ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES
The Extent of Susceptibility of Power Indices
Ramoni O. Lasisi and Vicki H. Allan
Department of Computer Science, Utah State University, Logan, UT 84322-4205, U.S.A.
Keywords:
Agents, Manipulations, Annexation, Merging, Power indices.
Abstract:
This paper discusses weighted voting games and two methods of manipulating those games, called annexation
and merging. These manipulations allow either an agent, called an annexer to take over the voting weights of
some other agents in the game, or the coming together of some agents to form a bloc of manipulators to have
more power over the outcomes of the games. We evaluate the extent of susceptibility to these manipulations
in weighted voting games of the following prominent power indices: Shapley-Shubik, Banzhaf, and Deegan-
Packel indices. We found that for unanimity weighted voting games of n agents and for the three indices: the
manipulability, (i.e., the extent of susceptibility to manipulation) via annexation of any one index does not
dominate that of other indices, and the upper bound on the extent to which an annexer may gain while annexing
other agents is at most n times the power of the agent in the original game. Experiments on non unanimity
weighted voting games suggest that the three indices are highly susceptible to manipulation via annexation
while they are less susceptible to manipulation via merging. In both annexation and merging, the Shapley-
Shubik index is the most susceptible to manipulation among the indices.
1 INTRODUCTION
Weighted voting games (WVGs) are mathematical ab-
stractions of voting systems. In a voting system, vot-
ers express their opinions through their votes by elect-
ing candidates to represent them or inﬂuence the pas-
sage of bills. Each member of the set of voters, V, has
an associated weight w :V Q
+
. A voter’s weight is
the number of votes controlled by the voter, and this
is the maximum number of votes she is permitted to
cast. The homogeneous voting system is a special case
in which all voters have unit weight (Levchenkova
and Levchenkov, 2002). In our context, a subset of
agents, called the coalition, wins in a WVG, if the
sum of the weights of the individual agents in the
coalition meets or exceeds a certain threshold called
the quota. In the more traditional homogeneous vot-
ing system, the winning coalition is determined by the
majority of the agents. However, in WVGs with all
agents having different weights, a coalition with sum
of the individual agents’ weights meeting or exceed-
ing the quota determines the winning coalition.
It is natural to naively think that the numeri-
cal weight of an agent directly determines the cor-
responding strength of the agent in a WVG. The
measure of the strength of an agent is termed its
power. This is the ability of an agent to inﬂuence
the decision-making process. 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 all the three vot-
ers is needed to win the game. Thus, each of the
voters are of equal importance in achieving the win-
ning coalition. Hence, they each have equal power
irrespective of their weight distribution, in that ev-
ery voter is necessary for a win. The three prominent
power indices for measuring agents’ power are the
Shapley-Shubik, Banzhaf, and Deegan-Packel indices
(Matsui and Matsui, 2000). All the three methods as-
sign equal power to the voters in this example.
This paper discusses WVGs and two methods
of manipulating those games, called annexation and
merging (Aziz and Paterson, 2009). In annexation,
a strategic agent, termed an annexer, may alter a
game by taking over the voting weights of some other
agents in the game in order to use the weights in
her favor. As a straightforward example of annexa-
tion, consider when a shareholder buys up the vot-
ing shares of some other shareholders (Machover and
Felsenthal, 2002). We refer to the agents whose vot-
ing shares were bought as the assimilated voters. The
new game consists of the previous agents in the orig-
124
O. Lasisi R. and H. Allan V..
ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices.
DOI: 10.5220/0003177201240133
In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 124-133
ISBN: 978-989-8425-41-6
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
inal game whose weights were not annexed by the
strategic agent 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 the 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 co-
ordinated 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 pre-
vious agents in the original game that are not assim-
ilated as well as the bloc formed by the assimilated
voters. The power of the bloc in the new game is com-
pared to the sum of the individual powers of all mem-
bers of the assimilated bloc in the original game. No
annexation costs occur as individual voters in the bloc
are compensated via power. All the agents in the bloc
beneﬁt from the merging in case of power increase,
having agreed on how to distribute the gains from
their collusion.
In both annexation and merging, strategic agents
anticipate that the value of their power in the new
games to be at least the value of their power in
the original games. (Machover and Felsenthal, 2002)
show that this anticipation of power increase due
to annexation is always achieved by the annexer
when the Shapley-Shubik index is used to compute
power. This is not true for the Banzhaf index as the
power of the annexer can decrease compared to its
power in the original game. For the case of merging
and for both the Shapley-Shubik and Banzhaf indices,
there are situations where the power of the bloc de-
creases compared to the sum of the individual powers
of all members of the assimilated bloc in the original
games.
