Information Technology Reference
In-Depth Information
6.1
Simulation Environment
We perform experiments to evaluate the effects of manipulation via annexation and
merging by agents using each of the three power indices. To facilitate comparison, we
have
15
agents in each of the original WVGs. The weights of agents in these games are
chosen so that all weights are integers not larger than ten. These weights are reflective
of realistic voting procedures as the weights of agents in real votings are not too large
[4]. When creating a new game, all agents are randomly assigned weights, and the
quota of the game is also generated to satisfy the inequality of non unanimity WVGs
of Subsection 5.2. For the case of manipulation via annexation, we randomly generate
WVGs and assume that only the first agent in the game is engaging in the manipulation,
i.e., the annexer. Then, we determine the power derived by each of the three power
indices (i.e., Shapley-Shubik, Banzhaf, and Deegan-Packel power index) for this agent
in the game. After this, we consider annexation of at least one agent in the game by the
annexer, while the weights of other agents not annexed remain the same in the altered
games. For a particular game, the annexer may annex
1
10
other agents; we refer
to
i
as the
bloc size
. The bloc size and the members of the bloc are randomly
1
generated
for each game. The weight of the annexer in the new game is the sum of the weights of
the agents it annexed plus the annexer's initial weight in the original game. We compute
the new power index of the annexer in the altered games next. Now, we determine the
factor of increment by which the annexer gains or loses in the manipulation for the
corresponding bloc sizes
i
, in the range
1
≤
i
≤
10
.
We use the same procedure as described above for the case of manipulation via merg-
ing with the following modifications. Since merging requires coordinated action of the
manipulators, we randomly select strategic agents among the agents in the WVGs to
form the blocs of manipulators. The bloc size
2
≤
i
≤
10
, for mergng is also randomly
generated for each game. The weight of a bloc in a new game is the sum of the weights
of the assimilated agents in the bloc. The bloc participates in the new game as though a
single agent. We compute the new power index of the bloc in the altered games next. We
determine the factor of increment by which the bloc gains or loses in the manipulation
for the corresponding bloc sizes. Unlike in annexation, the power of the bloc is com-
pared with the sum of the original powers of the individual agents in the bloc. For our
study, we generate
2
,
000
original WVGs for various bloc sizes and allow manipulation
by the annexer or the bloc of manipulators. For each game, we compute the factor of
increment by which the annexer or the bloc gains or loses. Finally, we compute the av-
erage value of these factors of increment over all the games for each bloc size. We use
2
,
000
WVGs in order to capture a variety of games that are representative of the non
unanimity WVGs and to minimize the standard deviation from the true factors when
we compute the average values. The average value of the factors of increment provides
the extent of susceptibility to manipulation by each of the three indices. We estimate
the domination of manipulability among the three indices by comparing their average
factors of increment simultaneously in similar games.
≤
i
≤
1
We note that randomly generating members of the blocs fails to consider the benefits of a more
strategic approach to manipulation. We plan to address this in future work.
