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Social threshold aggregations
Fuad T. Aleskerov,
Vyacheslav V. Chistyakov,
Valery A. Kalyagin
Higher School of Economics
2
Examples
• Apartments
• Three students – whom we hire
• Refereeing process in journals
3
- 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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