To date, the more detailed analysis of players
merging into blocs remains unexplored (Aziz and Pa-
terson, 2009). This paper evaluates the susceptibility
to manipulation via annexation and merging in WVGs
of the following power indices: Shapley-Shubik,
Banzhaf, and Deegan-Packel indices. This is the ex-
tent to which strategic agents may gain power with re-
spect to the original games they manipulate. We pro-
vide empirical analysis of susceptibility to annexation
and merging in WVGs among the three indices. The
main results of this paper are the following:
1. For any unanimity WVGs of n agents:
a. Contrary to (Aziz and Paterson, 2009) that for
both Shapley-Shubik and Banzhaf indices it is
advantageous for a player to annex, we show
that this is not true in its entirety. 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 re-
duce the beneﬁt the agent thought it gained.
b. Using the Shapley-Shubik, Banzhaf, and
Deegan-Packel indices to compute power, the
manipulability of any one index does not dom-
inate the manipulability of other indices.
c. The upper bound on the extent to which a
strategic agent may gain (i.e., the factor of in-
crement) while annexing other agents in the al-
tered game is at most n times the power of the
agent in the original game. The result holds for
the Shapley-Shubik, Banzhaf, and the Deegan-
Packel power indices.
2. The Shapley-Shubik, Banzhaf, and the Deegan-
Packel indices are all highly susceptible to
manipulation via annexation in non unanimity
WVGs. However, the Shapley-Shubik index is the
most susceptible of the three indices.
3. Unlike manipulation via annexation in the non
unanimity WVGs, the Shapley-Shubik, Banzhaf,
and the Deegan-Packel indices are all less sus-
ceptible to manipulation via merging. Again, the
Shapley-Shubik index is the most susceptible of
the three indices.
4. Finally, the Shapley-Shubik index manipulability
dominates that of the Banzhaf index, which in
turn dominates that of the Deegan-Packel index
for both manipulation via annexation and merg-
ing in non unanimity WVGs.
The remainder of the paper is organized as fol-
lows. Section 2 discusses related work. Section 3 pro-
vides the deﬁnitions and notations used in the pa-
per. In Section 4, we provide examples using the three
power indices to illustrate manipulation via annexa-
tion and merging in WVGs. Section 5 considers una-
nimity and non unanimity WVGs. We also provide
evaluation of manipulation via annexation for una-
nimity WVGs. Section 6 provides empirical evalua-
tion of susceptibility of the three power indices to ma-
nipulations via annexation and merging for non una-
nimity WVGs. We conclude in Section 7.
2 RELATED WORK
Weighted voting games and power indices are widely
studied (Matsui and Matsui, 2000; Leech, 2002;
Alonso-Meijide and Bowles, 2005; Bachrach et al.,
ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices
125
2008; Aziz and Paterson, 2009). WVGs have many
applications, including economics, political science,
neuroscience, threshold logic, reliability theory, dis-
tributed systems (Aziz et al., 2007), and multia-
gent systems (Bachrach and Elkind, 2008). Prominent
real-life situations where WVGs have found applica-
tions include the United Nations Security Council, the
Electoral College of the United States and the Interna-
tional Monetary Fund (Leech, 2002; Alonso-Meijide
and Bowles, 2005).
The study of WVGs has also necessitated the
need to fairly determine the power of players in a
game. This is because the power of a player in a game
provides information about the relative importance of
that player when compared to other players. To eval-
uate players’ power, prominent power indices such
as Shapley-Shubik, Banzhaf, and Deegan-Packel in-
dices are commonly employed (Matsui and Matsui,
2000). These indices satisfy the axioms that charac-
terize a power index, have gained wide usage in po-
litical arena, and are the main power indices found in
the literature (Laruelle, 1999). These power indices
have been deﬁned on the framework of subsets of
winning coalitions in the game they seek to evalu-
ate. A wide variation in the results they provide can
be observed. This is due to the different deﬁnitions
and methods of computation of the associated subsets
of the winning coalitions. Then, comes the question
of which of the power indices is the most resistant to
manipulation in a WVG. The choice of a power index
depends on a number of factors, namely, the a priori
properties of the index, the axioms characterizing the
index, and the context of decision making process un-
der consideration (Laruelle, 1999).
The three indices we consider measure the in-
ﬂuence of voters differently. There are many situ-
ations where their values are the same for similar
games. However, there exists an important example
of the US federal system while using the Shapley-
Shubik and Banzhaf indices where they do not agree
(Kirsch and Langner, 2010). According to (Laruelle
and Valenciano, 2005), and (Kirsch, 2007), the deci-
sion of which index to use in evaluating a voting situ-
ation is largely dependent on the assumptions about
the voting behavior of the voters. When the voters
are assumed to vote completely independently of each
other, the Banzhaf index has been found to be ap-
propriate. On the other hand, Shapley-Shubik index
should be employed when all voters are inﬂuenced by
a common belief on their choices. Deegan-Packel in-
dex is appealing in that it assigns powers based on
size of the winning coalition, thus giving preference
to smaller coalitions (which may be easier to form).
