Social threshold aggregations Fuad T. Aleskerov, Vyacheslav V. Chistyakov, Valery A. Kalyagin Higher School of Economics
Jan 17, 2016
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
4
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
5
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
6
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
7
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
8
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.
9
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
10
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
11
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
15
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).
16
Thank you
17