Social threshold aggregations

Fuad T. Aleskerov,

Vyacheslav V. Chistyakov,

Valery A. Kalyagin

Higher School of Economics

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Examples

• Apartments

• Three students – whom we hire

• Refereeing process in journals

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- alternatives ,

, - agents,

- set of ordered grades

with .

An evaluation procedure

assigns to and a grade

, i.e.,

where for each is the set of all -dimentional vectors with components from

X 2, 1,2,...,X N n 2n 1,2,...,M m 1 2 ... m

3m:E X N M

i N( , )ix E x i M

x X

n

1ˆ ( , ) ,..., nnX x x E x x x M

1,..., :nn iM x x x M i N

M

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We assume , so

the number of grades in the vector

Note that

Let

ˆ nX X M

1ˆ iff ,..., with nn ix X x x x x M x M

( )jv x j 1,... :nx x x

( ) : .j iv x i N x j

0 ( ) for all and andjv x n x X j M

1 21

( ) ( ) ( ) ... ( ) for all .m

j mj

v x v x v x v x n x X

01

( ) ( ) if 1 and ( ) 0k

k jj

V x v x k m V x

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Social decision function

Social decision function on

satisfying

(a) iff is socially (strictly) more

preferable than , and

(b) iff and are socially

indifferent

X: X R

( ) ( )x y

x( ) ( )x y y

xy

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Axioms( .1) (Pairwise Compensation): if , and

( ) ( ) for all 1 1, then ( ) ( ).

( .2) (Pareto Domination): if , and ,

then ( ) ( ).

( .3) (Noncompensatory Threshold and Contraction):

f

j j

A x y X

v x v y j m x y

A x y X x y

x y

A

1 1

2 1

or each natural number 3 the following

condition holds:

( .3. ) if , , ( ) ( ) for all 1

(if , this condition is omitted),

( ) 1 ( ) ( ),

( ) and ( )

j j

m k m k m k

m k m k

k m

A k x y X v x v y j m k

k m

v x v y n V y

V x n V y

( ) , then ( ) ( )mv y n x y

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The binary relation on

is said to be the lexicographic ordering if, given

and from , we have:

in iff there exists an such that

for all (with no condition if ) and

Construction of social ordering – threshold rule

compare vectors and

k

kR

kR

kR

1,..., ku u u 1,..., kv v vu v 1 i k j ju v

1 1j i 1i i iu v

x y

1 1

1 1 2 2

if ( ) ( )

if ( ) ( ) compare ( ) and ( )

...

v y v x xRy

v y v x v x v y

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Theorem:

A social decision function on X satisfies the axioms Pairwise Compensation, Pareto Domination, Noncompensatory Threshold and Contraction iff its range is the set of binary relations on X generated by the threshold rule.

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Let be a set of three different

alternatives, a set of

voters and the set of grades

(i.e., ).

, ,X x y z

1,...,13N 13n 1,2,3M

3m

3 voters 4 voters 6 voters

Simple Majority Rule Bord

a Threshold

ran

k

3

x x x y zy

y

2

1

y x

z

z

x

yz

z y z x

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The dual threshold aggregation

1 1

Compare ( ) and ( ).

If ( ) ( ), then

If ( ) ( ),

then compare ( ) ( ), etc.

m m

m m

m m

m m

v x v y

v x v y xRy

v x v y

v x v y

Manipulability of Threshold Rule

• Computational Experiments

• Multiple Choice Case

• Several Indices of Manipulability

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Indices

- better off- worse off- nothing changed

)1!()!(1

)!(1

1

mnmI

n

ijni

mj

n

ij

ij

ij0

1!0 mijijij

Applications

• Development of Civil Society in Russia

• Performance of Regional Administrations in Implementation of Administrative Reform

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References• Aleskerov, F.T., Yakuba, V.I., 2003. A method for aggregation of rankings of special form. Abstracts of

the 2nd International Conference on Control Problems, IPU RAN, Moscow, Russia.

• Aleskerov, F.T., Yakuba, V.I., 2007. A method for threshold aggregation of three-grade rankings. Doklady Mathematics 75, 322--324.

• Aleskerov, F., Chistyakov V., Kaliyagin V. The threshold aggregation, Economic Letters, 107, 2010, 261-262

• Aleskerov, F., Yakuba, V., Yuzbashev, D., 2007. A `threshold aggregation' of three-graded rankings. Mathematical Social Sciences 53, 106--110.

• Aleskerov, F.T., Yuzbashev, D.A., Yakuba, V.I., 2007. Threshold aggregation of three-graded rankings. Automation and Remote Control 1, 147--152.

• Chistyakov, V.V., Kalyagin, V.A., 2008. A model of noncompensatory aggregation with an arbitrary collection of grades. Doklady Mathematics 78, 617--620.

• Chistyakov V.V., Kalyagin V.A. 2009. An axiomatic model of noncompensatory aggregation. Working paper WP7/2009/01. State University -- Higher School of Economics, Moscow, 2009, 1-76 (in Russian).

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Thank you

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