Under certain assumptions in the WVGs, com-
puting the power indices of voters using any of
Shapley-Shubik, Banzhaf, or Deegan-Packel indices
is NP-hard (Matsui and Matsui, 2000). (Deng and
Papadimitriou, 1994) also show that computing the
Shapley value in WVGs is #P-complete. However,
the power of voters using any of the three indices
can be computed in pseudo-polynomial time by dy-
namic programming (Garey and Johnson, 1979; Mat-
sui and Matsui, 2000). There are also approxima-
tion algorithms for computing the Shapley-Shubik
and Banzhaf power indices (Bachrach et al., 2008).
(Bachrach and Elkind, 2008) have studied a form
of manipulation in WVGs called false name manipu-
lation. In false name manipulation, a strategic agent
may alter a game in anticipation of power increase by
splitting its weight among several false identities that
are not in the original game. They use the Shapley-
Shubik index to evaluate agents’ power and consider
the case when an agent splits into exactly two false
identities. The extent to which agents increase or de-
crease their Shapley power is also bounded. Similar
results using Banzhaf index were obtained by (Aziz
and Paterson, 2009). Furthermore, (Lasisi and Al-
lan, 2010) extends existing work by (Bachrach and
Elkind, 2008), and (Aziz and Paterson, 2009). Their
work empirically considers the effects of false name
manipulation in WVGs when an agent splits into
more than two identities. Results of their experi-
ments suggest that the three indices are susceptible
to false name manipulation in WVGs. However, that
the Deegan-Packel index is more susceptible than the
Shapley-Shubik and Banzhaf indices.
As mentioned in the introduction, very little work
exists on manipulation via annexation and merging
in WVGs, and the more detailed analysis of players
merging into blocs, until now, has remained unex-
plored (Aziz and Paterson, 2009). We discuss some
notable exceptions. (Machover and Felsenthal, 2002),
prove that if a player annexes other players, then the
annexation is always advantageous for the annexerus-
ing the Shapley-Shubik index. The annexation can be
index. For the case of merging, and for both the
Shapley-Shubik and Banzhaf indices, merging can be
son, 2009) show that for some classes of WVGs, and
for both Shapley-Shubik and Banzhaf indices, it is
disadvantageous for a coalition to merge, while ad-
vantageous for a player to annex. They also prove
some NP-hardness results for annexation and merging
in WVGs. They show that for both Shapley-Shubik
and Banzhaf indices, ﬁnding a beneﬁcial annexation
is NP-hard. Also, determining if there exists a beneﬁ-
cial merge is NP-hard for the Shapley-Shubik index.
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
126
(Machover and Felsenthal, 2002), and (Aziz and
Paterson, 2009) have shown that it can be advanta-
geous for strategic agents to engage in annexation or
merging for Shapley-Shubik and Banzhaf indices in
some classes of WVGs. The authors stop short of ad-
dressing the question of upper bounds on the extent
to which strategic agents may gain with respect to the
games they manipulate. In view of this, our work dif-
fer from those of these authors. We extend the work of
(Lasisi and Allan, 2010), as we study the susceptibil-
ity of the three indices to manipulation via annexation
and merging. We empirically consider the extent to
which strategic agents may gain by engaging in such
manipulation and show how the susceptibility among
the indices compares for different WVGs.
3 DEFINITIONS & NOTATIONS
We give the following deﬁnitions and notations used
throughout the paper.
Weighted Voting Game. Let I = {1, ··· , n} be a
set of n agents. Let w = {w
1
, ··· , w
n
} be the corre-
sponding positive integer weights of the agents. Let S
be a non empty set of agents. S I is a coalition. A
WVG G with quota q involving agents I is deﬁned as
G = [w
1
, ··· , w
n
;q]. Denote by w(S), the weight of a
coalition S derived from the summation of the indi-
vidual weights of agents in S i.e., w(S) =
iS
w
i
. A
coalition, S, wins in the game G if w(S) q oth-
erwise it loses. q is constrained as follows
1
2
w(I) <
q w(I). Thus, disjoint winning coalitions cannot
emerge.
Simple Voting Game. Each of the coalitions S I
has an associated value function v : S {0, 1}. The
value 1 implies a win for the coalition and 0 a loss. In
the game G, v(S) = 1 if w(S) q and 0 otherwise.
Dummy and Critical Agents. An agent i S is
dummy if its weight in S is not needed for S to be a
winning coalition, i.e., w(S\{i}) q. Otherwise, it is
critical to S, i.e., w(S) q and w(S\{i}) < q.
Unanimity Weighted Voting Game. A WVG in
which there is a single winning coalition and every
agent is critical to the coalition is unanimity weighted
voting game.
