A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING

IN WEIGHTED VOTING GAMES

Ramoni O. Lasisi and Vicki H. Allan

Department of Computer Science, Utah State University, Logan, U.S.A.

Keywords:

Agents, Weighted voting games, Power indices, False-name manipulation, Annexation, Merging.

Abstract:

Weighted voting games are classic cooperative games which provide a compact representation for coalition

formation models in multiagent systems. We consider manipulation in weighted voting games via annexation

and merging, which involves an agent or some agents misrepresenting their identities in anticipation of gaining

more power at the expense of other agents in a game. We show that annexation and merging in weighted voting

games can be more serious than as presented in the previous work. Speciﬁcally, using similar assumptions as

employed in a previous work, we show that manipulators need to do only a polynomial amount of work

to ﬁnd a much improved power gain, and then present two search-based pseudo-polynomial algorithms that

manipulators can use. We empirically evaluate our search-based method for annexation and merging. Our

method is shown to achieve signiﬁcant improvement in beneﬁts for manipulating agents in several numerical

experiments. While our search-based method achieves improvement in beneﬁts of over 300% more than those

of the previous work in annexation, the improvement in beneﬁts is 28% to 45% more than those of the previous

work in merging for all the weighted voting games we considered.

1 INTRODUCTION

False-name manipulation in weighted voting games

(WVGs), which involves an agent or some agents

misrepresenting their identities in anticipation of

power increase, has been identiﬁed as a problem. This

is because the anticipated power gain by manipulat-

ing agents is at the expense of other agents in the

game. The menace can take different forms. With an-

nexation, an agent, termed, an annexer, takes over the

voting weights of some agents in a game. Power is not

shared with the annexed agents. Forming an alliance

or merging involves voluntary merging of weights by

two or more agents to form a single bloc (Machover

and Felsenthal, 2002; Aziz et al., 2011; Lasisi and Al-

lan, 2011). Merged agents expect to be compensated

with their share of the power gained by the bloc. The

agents whose voting weights are taken over or merged

into a bloc are referred to as assimilated agents. When

agents engage in these manipulations, it becomes dif-

ﬁcult to establish or maintain trust, and more impor-

tantly it becomes difﬁcult to assure fairness in such

games.

WVGs are classic cooperative games which pro-

vide a compact representation for coalition formation

models in multiagent systems. Each agent in a WVG

has an associated weight. A subset of agents whose

total weight meets or exceeds a speciﬁed quota is

called a winning coalition. The weights of agents in a

game correspond to resources or skills available to the

agents, while the quota is the amount of resources or

skills required for a task to be accomplished. For ex-

ample, in search and rescue, robotic agents put their

resources (i.e., weights) together in large natural dis-

aster environments to reach the necessary levels (i.e.,

quota) to save life and property.

We are concerned with the ways in which agents

that complete a task are compensated from their

jointly derived payoff, taking into account each

agent’s resource (weight) contribution. The relative

power of each agent reﬂects its signiﬁcance in the

elicitation of a winning coalition. Although a larger

weight by an agent makes it more likely that an agent

can affect the outcome of a WVG, the weight of an

agent in a game is not always proportional to its power

(Aziz et al., 2011). A widely accepted method for

measuring such relative power in WVGs uses power

indices. The two best-known and most used indices

for measuring power in WVGs are Shapley-Shubik

(Shapley and Shubik, 1954) and Banzhaf (Banzhaf,

1965) power indices.

WVGs can be viewed as a form of competi-

tion among agents to share the available ﬁxed power

whose total value is always assumed to be 1. Agents

O. Lasisi R. and H. Allan V..

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES.

DOI: 10.5220/0003741300440053

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 44-53

ISBN: 978-989-8425-96-6

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

may thus resort to a form of false-name manipula-

tion (annexation or merging) to improve their inﬂu-

ence in anticipation of gaining more power. This pa-

per continues the work studied originally by (Ma-

chover and Felsenthal, 2002), (Aziz et al., 2011), and

(Lasisi and Allan, 2011) on annexation and merging

in WVGs. We extend the framework of (Lasisi and

Allan, 2011) on susceptibility of power indices to an-

nexation and merging in WVGs to consider a much

improved power gain or beneﬁt for manipulators.

