Information Technology Reference
In-Depth Information
5
Weighted Voting Games
5.1
Unanimity Weighted Voting Games
Recall that a WVG in which there is a single winning coalition and every agent is
critical to the coalition is a unanimity WVG. Manipulation via annexation and merg-
ing in unanimity WVGs is less interesting compared to the non unanimity WVGs that
provides more complex and realistic scenarios that are not well-understood. Aziz and
Paterson [2] show that for unanimity WVGs, for both the Shapley-Shubik and Banzhaf
indices:
it is disadvantageous for a coalition to merge and advantageous for a player
to annex other players
. These results extend to the Deegan-Packel index too [13]. Con-
trary to Aziz and Paterson [2] however, for the case of annexation, Lasisi and Allan [13]
contend that it is not true in its entirety that
it is advantageous for an annexer to annex
other players in unanimity
WVGs. They argue that apart from the fact that annexation
always increases the power of other agents that are not annexed by the same factor of
increment as the annexer achieved, the annexer also incurs annexation costs that reduce
the benefit the annexer thought it gained. Finally, Lasisi and Allan [13] bound the extent
to which a strategic agent may gain in annexation. They show that for any unanimity
WVG of
n
agents, the upper bound on the extent to which a strategic agent may gain
while annexing other agents is at most
n
times the power of the agent in the original
game using any of the three indices.
5.2
Non Unanimity Weighted Voting Games
For the sake of simplicity, we assume that only one of the agents is engaging in annex-
ation at a time. However, we are not oblivious of the fact that other agents also have
similar motivations to engage in annexation in anticipation of power increase. For the
case of manipulation via merging, we assume that the assimilated agents in the bloc can
easily distribute the gains from their collusion among themselves in a fair and stable
way. Thus, paving way for manipulation.
Consider a WVG
G
of
I
agents with quota
q
. If any agent
i
q
,
then the agent will always win without forming coalitions with other agents. The more
interesting games we consider are those for which
w
i
<q
, and such that
q
satisfies the
inequality
q
∈
I
has weight
w
i
≥
m
,where
m
is chosen randomly to be the weight of exactly one
of the agents in the game. When the grand coalition (i.e., a coalition of all the agents)
emerges, it will always contain some agents that are not critical in the coalition. It is easy
to see that all the winning coalitions in this type of games are non unanimity. In order
to evaluate the behaviors of the power indices for non unanimity WVGs, we conduct
experiments to evaluate the effects of manipulation when a strategic agent annexes other
agents in the games or when manipulators merge to form blocs using each of the three
indices.
≤
w
(
I
)
−
6
Experiments
This section provides detail descriptions of the simulation environment used for the
conduct of experiments and analysis of the experimental results used for the evaluation
of the effects of annexation and merging in non unanimity WVGs.