Shapley-Shubik Power Index. The Shapley-
Shubik power index is one of the oldest power
indices and has been used widely to analyze political
power. The index quantiﬁes the marginal contribution
of an agent to the grand coalition. Each agent in a
permutation is given credit for the win if the agents
preceding it do not form a winning coalition but by
adding the agent in question, a winning coalition is
formed. The power index is dependent on the number
of permutations for which an agent is critical. For
the n! permutations of agents used in determining
the Shapley-Shubik index, there exists exactly one
critical agent in each of the permutations. Denote by
Π the set of all permutations of n agents in a WVG G.
Let π Π deﬁne a one-to-one mapping where π(i) is
the position of the ith agent in the permutation order.
Denote by S
π
(i), the predecessors of agent i in π, i.e.,
S
π
(i) = { j : π( j) < π(i)}. The Shapley-Shubik index,
ϕ
i
(G), of agent i in G is given by
ϕ
i
(G) =
1
n!
πΠ
[v(S
π
(i) {i}) v(S
π
(i))] (1)
Banzhaf Power Index. Another index that has also
gained wide usage in the political arena is the Banzhaf
power index. Unlike the Shapley-Shubik index, its
computation depends on the number of winning coali-
tions in which an agent is critical. There can be more
than one critical agent in a particular winning coali-
tion. The Banzhaf index, β
i
(G), of agent i in the same
game, G, as above is given by
β
i
(G) =
η
i
(G)
iI
η
i
(G)
(2)
where η
i
(G) is the number of coalitions in which i is
critical in G.
Deegan-Packel Power Index. The Deegan-Packel
power index is also found in the literature for com-
puting power indices. The computation of this power
index for an agent i takes into account both the num-
ber of all the minimal winning coalitions (MWCs) in
the game as well as the sizes of the MWCs having i as
a member (Matsui and Matsui, 2000). Thus, it is more
impressive to be one in three (who elicited the win)
rather than one in ten. A winning coalition C I is a
MWC if every proper subset of C is a losing coalition,
i.e., w(C) q and T C, w(T) < q. The Deegan-
Packel power index, γ
i
(G), of an agent i in G is given
by
γ
i
(G) =
1
|MWC|
SMWC
i
1
|S|
(3)
where MWC
i
are the sets of all MWCs in G that in-
clude i.
Susceptibility of Power Index to Manipulation.
Consider a coalition S I, let &S deﬁnes a bloc of
assimilated voters formed by agents in S.
ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices
127
Annexation : Let Φ be a power index. Denote by
Φ
i
(G), the power of an agent i in a WVG G. Suppose
i alters G by annexing a coalition S. Let G
be the
resulting game after the annexation. We say that Φ
is susceptible to manipulation via annexation if there
exists a G
, such that Φ
&(S∪{i})
(G
) > Φ
i
(G); the an-
nexation is termed advantageous. If Φ
&(S∪{i})
(G
) <
Φ
i
(G), then the annexation is disadvantageous.
Merging : Let Φ be a power index. Denote by
Φ
i
(G), the power of an agent i in a WVG G. Suppose
a coalition, S, alters G by merging into a bloc. Let
G
be the resulting game after the merging. We say
that Φ is susceptible to manipulation via merging if
there exists a G
, such that Φ
&S
(G
) >
iS
Φ
i
(G);
the merging is termed advantageous. If Φ
&S
(G
) <
iS
Φ
i
(G), then the merging is disadvantageous.
Factor of Increment (Decrement). Let Φ be a
power index. Denote by Φ
i
(G), the power of an agent
i in a WVG G. Let G
be the resulting game when i al-
ters G by manipulation. The factor of increment (resp.
decrement) of the original power from the manipula-
tion is
Φ
i
(G
)
Φ
i
(G)
. The value represents an increment (or
gain) if it is greater than 1 and decrement (or loss) if
it is less than 1. The factor of increment provides an
indication of the extent of susceptibility of power in-
dices to manipulation. A higher factor of increment
by a power index in a game indicates that the index is
more susceptible to manipulation in that game.
Domination of Manipulability. Let Φ and Θ be
two different power indices. Denote by Φ
i
(G) and
Θ
i
(G), the respective power of an agent i in a WVG G
as determined by Φ and Θ. Let i be an annexer. Sup-
pose the corresponding power of the agent in a new
game G
when i alters G by assimilating agents S
are Φ
&(S∪{i})
(G
) and Θ
&(S∪{i})
(G
). We say that the
manipulability of one index say Φ
G
G
, dominates the
manipulability of another index Θ
G
G
for a particu-
lar game G, if the factor by which i gain in Φ is
greater than the factor by which it gain in Θ, i.e.,
Φ
&(S∪{i})
(G
)
Φ
i
(G)
>
Θ
&(S∪{i})
(G
)
Θ
i
(G)
, and hence, Φ is more sus-
ceptible to manipulation than Θ in G. The domination
of manipulability can be similarly deﬁne for manipu-
lation via merging.
