Information Technology Reference
In-Depth Information
Banzhaf Power Index.
Another index that has also gained wide usage in the political
arena is the Banzhaf power index. Unlike the Shapley-Shubik index, its computation
depends on the number of winning coalitions in which an agent is critical. There can
be more than one critical agent in a particular winning coalition. The Banzhaf index,
β
i
(
G
)
, of agent
i
in a game
G
,isgivenby
η
i
(
G
)
i∈I
η
i
(
G
)
β
i
(
G
)=
(2)
where
η
i
(
G
)
is the number of coalitions in which
i
is critical in
G
.
Deegan-Packel Power Index.
The Deegan-Packel power index is also found in the lit-
erature for computing power indices. The computation of this power index for an agent
i
takes into account both the number of all the minimal winning coalitions (MWCs) in
the game as well as the sizes of the MWCs having
i
as a member [16]. A winning coali-
tion
C
⊆
I
is a MWC if every proper subset of
C
is a losing coalition, i.e.,
w
(
C
)
≥
q
and
∀
T
⊂
C,w
(
T
)
<q
. The Deegan-Packel power index,
γ
i
(
G
)
, of an agent
i
in a
game
G
,isgivenby
1
1
γ
i
(
G
)=
(3)
|
MWC
|
|
S
|
S
∈
MWC
i
where MWC
i
are the sets of all MWCs in
G
that include
i
.
Susceptibility of Power Index to Manipulation.
Consider a coalition
S
I
,let
&
S
de-
fine a bloc of assimilated voters formed by agents in
S
.Let
Φ
be a power index. Denote
by
Φ
i
(
G
)
, the power of an agent
i
in a WVG
G
.Let
G
be the resulting game when a
WVG
G
is manipulated via annexation or merging.
Annexation: Let an agent
i
alter
G
by annexing a coalition
S
. We say that
Φ
is sus-
ceptible to manipulation via annexation if there exists a
G
, such that
Φ
&(
S∪{i}
)
(
G
)
>
Φ
i
(
G
)
; the annexation is termed
advantageous
.If
Φ
&(
S∪{i}
)
(
G
)
<Φ
i
(
G
)
,thenitis
disadvantageous
.
Merging: Let a coalition
S
alter
G
by merging into a bloc
&
S
. We say that
Φ
is
susceptible to manipulation via merging if there exists a
G
, such that
Φ
&
S
(
G
)
>
⊂
i∈S
Φ
i
(
G
)
; the merging is termed
advantageous
.If
Φ
&
S
(
G
)
<
i∈S
Φ
i
(
G
)
,thenit
is
disadvantageous
.
Factor of Increment (Decrement).
The factor of increment (resp. decrement) of the
original power from a manipulation is
Φ
i
(
G
)
Φ
i
(
G
)
. The value represents an increment (or
gain) if it is greater than
1
and decrement (or loss) if it is less than
1
. The factor of
increment provides an indication of the extent of susceptibility of power indices to
manipulation. A higher factor of increment indicates that the index is more susceptible
to manipulation in that game.
