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equal
groups using the quotas. We refer to the groups as
Range1, Range2, Range3,
and
Range4
. First, we consider the effect of the quota on annexation. We generate
400
original WVGs, which in turn generate approximately
3
,
800
altered WVGs in each of
the four ranges. As expected for annexation (based on results of previous sections), all
the altered WVGs are advantageous for the three indices with variations in the factors
of increment achieved by annexers. So, we evaluate the factors of increment for the
ranges.
Fig. 4.
Effect of quota on annexation for WVGs using average factor of increment
Figure 4 shows the average factor of increment for manipulation via annexation
among the three power indices in the altered WVGs for the four ranges. The
x
-axis
indicates the ranges while the
y
-axis is the average factor of increment. The figure
shows that the average factor of increment is high for the altered WVGs in Range1. Our
explanation for this is as follow. The quotas of all the altered WVGs in Range1 have
the least values compared to other ranges. So, it is easy for annexers to quickly acquire
the quota requirement of the games. More importantly, we observe many cases where
the weight of the blocs of assimilation due to annexation are greater than the quota for
this range. In such cases, an annexer claims the maximum power of
1
that is available
in the game. However, as the quota increases into other ranges, it becomes increasingly
difficult for an annexer to achieve high values since the annexer needs to form coalitions
with other agents to gain some power. Summarily, we observe a fairly uniform degra-
dation of the factors of increment from Range1 to Range4, and an inverse relationship
between the quotas of the games and the factors of increment using the three power
indices; at least for the non unanimity WVGs that we consider.
Now, we consider the effect of the quota on merging. Figure 5 provides similar re-
sults as above for manipulation via merging. Unlike annexation, where the factors of
increment is very high for Range1 i.e., greater than
15
for both Banzhaf and Shapley-
Shubik indices, the corresponding values for Range1 here is less than a factor of
1
.
5
for both Banzhaf and Shapley-Shubik indices. Similar explanations for degradation of
average factors of increment as the quotas increase into other ranges observed for an-
nexation above hold here too.