4 ANNEXATIONS & MERGING
This section provides examples illustrating manipula-
tion via annexation and merging in WVGs. The power
of the strategic agents, i.e., the annexer or the bloc of
manipulators, and the factor of increment (decrement)
are also summarized in a table for each example using
the three power indices.
4.1 Manipulation via Annexation
Example 1. Annexation Advantageous. Let G =
[5, 8, 3, 3, 4, 2, 4;18] be a WVG. The assimilated
agents are shown in bold, with agent 1 being the an-
nexer. In the original game, the Deegan-Packel in-
dex of the annexer is γ
1
(G) = 0.1722. In the new
game, G
= [9, 8, 3, 3, 2, 4;18], its Deegan-Packel in-
dex is γ
1
(G
) = 0.2604, a factor of increase of 1.51.
Table 1: The annexer power in the original
game G = [5, 8, 3, 3, 4, 2, 4;18], the altered game
G
= [9, 8, 3,3, 2, 4;18], and the factor of increment
for the three indices.
Power Index G G
Factor
Shapley-Shubik 0.1714 0.3500 2.04
Banzhaf 0.1712 0.3400 1.99
Deegan-Packel 0.1722 0.2604 1.51
Example 2. Annexation Disadvantageous. Let
G = [8, 9, 9, 5, 7, 3, 9;29] be a WVG. The assimilated
agents are shown in bold, with agent 1 being the an-
nexer. In the original game, the Deegan-Packel index
of the annexer is γ
1
(G) = 0.1711. In the new game,
G
= [11, 9, 9, 5, 7, 9;29], its Deegan-Packel index is
γ
1
(G
) = 0.1591, a factor of decrease of 0.93.
Table 2: The annexer power in the original
game G = [8, 9, 9, 5, 7, 3, 9;29], the altered game
G
= [11, 9, 9, 5, 7, 9;29], and the factor of increment
(decrement) for the three indices.
Power Index G G
Factor
Shapley-Shubik 0.1786 0.2167 1.21
Banzhaf 0.1774 0.2167 1.22
Deegan-Packel 0.1711 0.1591 0.93
4.2 Manipulation via Merging
Example 3. Merging Advantageous. Let G =
[4, 2, 1, 1, 8, 7, 4;17] be a WVG. The assimilated
agents are shown in bold. In the original game, the
Deegan-Packel indices of these agents are, γ
2
(G) =
0.0926, γ
6
(G) = 0.1889, and γ
7
(G) = 0.1704. Their
cummulative power is 0.4519. In the new game, G
=
[13, 4, 1, 1, 8;17], the Deegan-Packel index of the bloc
is γ
1
(G
) = 0.5000, a factor of increase of 1.11.
Example 4. Merging Disadvantageous. Let G =
[5, 8, 3, 4, 9, 1, 5;30] be a WVG. The assimilated
agents are shown in bold. In the original game, the
Deegan-Packel indices of these agents are, γ
2
(G) =
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
128
Table 3: The cummulaive power of the assimilated agents
in the original game G = [4, 2, 1, 1, 8, 7, 4;17], the power of
the bloc in the altered game G
= [13, 4, 1, 1, 8;17], and the
factor of increment for the three indices.
Power Index G G
Factor
Shapley-Shubik 0.4881 0.6667 1.37
Banzhaf 0.4851 0.6000 1.24
Deegan-Packel 0.4519 0.5000 1.11
0.1833, γ
5
(G) = 0.1333, and γ
7
(G) = 0.1417. Their
cummulative power is 0.5083. In the new game, G
=
[22, 5, 3, 4, 1;30], the Deegan-Packel index of the bloc
is γ
1
(G
) = 0.3056, a factor of decrease of 0.60.
Table 4: The cummulaive power of the strategic agents in
the original game G = [5, 8, 3, 4, 9, 1, 5;30], the power of the
bloc in the altered game G
= [22, 5, 3, 4, 1;30], and the fac-
tor of decrement for the three indices.
Power Index G G
Factor
Shapley-Shubik 0.6762 0.4667 0.69
Banzhaf 0.5789 0.3684 0.64
Deegan-Packel 0.5083 0.3056 0.60
5 WEIGHTED VOTING GAMES
This section considers manipulatons via annexation
and merging for both unanimity and non unanim-
ity WVGs. For the sake of simplicity in our discus-
sion, we assume that for the case of manipulation
via annexation, the annexer has enough resources to
cover the annexation costs for all the agents it an-
nexes. Also, we assume that only one of the agents
is engaging in the annexation 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 manip-
ulation 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, making them agree to engage in the ma-
nipulation if it is proﬁtable.
