Information Technology Reference
In-Depth Information
6.2
Simulation Results
Experiments confirm the existence of advantageous annexation and merging for the non
unanimity WVGs when agents engage in manipulation using the three indices. How-
ever, the extent to which agents gain varies with both annexation and merging, and
among the indices. Consider manipulation via annexation first. We provide a compari-
son of susceptibility to manipulation among the three power indices by comparing the
population of factors of increment attained by strategic agents in different games for
each of the indices. A summary of susceptibility to manipulation via annexation among
the three indices for
2
,
000
WVGs is shown in Figure 1. The
x
-axis indicates the bloc
sizes while the
y
-axis is the average factor of increment achieved by agents in the
2
,
000
WVGs for corresponding bloc sizes.
Average Factor of Increment vs Bloc Size
16.0
14.0
e
t
r
f
I
e
t
12.0
10.0
8.0
6.0
4.0
2.0
0.0
0
1
2
3
4
5
6
7
8
9
10
11
Bloc Size
Deegan-Packel
Banzhaf
Shapely-Shubik
Fig. 1.
Susceptibility to manipulation via annexation among Shapley-Shubik, Banzhaf, and
Deegan-Packel indices for non unanimity WVGs
The effect of manipulation via annexation is pronounced for the three power indices,
as all the indices are highly susceptible to manipulation. However, the higher suscep-
tibility of the Shapley-Shubik and Banzhaf indices than the Deegan-Packel index can
be observed from Figure 1. While the average factor of increment for manipulation
rapidly grows with the bloc sizes for the Shapley-Shubik and Banzhaf indices, that of
the Deegan-Packel index grows more slowly. By the average factor of increment, the
Shapley-Shubik index manipulability dominates that of Banzhaf index, which in turn
dominates that of Deegan-Packel index. Also, there is a positive correlation between the
average factor of increment and the bloc sizes for the three indices. The average factor
of increment increases with the bloc sizes. This analysis suggests that the Shapley-
Shubik and Banzhaf power indices are more susceptible to manipulation via annexation
than the Deegan-Packel power index. Since all the three power indices are susceptible
to manipulation via annexation, this provides some motivations for strategic agents to
generally engage in such manipulation for non unanimity WVGs when they are being
evaluated using any of the three power indices, and in particular, when the Shapley-
Shubik power index is employed.