Consider a WVG of n agents. The simulation of

(Lasisi and Allan, 2011) for annexation and merging

is based on a random approach where some agents,

say k < n, in the game are randomly selected to

be assimilated in annexation, and to form a volun-

tary bloc of manipulators in merging. This simple

random approach shows that on average, annexa-

tions can be effective for manipulators using both the

Shapley-Shubik and Banzhaf power indices to com-

pute agents’ power. Their results also show that merg-

ing only has a minor effect on the power gained for

manipulators using the Shapley-Shubik index, while

it is typically non-beneﬁcial (i.e., no power is gained)

for manipulators using the Banzhaf index. We note

that randomly selecting the k agents to be assimilated

for both annexation and merging this way fails to con-

sider the beneﬁts of a more strategic approach.

We show that manipulation via annexation and

merging can be more serious than as presented in

the previous work. Speciﬁcally, we show, using sim-

ilar assumptions for annexation and meging as em-

ployed in the simple random simulation of (Lasisi

and Allan, 2011), that manipulators need to do only a

polynomial amount of work to ﬁnd a much improved

power gain during manipulation. Given that the prob-

lem of computing the Shapley-Shubik and Banzhaf

power indices of agents is already NP-hard, and only

pseudo-polynomial or approximation algorithms are

available to compute agents’ power, we then present

two search-based pseudo-polynomialtime algorithms

that manipulators can use to ﬁnd a much improved

power gain. Furthermore, for reasons of efﬁciency, we

do not implement the two algorithms exactly. Rather,

we employ informed heuristic search strategies to

complement the performance of the algorithms, while

taking into consideration the two power indices in the

design of the heuristics.

We empirically evaluate our search-based method

for annexation and merging. Our method is shown

to achieve signiﬁcant improvement in beneﬁts over

previous work for manipulating agents in several nu-

merical experiments. Thus, unlike the simple random

simulation of (Lasisi and Allan, 2011) where merging

has little or no beneﬁts for manipulators using both

the Shapley-Shubik and Banzhaf indices, results from

our experiments suggest that manipulation via merg-

ing can be highly effective for manipulators. The sim-

ple random approach to manipulation via annexation

and merging seems unintelligent, thus, it is imprac-

tical that strategic agents would be keen in employ-

ing such method. In view of this, we modify the sim-

ple random approach to select the best power gain or

beneﬁt from three random choices (which we refer to

as best-of-three) and compare with our search-based

method. We note that this simple modiﬁcation pro-

vides higher average beneﬁts to the manipulators than

those of the simple random approach.

The remainder of the paper is organized as fol-

lows. Section 2 provides some preliminaries. Section

3 demonstrates examples of annexation and merg-

ing in WVGs. We present our search-based approach

to annexation and merging in Section 4. In Section

5, we consider informed heuristic search strategies

to complement the performance of the search-based

method. In Section 6, we present results of empirical

evaluation of our search-based method. Section 7 dis-

cusses related work. We conclude in Section 8.

2 PRELIMINARIES

2.1 Weighted Voting Games

Let I = {1, ··· , n} be a set of n agents and the cor-

responding positive weights of the agents be w =

, · ·· , w

}. Let a coalition S ⊆ I be a non-empty

subset of agents. A WVG G with quota q involving

agents I is represented as G = [w

, · ·· , w

;q]. Denote

by w(S), the weight of a coalition, S, derived as the

summation of the weights of agents in S, i.e., w(S) =

∑

j∈S

. A coalition, S, wins in game G if w(S) ≥ q,

otherwise it loses. WVGs belong to the class of simple

voting games. In simple voting games, each coalition,

S, has an associated function v : S → {0, 1}. The value

1 implies a win for S and 0 implies a loss. So, v(S) = 1

if w(S) ≥ q and 0 otherwise.

2.2 Power Indices

We provide brief descriptions of the two power

indices we use in computing agents’ power in

WVGs. For further discussion, we refer the reader to

(Felsenthal and Machover, 1998; Laruelle, 1999).