5.1 Unanimity Weighted Voting Games
We recall that a WVG in which there is a single win-
ning coalition and every agent is critical to the coali-
tion is unanimity WVG. (Aziz and Paterson, 2009)
have shown that for unanimity WVGs and for both the
Shapley-Shubik and Banzhaf indices: it is disadvan-
tageous for a coalition to merge and advantageous
for a player to annex other players. We observe that
these results naturally extend to the Deegan-Packel
index too. This is because for unanimity WVGs, the
deﬁnitions of the Shapley-Shubik, Banzhaf, and the
Deegan-Packel indices using Formulas 1, 2, and 3,
respectively, are equivalent. In fact, the power of all
agents in any unanimity WVGs is the same for the
three indices. In view of the annexation result of (Aziz
and Paterson, 2009) above, and the fact that strategic
agents are interested in annexations and merging that
improve their power, we consider only manipulation
via annexation for the unanimity WVGs.
Note that (Aziz and Paterson, 2009) have not con-
sidered the bounds on the extent to which strate-
gic agents may gain with respect to games they ma-
nipulate. This is important as it provides motiva-
tions for strategic agents to engage in manipulation
when derivable gains are appreciable. Apart from this,
the gains or the factor of increments show the ex-
tent of susceptibility to manipulation and provide a
measure of domination of manipulability among the
indices. The magnitude of this gain for unanimity
WVGs, as we shall see shortly, depends on the num-
ber of agents in the original game, the number of
agents the annexer is able to annex, as well as the
annexation costs. Example 5 illustrates a unanim-
ity WVG where an annexer appears to achieve three
times its original power annexing other agents.
Example 5. Annexation Advantageous:
Unanimity WVGs. Consider G =
[7, 6, 9, 2, 5, 3, 1, 1, 8, 2, 2, 8, 4, 9, 6;73], a unanim-
ity WVG of 15 agents. The Deegan-Packel index
of any agent in the game is 0.0667. Suppose the
ﬁrst agent with weight 7, alters G by annexing
the next ten agents in the game. The new game
G
= [46, 8, 4, 9, 6;73]. The annexer has improved its
weight to 46. The Deegan-Packel index of the annexer
in G
is γ
1
(G
) = 0.2000. The agent beneﬁts from the
annexation and increases its power by a factor of 3.
Table 5: The annexer power in the original game
G = [7, 6, 9, 2, 5, 3, 1, 1, 8, 2, 2, 8,4, 9, 6;73], the altered game
G
= [46, 8, 4, 9, 6;73, and the factor of increment for the
three indices.
Power Index G G
Factor
Shapley-Shubik 0.0667 0.2000 3.00
Banzhaf 0.0667 0.2000 3.00
Deegan-Packel 0.0667 0.2000 3.00
It appears that the annexer has achieved a gain of
three times its original power while annexing other
agents, but this is not true in its entirety. We provide
the following arguments. Since the original and the
altered games are unanimity, the power of all agents
in each game is the same. While the annexer has im-
provedits weight, and consequentlyits power by three
ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices
129
times its original power, other agents that were not as-
similated have also had their power increased by the
same factor, even though their weights in the origi-
nal and altered games remain the same. Clearly, these
agents do not incur any cost like the annexer whose
improved weight and power must have been achieved
at annexation costs. The annexation costs reduce the
beneﬁts the agent thought it gained, making the an-
nexer’s beneﬁt worse than the beneﬁts of other agents
not engaging in annexation. This weakens (Aziz and
Paterson, 2009) result that for unanimity WVG and
for both Shapley-Shubik and Banzhaf indices it is ad-
vantageous for a player to annex.
Now, suppose we assume that the annexer still ac-
crues some gains even after the application of the an-
nexation costs, then these gains are the same for the
three indices. We see that the extents of susceptibil-
ity to manipulation among the three indices are the
same. Hence, for any unanimity WVGs, the manipu-
lability of any one index does not dominate the ma-
nipulability of other indices.
Finally, the generalization of the upper bound on
the extent to which a strategic agent may gain with re-
spect to games it manipulate in any unanimity WVGs
follows from (Aziz and Paterson, 2009). For any una-
nimity WVG of n agents, the power of each agent
is
1
n
. If a strategic agent annexes k 1 other agents,
the power of the strategic agent as well as that of the
other agents in the new game is
1
nk+1
. Hence, the fac-
tor of increment for each agent is
n
nk+1
. This factor
of increment is the same for the three indices. When
k = 1, (i.e., the strategic agent is not annexing any
other agent), then the factor of increment is 1, and
this implies the same game we started with. Whereas,
when k = n, the strategic agent is able to annex the
remaining n 1 agents in the original game, then the
factor of increment is n times the power of the agent
in the original game. This is the upper bound on the
extent to which a strategic agent may achieve while
annexing other agents in any unanimity WVG. This
bound holds for the three power indices.
5.2 Non Unanimity Weighted Voting
Games
Manipulation by annexation and merging in the gen-
eral case of WVGs is more interesting as it pro-
vides more complex and realistic scenarios that are
not well-understood. As the structure of the WVGs
changes due to annexation and merging, the number
of winning coalitions as well as the minimal winning
coalitions in the games also changes.
