104/20/23

Aggregation of Binary Evaluations

without Manipulations

Dvir Falik Elad Dokow

2 04/20/23

“Doctrinal paradox”

• Majority rule is not consistent!

The defendant

killed the victim

The defendant was sane at the time

The defendant is guilty

Judge 1100

Judge 2010

Judge 3111

Majority110

q qp p

3 04/20/23

“Doctrinal paradox”

Assume that for solving this paradox the society decide only on p and q.

The defendant

killed the victim

The defendant was sane at the time

The defendant is guilty

Judge 1100

Judge 2010

Judge 3111

Majority111

q qp p

4 04/20/23

“Doctrinal paradox”

Judge 1 can declare 0 on p and manipulate the result of the third column .

The defendant

killed the victim

The defendant was sane at the time

The defendant is guilty

Judge 1000

Judge 2010

Judge 3111

Majority010

q qp p

Linear classification

5 04/20/23

)}1,0,1,0(),0,1,0,1{(\}1,0{ 4X

)1,1,1,0(

6 04/20/23

“Condorcet paradox” (1785)

• Majority rule is not consistent!

IS a>bIS b>cIS c>a

Judge 1110

Judge 2101

Judge 3011

Majority111

• Arrow Theorem: There is no function which is IIA paretian and not dictatorial.

a>b>cc>a>bb>c>a

)}1,1,1(),0,0,0{(\}1,0{ 3X

Example :

04/20/237

)}1,1,1(),0,0,0{(\}1,0{ 3X

100

001

011

101

110

010

My opinion

Social aggregator

Facility location

Example :

04/20/238

)}1,1,1(),0,0,0{(\}1,0{ 3X

100

001

011

101

110

010

My opinion

Social aggregator

Full Manipulation

Example :

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)}1,1,1(),0,0,0{(\}1,0{ 3X

100

001

011

101

110

010

My opinion

Social aggregator

Full Manipulation Partial Manipulation

Example :

04/20/2310

)}1,1,1(),0,0,0{(\}1,0{ 3X

100

001

011

101

110

010

My opinion

Social aggregator

Full Manipulation Partial ManipulationHamming manipulation

04/20/23

Gibbard Satterhwaite theorem:

Social choice function: AARf n )(:

)()(: ARARf n Social welfare function:

GS theorem: For any , there is no Social choice function which is onto A, and not manipulatable.

3|| A

11

Example:GS theorem

04/20/2312

)}1,1,1(),0,0,0{(\}1,0{ 3X

100

001

011

101

110

010

My opinion: c>a>bSocial aggregator

a

b

c

1304/20/23

The model A finite, non-empty set of issues K={1,…,k} A vector is an evaluation. The evaluations in are called feasible,

the others are infeasible. In our example, (1,1,0) is feasible ; but (1,1,1) is

infeasible.

kmxxx }1,0{),...,( 1

kX }1,0{

1404/20/23

• A society is a finite set .

• A profile of feasible evaluations is an matrix all of whose rows lie in X.

• An aggregator for N over X is a mapping .

mnnij

N Xxx )(

},...,1{ nN

XXf n :

1504/20/23

Different definitions of Manipulation

Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that:

ix),( iiN xxx

Xy

ijj

iijj

N xxyfandxxf ),()(

partialPartial

1604/20/23

Different definitions of Manipulation

Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that:

ix),( ii xx

Xy

ijj

iijj

N xxyfandxxf ),()(ijj

iijj

N xxyfxxfj ),()(,

fullFull

And:

We denote by and say that c is between a and b if . We denote by the set .

},|{],[ iiii bcoracicba ],[ bac

},{\],[ baba),( ba

1704/20/23

Different definitions of Manipulation

Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that:

ix),( ii xx

Xy

))(,[),( Nii xfxxyf

fullFull

1804/20/23

Different definitions of Manipulation

•Any other definition of manipulation should be between the partial and the full manipulation.•If is not partial manipulable then f is not full manipulable .

XXf n :

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Hamming Manipulation

Hamming manipulation: An aggregator f is Hamming manipulatable if there exists a judge i, an opinion , an evaluation , and a profile such that:

ixNxXy

)),,(()),(( iiiN xyxfdxxfd

•Hamming distance: i

ii yxyxd ||),(

Theorem (Nehiring and Puppe, 2002): Social aggregator f is not partial manipulatable

if and only if f is IIA and monotonic.Theorem (Nehiring and Puppe, 2002):Every Social aggregator which is IIA, paretian

and monotonic is dictatorial if and only if X is Totally Blocked.

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Partial Manipulation

Corollary (Nehiring and Puppe, 2002):Every Social aggregator which is not partial

manipulable and paretian is dictatorial if and only if X is Totally Blocked.

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Partial Manipulation

22 04/20/23

IIA• An aggregator is independent

of irrelevant alternatives (IIA) if for every and any two profiles and satisfying for all , we have

Ny

)()( Nj

Nj

Nj

Nj yfxfyx

XXf n :Jj

Nx ij

ij yx

Ni )()( Nj

Nj yfxf

123

Judge 1

Judge 2

Judge 3

aggregator

23 04/20/23

Paretian• An aggregator is Paretian if

we have whenever the profile is such that for all .

xxf N )(XXf n :

xxi Ni

Nx

123

Judge 11

Judge 21

Judge 31

aggregator 1

24 04/20/23

Monotonic• An aggregator is IIA and Monotonic

if for every coordinate j, if then for every we have .

XXf n :

123

Judge 11

Judge 20

Judge 30

aggregator 1

],[ cxy Nj

cxf Njj )(

cyf j )(

25 04/20/23

Monotonic• An aggregator is IIA and Monotonic

if for every coordinate j, if then for every we have .

