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in some classes of WVGs. The authors stop short of addressing the question of upper
bounds on the extent to which strategic agents may gain with respect to the games they
manipulate. In view of this, our work differ from those of these authors. We extend
the work of Lasisi and Allan [12], as we study the susceptibility of the three power in-
dices to manipulation via annexation and merging. We empirically consider the extent
to which strategic agents may gain by engaging in such manipulation and show how the
susceptibility among the indices compares for different WVGs.
3
Definitions and Notations
Weighted Voting Game.
Let
I
=
{
1
,
···
,n
}
beasetof
n
agents. Let
w
=
{
w
1
,
···
,w
n
}
be the corresponding positive integer weights of the agents. Let a
coalition
S
I
be a
non empty subset of agents. A WVG
G
with
quota
q
involving agents
I
is represented
as
G
=[
w
1
,
⊆
···
,w
n
;
q
]
. Denote by
w
(
S
)
, the weight of a coalition
S
derived from the
summation of the individual weights of agents in
S
i.e.,
w
(
S
)=
i∈S
w
i
. A coalition
S
, wins in the game
G
if
w
(
S
)
≥
q
otherwise it loses.
q
is constrained as follows
1
2
w
(
I
)
<q
≤
w
(
I
)
.
Simple Voting Game.
Each coalition
S
, has an associated value function
v
:
S
→
{
0
,
1
}
.
The value
1
implies a win for
S
and
0
a loss. In the WVG
G
above,
v
(
S
)=1
if
w
(
S
)
≥
q
and
0
otherwise.
Dummy and Critical Agents.
An agent
i
∈
S
is
dummy
if its weight in
S
is not needed
for
S
to be a winning coalition, i.e.,
w
(
S
\{
i
}
)
≥
q
.Otherwise,itis
critical
to
S
, i.e.,
w
(
S
)
≥
q
and
w
(
S
\{
i
}
)
<q
.
Unanimity Weighted Voting Game.
A WVG in which there is a single winning coalition
and every agent is critical to the coalition is a
unanimity
WVG.
Shapley-Shubik Power Index.
This index quantifies the marginal contribution of an
agent to the grand coalition. Each agent in a permutation is given credit for a win if the
agents preceding it do not form a winning coalition but, by adding the agent in question,
a winning coalition is formed. The index is dependent on the number of permutations
for which an agent is critical. Denote by
Π
the set of all permutations of
n
agents in a
WVG
G
.Let
π
Π
define a one-to-one mapping onto itself where
π
(
i
)
is the position
of the
i
th
agent in the permutation order. Denote by
S
π
(
i
)
, the predecessors of agent
i
in
π
, i.e.,
S
π
(
i
)=
∈
{
j
:
π
(
j
)
<π
(
i
)
}
.
The Shapley-Shubik index,
ϕ
i
(
G
)
, of agent
i
in
G
is
n
!
π∈Π
ϕ
i
(
G
)=
1
[
v
(
S
π
(
i
)
∪{
i
}
)
−
v
(
S
π
(
i
))]
(1)