Shapley-Shubik Power Index

The Shapley-Shubik index quantiﬁes the marginal

contribution of an agent to the grand coalition (i.e.,

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES

a coalition of all the agents). Each permutation (or or-

dering) of the agents is considered. We term an agent

to be pivotal in a permutation if the agents preceding

it do not form a winning coalition, but by including

this agent, a winning coalition is formed. Shapley-

Shubik index assigns power to each agent based on

the proportion of times it is pivotal in all permuta-

tions. We specify the computation of the power in-

dex using notation of (Bachrach et al., 2010). Denote

by π a permutation of the agents, so π : {1, . . . , n} →

{1, . . . , n}, and by Π the set of all possible permuta-

tions. Denote by S

(i) the predecessors of agent i in

π, i.e., S

(i) = { j : π( j) < π(i)}. The Shapley-Shubik

index, ϕ

(G), for each agent i in a WVG G is

(G) =

∑

π∈Π

[v(S

(i) ∪ {i}) − v(S

(i))]. (1)

Banzhaf Power Index

The Banzhaf power index bases power on an agent

being able to turn a losing coalition into a winning

coalition by its vote. An agent i ∈ S is referred to as

being critical in a winning coalition, S, if w(S) ≥ q

and w(S\{i}) < q. The Banzhaf power index compu-

tation for an agent i is the proportion of times i is crit-

ical compared to the total number of times any agent

in the game is critical. The Banzhaf index, β

(G), for

each agent i in a WVG G is given by

(G) =

(G)

∑

j∈I

(G)

(2)

where η

(G) is the number of coalitions for which

agent i is critical in G.

2.3 Annexation and Merging

Let G be a WVG. Let Φ be any of Shapley-Shubik or

Banzhaf power indices. We denote the power index of

an agent i in G by Φ

(G). Also, consider a coalition

S ⊆ I, we denote by &S a bloc of assimilated voters

formed by agents in S. We say that a power index

Φ is susceptible to manipulation whenever a WVG

G is altered by an agent i (in the case of annexation

or some agents in the case of merging) and such that

there exists a new game G

′

where Φ

′

) > Φ

(G). In

other words, Φ is susceptible to manipulation when

the power index of the agent in the altered game is

more than its power index in the original game.

Deﬁnition 1. (Manipulation by Annexation).

Let agent i alter game G by annexing a coalition S

(i assimilates the agents in S to form a bloc &(S ∪

{i})). We say that Φ is susceptible to manipula-

tion via annexation if there exists a new game G

′

such that Φ

&(S∪{i})

′

) > Φ

(G); the annexation is

termed advantageous. The factor of increment by

which the annexer gains is given by

&(S∪{i})

′

)

(G)

. If

&(S∪{i})

′

) < Φ

(G), then the annexation is disad-

vantageous.

Deﬁnition 2. (Manipulation by Merging).

Let a manipulators’ coalition, S, alter G by merging

into a bloc &S. We say that Φ is susceptible to ma-

nipulation via merging if there exists a new game

′

such that Φ

′

) >

∑

j∈S

(G); the merging

is termed advantageous. The factor of increment by

which the manipulators gain is given by

′

)

∑

j∈S

(G)

. If

′

) <

∑

j∈S

(G), then the merging is disad-

vantageous. The agents in a bloc formed by merging

are assumed to be working cooperatively and have

transferable utility. For the sake of simplicity in our

analysis, we also refer to the factor of increment as

power gain or beneﬁt.

3 EXAMPLES OF ANNEXATION

& MERGING IN WVGs

We provide examples to illustrate annexation and

merging in WVGs. We have used Banzhaf power in-

dex as a reference for these examples. The annexer

and assimilated agents are all shown in bold.

Example 1. (Manipulation by Annexation).

Let G = [12, 16, 18, 19, 23, 26, 43, 46, 50;195] be a

WVG. The power index of agent 1 with weight 12

is β

(G) = 0.026. Suppose the agent annexes agents

3 and 4 with weights 18 and 19. An assimilated

bloc of weight 49 is formed in the new game G

′

[49, 16, 23, 26, 43, 46, 50;195]. The new power index

of the annexer β

′

) = 0.177 > β

(G). The agent

gains from the annexation and increases its power in-

dex by a factor of

0.177

0.026

= 6.81.

Example 2. (Manipulation by Merging).

