 f(x)=1
x =0 x =1 Pr[ ]=1 , Pr[ ]=0 ( )
f(x)=1 f(x)=1
i
i
f x x
f x x
<f,uS>=0 S: |S|>1f is not constant=> f depends on one variable
Proof of Lemma 6.1
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46
Proof of Arrow’s theorem (assuming neutrality)
From lemma 6.1 one can prove Arrow’s theorem for neutral social choice function
Instead we will use a generalization of this lemma to prove a generalization of Arrow’s theorem.
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47
1 2 3
1 2 3 1 2 3
| | 1
[ ]
2| | 1
[ ]
F is neutral
f=g=h p p p ½
(1 )(1 )(1 ) ( , , , ) / 3
, : ( ) ( )( 1/ 3)
0 ¼ , ¼ ( )( 1/ 3)
¼ ( )
S
s n
S
s n
p
W p p p p p p f g g h h g
f g f S g S
W f f f S
f S
| | 1
2 2
| | 1
2 2
2 | | 1 2
, ( 1/ 3) ( )
¼ ( ) * ( 1/ 3) ( )
( ) ¼ , ( 1/ 3) ( ) ¼
¼ ¼ *¼ 0
SS S
s s
SS S
s s
SS S
s s
u f S u
f S u f S u
f S u p f S u p
Proof of Arrow’s theorem using lemma 6.1
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48
| | 1
i
: ( ) ( 1/ 3) ( )
( ) 0 S:|S|>1
( ) 0 S:|S|>1
f is not constant, f(0,...,0)=0. Hence i f(x)=x
SS S
s s
f S u f S u
f S
f S
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49
2
2
| | 1
2
1 2 n i 1
Theorem 7.1: There exists constants K and K' s.t.
For every f a boolean function and every , if
f =p
( )
Then one of the following cases holds
p<K' or p>1-K'
i f(x , x ,..., x ) x <K or f(x , x
S
f S
2
2 n i,..., x ) (1 x ) <K
Notice that for =0 we get Lemma 6.1
Proofs of this thorem are the issue of
"Boolean Functions whose Fourier Transform is Concentrated on
the First Two Levels" / Friedgut, Kalai and
Naor.
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50
Generalized Arrow’s Theorem Theorem 7.2: For every ε>0 and for
every neutral social choice function on three alternatives:
If the probability the social choice function if non-rational≤ε
Then there is a dictator such that the probability that the social choice differs from the dictator’s choice is smaller than Kε
Notice that for ε=0 we get Arrow’s theorem.
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51
1 2 3
1 2 3 1 2 3
| | 1
[ ]
2| | 1
[ ]
2
F is neutral
f=g=h p p p ½
(1 )(1 )(1 ) ( , , , ) / 3
, : ( ) ( )( 1/ 3)
¼ , ¼ ( )( 1/ 3)
¼ (
S
S n
S
S n
p
W p p p p p p f g g h h g
f g f S g S
W f f f S
f S
2| | 1
[ ] | | 1
22 | | 1
| | 1
2 2 2| | 1 3 1
| | 1 | | 1 | | 1
) ( )[( 1/ 3) 1]
¼ (½-½ ) ( )[( 1/ 3) 1]
( )[1 ( 1/ 3) ] ( )[1 (1/ 3) ] 8 / 9 ( )
S
S n S
S
S
S
S S S
f S
f S
f S f S f S
Proof of theorem 7.2 using theorem 7.1
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52
2
| | 1
2 2
1 2 n i 1 2 n i
2
1 2 n i
( ) 9 / 8
1/ 2
i f(x , x ,..., x ) x <K9 / 8 or f(x , x ,..., x ) (1 x ) <9 / 8K
(If we assume pareto optimality f(x , x ,..., x ) x <9 / 8K )
S
f S
p
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53
Corollary
For fm a balanced social choice family on m alternatives For every ε>0, as m tends to infinity,
If for every pair of alternatives there is no dictator with probability (1- ε)
Then, the probability for a rational outcome tends to zero
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54
The End
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55
Proposition 5.2
If the social choice function is neutral then the probability of a rational outcome is at least 3/4
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56
Proof of Proposition 5.2
1 2 3
1 2 3 1 2 3
| | 1
[ ]
2
[ ]
F is neutral
f=g=h p p p ½ f(1-x)=f(x)
(1 )(1 )(1 ) ( , , , ) / 3
, : ( ) ( )( 1/ 3)
¼ , ¼ ( )( 1/ 3)
S
S n
S n
p
W p p p p p p f g g h h g
f g f S g S
W f f f S
C
| | 1
|R S| |R S| C |R S|
R [n] R [n]
|R S| |S|-| S|
R [n]
|R S| | S| |R S|
R [n] R [n]
f (S)=0 S : |S| is even
x S
f (S)= f(R)(-1) ½ f(R)(-1) f(R )(-1)
½ f(R)(-1) (1 f(R))(-1)
½ f(R)(-1) (1 f(R))(-1) ½ (-1)
½ (-
S
R
R
|R S| |R {x} S| |R S| |R S|
R [n]\{x} R [n]\{x}
1) +(-1) ½ (-1) -(-1) 0
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57
Proof of Proposition 5.2
2 | | 1
[ ]
2 | | 1
[ ] |S| is odd
2 | | 1
[ ] |S| is odd
¼ ( )( 1/ 3)
¼ ( )( 1/ 3)
¼ ( )1/ 3 ¼
S
S n
S
S n
S
S n
W f S
f S
f S