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Domination of Manipulability.
Let
Θ
be another power index other than
Φ
. Let agent
i
alter a game
G
by annexing agents
S
. Suppose the power of the agent in a new game
G
are
Φ
&(
S∪{i}
)
(
G
)
and
Θ
&(
S∪{i}
)
(
G
)
as determined by
Φ
and
Θ
respectively. We
say that the manipulability of one index say
M
Φ
i
(
G
,G
)
, dominates the manipulability
of another index
M
Θ
i
(
G
,G
)
for a particular game
G
, if the factor by which
i
gain in
Φ
is greater than the factor by which it gain in
Θ
, i.e.,
M
Φ
i
(
G
,G
)
>M
Θ
i
(
G
,G
)
which implies that
Φ
&(
S
∪{
i
}
)
(
G
)
Φ
i
(
G
)
Θ
&(
S
∪{
i
}
)
(
G
)
Θ
i
(
G
)
. Hence,
Φ
is more susceptible to
manipulation via annexation in
G
than
Θ
. The domination of manipulability can be
similarly defined for manipulation via merging.
>
4
Annexations and Merging
This section provides examples illustrating manipulation via annexation and merging in
WVGs. The power of the strategic agents i.e., the annexer or the bloc of manipulators,
and the factor of increment (decrement) are also summarized in a table for each example
using the three power indices.
4.1
Manipulation via Annexation
Example 1.
Annexation Advantageous.
Let
G
=[
5
,
8
,
3
,
3
,
4
,
2
,
4;18]
be a WVG. The assimilated agents are shown in bold,
with agent
1
being the annexer. In the original game, the Deegan-Packel index of the
annexer is
γ
1
(
G
)=0
.
1722
. In the new game,
G
=[
9
,
8
,
3
,
3
,
2
,
4;18]
, its Deegan-
Packel index is
γ
1
(
G
)=0
.
2604
, a factor of increase of
1
.
51
.
Ta b l e 1 .
The annexer power in the game
G
=[
5
,
8
,
3
,
3
,
4
,
2
,
4; 18], the altered game
G
=
[
9
,
8
,
3
,
3
,
2
,
4; 18], and the factor of increment for the three indices
Power Index
G G
Factor
Shapley-Shubik 0
.
1714 0
.
3500 2
.
04
Banzhaf 0
.
1712 0
.
3400 1
.
99
Deegan-Packel 0
.
1722 0
.
2604 1
.
51
Example 2.
Annexation Disadvantageous.
Let
G
=[
8
,
9
,
9
,
5
,
7
,
3
,
9;29]
be a WVG. The assimilated agents are shown in bold,
with agent
1
being the annexer. In the original game, the Deegan-Packel index of the
annexer is
γ
1
(
G
)=0
.
1711
. In the new game,
G
=[
11
,
9
,
9
,
5
,
7
,
9;29]
, its Deegan-
Packel index is
γ
1
(
G
)=0
.
1591
, a factor of decrease of
0
.
93
.