Let G = [12, 16, 18, 19, 23, 26, 33, 40, 45;155] be a

WVG. The last four agents in the game are des-

ignated as would-be manipulators. The Banzhaf

power indices of these agents are: β

(G) = 0.116,

(G) = 0.142, β

(G) = 0.174, and β

(G) =

0.200. So,

∑

j=6

(G) = 0.632. Suppose the agents

decide to merge their weights. A merged bloc of

weight 144 is formed in the new game G

′

[12, 16, 18, 19, 23, 144;155]. The power index of the

bloc β

′

) = 0.861 > 0.632. The manipulators gain

ICAART 2012 - International Conference on Agents and Artificial Intelligence

from the merging and increase their power indices by

a factor of

0.861

0.632

= 1.36.

There exist examples where the two forms of ma-

nipulation may not be beneﬁcial using the two power

indices. However, (Machover and Felsenthal, 2002)

haveshownthat, in the case of annexation, it is always

beneﬁcial for an annexerto assimilate other agents us-

ing Shapley-Shubik power index.

4 SEARCH-BASED APPROACH

TO ANNEXATION & MERGING

4.1 Overview

As noted in the introduction, randomly selecting

agents to be assimilated in annexation, or to form a

voluntary bloc in merging fails to consider the ben-

eﬁts of a more strategic approach to false-name ma-

nipulation. In this section, we extend the simple ran-

dom simulation of (Lasisi and Allan, 2011) for annex-

ation and merging in WVGs. We propose a search-

based approach for the two forms of manipulation

(annexation and merging) using the Shapley-Shubik

and Banzhaf indices to compute agents’ power.

In considering our search-based approach to an-

nexation and merging, we have implemented two

pseudo-polynomial manipulation algorithms, one for

each form of manipulation. To begin with, we recall

that the problem of calculating the Shapley-Shubik in-

dices and Banzhaf indices for WVGs is NP-hard, and

both admit pseudo-polynomial algorithms using dy-

namic programming (Matsui and Matsui, 2000; Mat-

sui and Matsui, 2001) or generating functions (Brams

and Affuso, 1976; Bilbao et al., 2000) assuming the

weights of agents in the games are polynomial in the

number of agents.

Given that computing the two power indices is

already NP-hard, and only pseudo-polynomial or

approximation algorithms are available to compute

agents’ power, it is reasonable that the manipulation

algorithms we propose are also pseudo-polynomial

since we necessarily need to use these power in-

dices in computing agents’ beneﬁts during manipu-

lation. (Aziz et al., 2011) have also shown that deter-

mining if there exists a beneﬁcial merge is NP-hard

using either the Shapley-Shubik or Banzhaf power in-

dices. The same is true for determining the existence

of beneﬁcial annexation using the Banzhaf index.

4.2 Manipulation Algorithm for

Merging

The brute force approach to determine a coalition

that yields the most improved beneﬁt in merging in

a WVG is to simply enumerate all the possible coali-

tions of agents in the game and compute for each of

these coalitions its beneﬁt. We can then output the

coalition with the highest value. Unfortunately, enu-

merating all the possible coalitions is exponential in

the number of agents. Also, computing the power in-

dices (to determine the factor of increment of each

coalition) naivelyfrom their deﬁnitions means that we

have two exponential time problems to solve. We pro-

vide an alternative approach.

Let procedure PowerIndex(G, i) be a pseudo-

polynomial algorithm for computing the power in-

dex of an agent i in a WVG G of n agents for any

of Shapley-Shubik and Banzhaf power indices ac-

cording to (Matsui and Matsui, 2000). We ﬁrst use

PowerIndex(G, i) as a subroutine in the construc-

tion of a procedure, GetMergeBeneﬁt(G,S). Proce-

dure GetMergeBeneﬁt(G, S) accepts a WVG G and

a would-be manipulators’ coalition, S. It ﬁrst com-

putes the sum of the individual power index of the

assimilated agents in S using PowerIndex(G,i). Then,

it alters G by replacing the sum of the weights of

the assimilated agents in G with a single weight in

a new game G

′

before computing the power of the

bloc &S in G

′

. Finally, GetMergeBeneﬁt(G, S) returns

the factor of increment of the merged bloc &S. Let

A(G) be the pseudo-polynomialrunning time of Pow-

erIndex(G, i). Now, since |S| ≤ |I| = n, procedure

GetMergeBeneﬁt(G,S) takes at most O(n·A(G)) time

which is pseudo-polynomial.