XXf n :

123

Judge 11

Judge 21

Judge 30

aggregator 1

],[ cxy Nj

cxf Njj )(

cyf j )(

2604/20/23

Dictatorial• An aggregator is

dictatorial if there exists an individual such that for every profile.

dNNN xxfXxNd )(

dN xxf )(

XXf n :Nd

Almost dictator function:

Fact: For any set is not Hamming/strong manipulatable.

04/20/2327

Almost dictator

1

211

12

11

211

12

11 ),...,,(),...,,(

)(~

xelse

xxxxXxxxxxD kkkkN

)(~

,1,0 Nk xDX

Question: what are the conditions on such that there exists an anonymous, Hamming\strong non-manipulatable social function?

kX 1,0

Let be the majority function (|N| is odd) on each column.

Let be an IIA and Monotonic function. Let be a function with the following

property: there isn’t any between and . Let be a function with the following

property: for every , .The sets of those function will be denoted by

Easy to notice that

04/20/2328

Majority Nearest Neighbor

Xy

kNXmaj }1,0{:

Xsnn k }1,0{:)(xsnnx

Xhnn k }1,0{:Xy ),()),(( xydxxhnnd

kNXm }1,0{:

hnnX

snnX

mX FFF ,,

snnX

hnnX FF

})(,|{)( yxgFgyxHNN hnnX

04/20/2329

Nearest Neighbor

Proof:

First column

Second column

Third column

Judge I111

Judge 2

Judge 3

m110

]),,([)(,.1 iiN xxymxmXy

/0

/0

/0/0

Proposition: For any set is not full manipulatable. Furthermore, if is annonymous, then is annonymous.

mX

hnnX

k FmFgX ,,1,0

mgfXXf N ,:m f

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Nearest Neighbor

Proof :

ix ),( ixym )( Nxm

XxmcaseThe N )(:.2

XxymcaseThe i ),(:.3

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Nearest Neighbor

Proof:

ix ),( ixym )( Nxm

XxymxmcaseThe iN ),(),(:.4

Proposition: For any set

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Hamming Nearest Neighbor

mX

hnnX

k FmFgX ,,1,0

XXmg N :

1. If then judge i can’t manipulate by choosing instead of .

0)],(),([ Xxymxm iN y ix

2. If then judge i can’t manipulate by choosing instead of .

)),(())(( iN xymMTxmMT y ix

04/20/2333

Hamming Nearest NeighborProof of part 1: Let

, Xxymxmt iN )],(),([

)),(,()),(,( iii xymtdxymxd )),(,()),(,()),(())(,( iiNNi xymtdxymtdtxmdxmxd )),(())(,( txmdxmxd NNi

))(),(())(,())(,( NNNiNi xmgxmdxmxdxmgxd

)),(,()),(),,(()),(,( iiiiii xymgxdxymxymgdxymxd

Conclusions:

04/20/2334

Hamming Nearest Neighbor

1. An Hamming Nearest Neighbor function is not manipulatable on .

}111,000/{}1,0{ 3X

2. Manipulation can’t be too ‘far’.

cX

3504/20/23

MIPE-minimally infeasiblepartial evaluation

• Let , a vector with entries for issues in J only is a J-evaluation.

• A MIPE is a J-evaluationfor some which is infeasible, but such that every restriction of x to a proper subset of J is feasible.

jJiixx }1,0{)(

KJ

Jiixx )(KJ

},|{ MipeaisxxyXyA JJJ

xJ

},|{)( JxJJ AyMipeaisxxyMT

Proposition: For any set

04/20/2336

Hamming Nearest Neighbor

mX

hnnX

k FmFgX ,,1,0

XXmg N :2. If then judge i can’t manipulate by choosing instead of .

)),(())(( iN xymMTxmMT y ix

),(1)( iT

N xymxm Proof: Let

0))((1)(,.2 TSxmHNNxmKS NS

N )),((1),())((1)(.3 i

SiN

SN xymHNNxymxmHNNxm

0))((.1 TSxmMTx NS

)),(,())(,(.4 iiNi xymgxdxmgxd

Proposition: For any set

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Hamming Nearest Neighbor

mX

hnnX

k FmFgX ,,1,0

XXmg N :

1. If then judge i can’t manipulate by choosing instead of .

0)],(),([ Xxymxm iN y ix

2. If then judge i can’t manipulate by choosing instead of .

)),(())(( iN xymMTxmMT y ix

What happens in intermediate cases?

38 04/20/23

Examplepqs(P or q)s

0000

0100

1000

1100

0010

1011

0111

1111

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Examplep2

q2

s4

(p or q)s3

0000

0100

1000

1100

0010

1011

0111

1111

Weighted columns:

My opinion:

)( Nxm 1 0 1 0

6

8

4

6

2

3

7

5

),( ixym 1 1 1 0

8

6

6

4

4

5

5

3

5

2

1100

0010

1011

Maj:1010

1100

0010

1111

Maj:1110

Nx

),( ixy

04/20/23

Conjectures:Let: }'|:{ anonymicandblemanipulatatisnfXXf n

X

What are the conditions on X such that What are the conditions on X such that 0X

Conjecture: For every set such that and there exists a weighting of the columns, such that for every

ck XstXxX ,,,}1,0{0),( Xst )()( sMTtMT

hnnXFg

))(,())(,(.2

),(),(.1

tgxdsgxd

txdsxd

Conjecture:0}g,m|{0 X

hnnX

mXX FFmg