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In-Depth Information
Ta b l e 2 .
The annexer power in the game
G
=[
8
,
9
,
9
,
5
,
7
,
3
,
9; 29], the altered game
G
=
[
11
,
9
,
9
,
5
,
7
,
9; 29], and the factor of increment (decrement) for the three indices
Power Index
G G
Factor
Shapley-Shubik 0
.
1786 0
.
2167 1
.
21
Banzhaf 0
.
1774 0
.
2167 1
.
22
Deegan-Packel 0
.
1711 0
.
1591 0
.
93
4.2
Manipulation via Merging
Example 3.
Merging Advantageous.
Let
G
=[4
,
2
,
1
,
1
,
8
,
7
,
4
;17]
be a WVG. The assimilated agents are shown in bold. In
the original game, the Deegan-Packel indices of these agents are,
γ
2
(
G
)=0
.
0926
,
γ
6
(
G
)=0
.
1889
,
and
γ
7
(
G
)=0
.
1704
. Their cummulative power is
0
.
4519
. In the new
game,
G
=[
13
,
4
,
1
,
1
,
8;17]
, the Deegan-Packel index of the bloc is
γ
1
(
G
)=0
.
5000
,
a factor of increase of
1
.
11
.
Ta b l e 3 .
The cummulative power of the assimilated agents in the original game
G
=
[4
,
2
,
1
,
1
,
8
,
7
,
4
; 17], the power of the bloc in the altered game
G
=[
13
,
4
,
1
,
1
,
8; 17],and
the factor of increment for the three indices
Power Index
G G
Factor
Shapley-Shubik 0
.
4881 0
.
6667 1
.
37
Banzhaf 0
.
4851 0
.
6000 1
.
24
Deegan-Packel 0
.
4519 0
.
5000 1
.
11
Example 4.
Merging Disadvantageous.
Let
G
=[5
,
8
,
3
,
4
,
9
,
1
,
5
;30]
be a WVG. The assimilated agents are shown in
bold. In the original game, the Deegan-Packel indices of these agents are,
γ
2
(
G
)=
0
.
1833
,
γ
7
(
G
)=0
.
1417
. Their cummulative power is
0
.
5083
.In
the new game,
G
=[
22
,
5
,
3
,
4
,
1;30]
, the Deegan-Packel index of the bloc is
γ
5
(
G
)=0
.
1333
,
and
γ
1
(
G
)=
0
.
3056
, a factor of decrease of
0
.
60
.
Ta b l e 4 .
The cumulative power of the strategic agents in the original game
G
=
[5
,
8
,
3
,
4
,
9
,
1
,
5
; 30], the power of the bloc in the altered game
G
=[
22
,
5
,
3
,
4
,
1; 30],and
the factor of decrement for the three indices
Power Index
G G
Factor
Shapley-Shubik 0
.
6762 0
.
4667 0
.
69
Banzhaf 0
.
5789 0
.
3684 0
.
64
Deegan-Packel 0
.
5083 0
.
3056 0
.
60
