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Average Factor of Increment vs Bloc Size
1.2
1.1
t
1.0
e
t
r
f
I
e
m
0.9
0.8
0.7
0.6
0.5
0.4
0
1
2
3
4
5
6
7
8
9
10
11
Bloc Size
Deegan-Packel
Banzhaf
Shapely-Shubik
Fig. 2.
Susceptibility to manipulation via merging among Shapley-Shubik, Banzhaf, and Deegan-
Packel indices for non unanimity WVGs
Figure 2 provides similar results for the non unanimity WVGs for manipulation via
merging. We again compare susceptibility to manipulation among the three power in-
dices. Unlike manipulation via annexation, only the Shapley-Shubik index appears to be
susceptible to manipulation for this type of game. Also, there appears not to be any cor-
relation between the average factor of increment achieved by the bloc of manipulators
and the bloc size for the three power indices. Thus, it is unclear to the would-be ma-
nipulators what bloc size would be advantageous or disadvantageous to the bloc, and to
what extent. It is easy to see from the trends of the three power indices in Figure 2, that,
using the average factor of increment over the games we consider, the Shapley-Shubik
index manipulability dominates that of the Banzhaf index, which in turn dominates that
of the Deegan-Packel index. Another positive result that is observable from Figure 2 is
that the highest average factor of increment for the three power indices is less than a fac-
tor of
1
.
2
as compared to a factor of
15
, found for the Shapley-Shubik index,
12
for the
Banzhaf index, and
6
for the Deegan-Packel index under manipulation via annexation
(see Figure 1).
In Figure 3, examination of the
2
,
000
non unanimity WVGs we consider reveals that
many of the games are advantageous for Shapley-Shubik index, few for the Banzhaf
index, and virtually none for the Deegan-Packel index. The figure shows the percentage
of advantageous and disadvantageous games for manipulation via merging among the
three indices. Even for the cases where the games are advantageous for the three indices,
the factor of increment achieved by the blocs of manipulators are not very high, and
in all cases are less than a factor of
2
. The experimental evidence suggests that the the
Shapley-Shubik index is more susceptible to manipulation via merging than the Banzhaf
and Deegan-Packel power indices for non unanimity WVGs, even though the factor of
increment is not high. Now, since only the Shapley-Shubik index is more susceptible to
manipulations via merging, and also, since the factor by which the bloc of manipulators
gains is very low, we suspect that this may provide less motivation for strategic agents
to generally engage in manipulation via merging for the non unanimity WVGs when
they are being evaluated using any of the three power indices, and in particular, when
the Deegan-Packel index is employed.
