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of advantageous games for Shapley-Shubik index is the highest in all cases. This result
is consistent with our results of susceptibility of manipulation via merging of Subsection
6.2 (see Figure 2).
8
Conclusions
We consider two methods of manipulating weighted voting games, called
annexa-
tion
and
merging
while focusing on the susceptibility of three of the power indices
used in evaluating agents' power in such games. The following prominent power in-
dices are used to evaluate agents' power: Shapley-Shubik, Banzhaf, and Deegan-Packel
indices. We consider the extent to which strategic agents may gain by engaging in
such form of manipulation and show how the susceptibility among the three indices
compares for non unanimity weighted voting games. Experiments on weighted voting
games suggest that the games are less vulnerable to manipulation via merging, while
they are extremely vulnerable to manipulation via annexation for the three power in-
dices. Also, while the average factor of increment of power due to manipulation grows
with bloc sizes for manipulation via annexation, there appears to be no correlation be-
tween the average factor of increment and the bloc size for manipulation via merg-
ing. Again, the Shapley-Shubik index manipulability (i.e., the extent of susceptibility
to manipulation) dominates that of the Banzhaf index, which in turn dominates that of
the Deegan-Packel index for both manipulation via annexation and merging. Hence, the
Shapley-Shubik index is more susceptible to manipulation via annexation and merging
than the Banzhaf and the Deegan-Packel indices, with Deegan-Packel index being the
least susceptible among the three power indices.
Further experiments on the effect of quotas suggest the existence of an inverse re-
lationship between the susceptibility of the three power indices to manipulation and
the quotas of weighted voting games for both manipulation via annexation and merg-
ing. Thus, weighted voting games with large quota values closer to the total weight of
agents in the games may be less vulnerable to manipulation via annexation and merging
than those with corresponding smaller quota values.
Acknowledgements.
This work is supported by NSF research grant #0812039 entitled
“Coalition Formation with Agent Leadership”.
References
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2. Aziz, H., Paterson, M.: False-name Manipulations in Weighted Voting Games: splitting,
merging and annexation. In: 8th International Conference on Autonomous Agents and Mul-
tiagent Systems, Budapest, Hungary, pp. 409-416 (2009)
3. Aziz, H., Paterson, M., Leech, D.: Combinatorial and Computational Aspects of Multiple
Weighted Voting Games. The Warwick Economics Research Paper Series (TWERPS) 823,
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