Consider a WVG G of I agents with quota q. If
any agent i I has weight w
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 in-
equality q < w(I) m, where m is at least 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; hence, all the games here are non una-
nimity WVGs. In order to evaluate the behaviors of
the indices for non unanimity WVGs, we conduct
experiments to evaluate the effects of manipulation
when a strategic agent annexes some other agents in
the games or when some manipulators merge to form
blocs using each of the three indices. The simulation
environment and simulation results are discussed in
Subsections 6.1 and 6.2, respectively.
6 EXPERIMENTS
This section provides detail descriptions of the sim-
ulation environment used for the conduct of experi-
ments, and analysis of the experimental results used
for the evaluation of the effects of manipulation via
annexation and merging in non unanimity WVGs.
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 origi-
nal WVGs. The weights of our agents in these games
are chosen so that no weight is larger than ten. These
weights are reﬂective of realistic voting procedures
as the weights of agents in real votings are not too
large (Bachrach and Elkind, 2008). 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. The least possible weight for any agent is one.
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 manip-
ulation, i.e., the annexer. Then, we determine the
three power indices (i.e., Shapley-Shubik, Banzhaf,
and Deegan-Packel power index) of 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 an-
nexer may annex 1 i 10 other agents; we refer to
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
130
i as the bloc size. The bloc size is randomly generated
for each game. The weight of the annexer in the new
game is the sum of the weights of the agents it an-
nexed plus the annexer’s initial weight in the original
game. We compute the new power index of the an-
nexer 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 merging with the fol-
lowing modiﬁcations. Since merging requires coor-
dinated action of the manipulators, we randomly se-
lect 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 in-
dex of the bloc in the altered games next. Again, we
determine the factor of increment by which the bloc
gains or loses in the manipulation for the correspond-
ing bloc sizes. Unlike in annexation, the power of the
bloc is compared 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 an-
nexer or the bloc gains or loses. Finally, we compute
the average value of these factors of increment overall
the games for each bloc size. We use 2, 000 WVGs in
order to capture a variety of games that are represen-
tative 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 suscepti-
bility to manipulation by each of the three indices. We
estimate the domination of manipulability among the
three indices by comparing their average factors of in-
crement simultaneously in similar games.
6.2 Simulation Results
We present the results of our simulations. Experi-
ments conﬁrm the existence of advantageous annexa-
tion and merging for the non unanimity weighted vot-
ing games when agents engage in manipulation us-
ing the three indices. However, the extent to which
agents gain varies with both annexation and merging,
and among the indices.
Consider manipulation by annexation in non una-
nimity WVGs ﬁrst. We provide a comparison of sus-
ceptibility to manipulation among the three indices by
comparing the population of factors of increment at-
tained by strategic agents in different games for each
of the indices. A summary of susceptibility to ma-
nipulation among the three indices for 2, 000 WVGs
is shown in Figure 1. The x-axis indicates the bloc
sizes while the y-axis is the average factor of incre-
ment achieved by agents in the 2, 000 WVGs for cor-
responding bloc sizes.
Figure 1: Susceptibility to Manipulation via Annexation
among the Shapley-Shubik, Banzhaf, and Deegan-Packel
indices for Non Unanimity WVGs.
The effect of manipulation via annexation is pro-
nounced for the three power indices, as all the in-
dices are highly susceptible to manipulation. How-
ever, the higher susceptibility of the Shapley-Shubik
and Banzhaf indices than the Deegan-Packel index
can be observed from Figure 1. While the aver-
age factor of increment for manipulation rapidly
grows with the bloc sizes for the Shapley-Shubik
and Banzhaf indices, that of the Deegan-Packel index
grows more slowly. By the average factor of incre-
ment, the Shapley-Shubik index manipulability dom-
inates that of Banzhaf index, which in turn dominates
that of Deegan-Packel index. Also, there is a positive
correlation between the average factor of increment
and the bloc sizes for the three indices. The average
factor of increment increases with the bloc sizes.
This analysis suggests that the Shapley-Shubik
and Banzhaf power indices are more susceptible to
manipulation via annexation than the Deegan-Packel
power index. Since all the three power indices are
susceptible to manipulation via annexation, this pro-
vides some motivation for strategic agents to gener-
ally engage in such manipulation for the non unanim-
ity WVGs when they are being evaluated using any
of the three power indices, and in particular, when the
Shapley-Shubik index is employed.
Figure 2 provides similar results for the non una-
nimity WVGs when there are coordinated efforts
among manipulators that culminate in merging. We
ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices
131
Figure 2: Susceptibility to Manipulation via Merging
among the Shapley-Shubik, Banzhaf, and Deegan-Packel
indices for Non Unanimity WVGs.
again compare susceptibility to manipulation among
the three power indices. Unlike manipulation via
annexation, only the Shapley-Shubik index appears
to be susceptible to manipulation for this type of
game. Also, there appears not to be any correlation
between the average factor of increment achieved by
the bloc of manipulatorsand the bloc size for the three
power indices. Thus, it is unclear to the would-be ma-
nipulators what bloc size would be advantageous or
disadvantageous to the bloc, and to what extent.
