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termed its
power
. Consider, for example, a WVG of three voters,
a
1
,a
2
,and
a
3
with
respective weights
6
,
3
,
and
1
. When the quota for the game is
10
, then a coalition
consisting of the three voters is needed to win the game. Thus, each of the voters are
of equal importance in achieving the winning coalition. Hence, they each have equal
power irrespective of their weight distribution. The power of each agent in a WVG re-
flects its significance in the elicitation of winning coalitions. A widely accepted method
for measuring such power is using
power indices
[9,10,15,16]. Three prominent power
indices are the
Shapley-Shubik
,
Banzhaf
,and
Deegan-Packel
indices [15,16].
This paper discusses WVGs and two methods of manipulating those games, called
annexation
and
merging
[2]. In annexation, a strategic agent, termed an
annexer
,may
alter a game by taking over the voting weights of some other agents in order to use
the weights in her favor. As a straightforward example of annexation, consider when
a shareholder buys up the voting shares of some other shareholders [15]. We refer to
agents whose voting shares were bought over as
assimilated voters
. The new game con-
sists of the previous agents in the original game whose weights were not annexed and
the
bloc
of agent made up of the annexer and the assimilated voters. The annexer also
incurs some
annexation cost
to allow purchasing the votes of the assimilated voters. In
this situation, only the annexer benefits from annexation as the power of the bloc in the
new game is compared to the power of the annexer in the original game. On the other
hand, merging is the voluntary coordinated action of would-be manipulators who come
together to form a bloc. The agents in the bloc are also assumed to be assimilated voters
since they can no more vote as individual voters in the new game, rather as a bloc. The
new game consists of the previous agents in the original game that were not assimilated
as well as the bloc formed by the assimilated voters. The power of the bloc in the new
game is compared to the sum of the individual powers of all members of the assimilated
bloc in the original game. No annexation costs occur as individual voters in the bloc are
compensated via increase of power. All the agents in the bloc benefit from the merg-
ing in the case of power increase, having agreed on how to distribute the gains of their
collusion. In both annexation and merging, strategic agents who agree to assimilation
anticipate that the value of their power in the new games to be at least the value of their
power in the original games.
We evaluate the susceptibility to manipulation via annexation and merging in WVGs
of the following power indices: Shapley-Shubik, Banzhaf, and Deegan-Packel indices.
Susceptibility to manipulation is the extent to which strategic agents may gain power
with respect to the original games they manipulate. We provide empirical analysis of
susceptibility to annexation and merging in WVGs among the three indices. The re-
mainder of the paper proceeds as follows. Section 2 discusses related work. Section 3
provides definitions and notations used in the paper. In Section 4, we provide examples
using the three indices to illustrate manipulation via annexation and merging. Section
5 considers unanimity and non unanimity WVGs. Section 6 provides empirical evalua-
tion of susceptibility of the indices to manipulation via annexation and merging for non
unanimity WVGs. In Section 7, we empirically evaluate the effects of quota of WVGs
on annexation and merging for non unanimity WVGs. We conclude in Section 8.