It is easy to see from the trends of the three power
indices in Figure 2, that, using the average factor of
increment over the games we consider, the Shapley-
Shubik index manipulability dominates that of the
Banzhaf index, which in turn dominates that of the
Deegan-Packel index. Another positive result that is
observable from Figure 2 is that the highest aver-
age factor of increment for the three power indices
is less than a factor of 1.2 as compared to a factor
of 15, found for the Shapley-Shubik index, 12 for
the Banzhaf index, and 6 for the Deegan-Packel in-
dex under the manipulation via annexation. See Fig-
ure 1. Again, examination of the 2, 000 WVGs reveals
that many of the games are advantageous for Shapley-
Shubik index, few for the Banzhaf index, and virtually
none for the Deegan-Packel index. Figure 3 shows
games for manipulation via merging among the three
indices for the 2, 000 non unanimity WVGs. Even for
the cases where the games are advantageous for the
three indices, the factor of increment achieved by the
blocs of manipulators are not very high, and in all
cases are less than a factor of 2.
The analysis suggests that the the Shapley-Shubik
index is more susceptible to manipulation via merging
than the Banzhaf and Deegan-Packel power indices
for non unanimity WVGs, even though the factor of
increment is not high. Now, since only the Shapley-
Shubik index is more susceptible to manipulations via
Figure 3: Percentage of Advantageous and Disadvanta-
geous Games for Manipulation via Merging among the
three indices for 2, 000 Non Unanimity WVGs.
merging, and also, since the factor by which the bloc
of manipulators gains is very low, we suspect that this
may provide less motivation for strategic agents to
generally engage in manipulation via merging for the
non unanimity WVGs when they are being evaluated
using any of the three power indices, and in particular,
when the Deegan-Packel index is employed.
7 CONCLUSIONS
We have considered the effects of manipulation by
annexation and merging in weighted voting games
focusing on the indices used in evaluating agents’
power in such games. The following prominent
power indices are used to evaluate the power of
agents: Shapley-Shubik, Banzhaf, and the Deegan-
Packelindices. We consider the extent to which strate-
gic agents may gain by engaging in such manipula-
tion and show how the susceptibility among the three
indices compares for unanimity and non unanimity
weighted voting games.
For unanimity weighted voting games of n agents,
we show that apart from the fact that annexation al-
ways increases the power of other agents that are
not annexed by the same factor of increment as the
annexer achieved, the annexer also incurs annexa-
tion costs that reduce the beneﬁt the agent thought
it gained, making the annexer’s beneﬁt worse than
the beneﬁts of other agents not engaging in annexa-
tion. Also, for the three power indices, the manipula-
bility of any one index does not dominate the manipu-
lability of other indices for manipulation via annexa-
tion. Finally, the upper bound on the extent to which a
strategic agent may gain while annexing other agents
in the altered game is at most n times the power of the
agent in the original game. This bound holds for the
three power indices.
For non unanimity weighted voting games, we
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
132
show that the games are less vulnerable to manip-
ulation via merging, while they are extremely vul-
nerable to manipulation via annexation for the three
power indices. Also, while the factor of increment
from manipulation grows with bloc sizes for manip-
ulation via annexation, there exists no correlation be-
tween the factor of increment and the bloc size for
manipulation via merging. Finally, we show that the
Shapley-Shubik index manipulability 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 an-
nexation and merging than the Banzhaf and Deegan-
Packel indices, with Deegan-Packel index being the
least susceptible among the indices.
We have some comments regarding these re-
sults. First, we found that our results of manipula-
tion via annexation and merging for non unanimity
weighted voting games are consistent with those of
(Lasisi and Allan, 2010). They consider false name
manipulation in weighted voting games. The manip-
ulation allows an agent to have more power over
the outcomes of the games by splitting into multiple
names and distributing its weights across all associ-
ated names. They showed that for the non unanimity
weighted voting games; the Deegan-Packel index is
more susceptible to false name manipulation than the
Banzhaf and Shapley-Shubik indices, with Shapley-
Shubik index being the least susceptible among the
three indices. The implication of this consistency is
that a scenario where splitting by a strategic agent is
disadvantageous corresponds to a scenario where it is
advantageous for several strategic agents to merge.
Second, we have assumed throughout this paper
that for the case of manipulation via merging, the as-
similated agents in the bloc can easily distribute the
gains from their collusion among themselves in a fair
and stable way. Thus, making them agree to engage in
the manipulation if it is proﬁtable. This assumption is
strong. Even at that, we see that all the three indices
are less vulnerable to manipulation via merging.
ACKNOWLEDGEMENTS
This work is supported by NSF research grant
#0812039 entitled “Coalition Formation with Agent
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