We now use GetMergeBeneﬁt(G, S) to construct

an algorithm that manipulators can use to determine

a coalition that yields a good beneﬁt in merging. We

ﬁrst argue that manipulators tend to prefer coalitions

which are small in size because they are easier to form

and manage. Also, intra-coalition coordination, com-

munication, and other overheads increase with coali-

tion size. Thus, we suggest a limit on the size of the

manipulators’ coalitions since it is unrealistic and im-

practical that all agents in a WVG will belong to the

manipulators’ coalition. This is also consistent with

the assumptions of the previous work on annexation

and merging (Aziz et al., 2011; Lasisi and Allan,

2011). We note, however, that limiting the manipu-

lators’ coalitions size this way does not change the

complexity class of the problem as ﬁnding the coali-

tion that yields the most improved beneﬁt remains

NP-hard even with such limitation.

Consider a WVG of n agents. Suppose the manip-

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES

ulators’ coalitions have a limit, k < n, on the size of

the members of the coalitions, i.e., the manipulators’

coalitions, S, are bounded as 2 ≤ |S| ≤ k. In this case,

the number of coalitions that the manipulators need

to examine is at most O(n

) which is polynomial in

n. Speciﬁcally, the total number of these coalitions is:









+ · ·· +





∑

j=2





. (3)

So, we have

∑

j=2





∑

j=2

n(n− 1)···(n− j + 1)

≤

∑

j=2

≤

∑

j=2

j−1

+ · ·· +

k−1

= O(n

Running GetMergeBeneﬁt(G,S) while updating

the most

improved beneﬁt found so far from each

of these coalitions requires a total running time of

O(n

· A(G)) which is pseudo-polynomial time, and

thus becomes reasonable to compute.

4.3 Manipulation Algorithm for

Annexation

Our pseudo-polynomial manipulation algorithm for

annexation provides a basic modiﬁcation of the merge

algorithm above. Speciﬁcally, we ﬁrst replace the

procedure GetMergeBeneﬁt(G, S) with another pro-

cedure, GetAnnexationBeneﬁt(G, i, S). The procedure

GetAnnexationBeneﬁt(G, i, S) accepts a WVG G, an

annexer, i, and a coalition S to be assimilated by i. The

procedure then returns the factor of increment or ben-

eﬁt of the assimilated bloc &(S∪ {i}).

Again, we use GetAnnexationBeneﬁt(G, i, S) to

construct an algorithm that the annexer can use to de-

termine the coalition that yields the most improved

beneﬁt in annexation. The method of construction of

the algorithm is the same as that of the previous ma-

nipulation algorithm for merging with the exception

that we add the weight of an annexer i to the weight

of each coalition S and compare the power index

&(S∪{i})

′

) of the assimilated bloc in a new game

′

to the power index Φ

(G) of the annexer in the

We refer to the most improved beneﬁt among the O(n

)

polynomial coalitions and not from the original 2

coali-

tions since we have restricted each manipulators’ coalition

size to a constant k < n.

original game G. The annexer examines a polynomial

number of coalitions of the agents assuming a limit

k < n on the size of each coalition. Since any of the n

agents can be an annexer and the annexer will belong

to any of the coalitions it annexes, the total number of

coalitions examined by all the annexers is:





n− 1





n− 1



+ ·· · +



n− 1

k− 1







k−1

∑

j=1



n− 1



. (4)

Bounding this equation using similar approach as

in Equation 3 shows that Equation 4 is O(n

). Thus,

as before, the manipulation algorithm for annexation

also runs in pseudo-polynomialtime, with a total run-

ning time of O(n

· A(G)).

5 INFORMED HEURISTIC

SEARCH STRATEGIES

To further improve the performance of the manipu-

lation algorithms, we use heuristics. In this section,

we provide descriptions of search infrastrutures and

enhancements to complement the performance of the

manipulation algorithms.

5.1 Merging Heuristics

The search space for the manipulation algorithm for

merging is the polynomial number of coalitions of

size at most k (see Equation 3) as described ear-

lier. It is important to point out that the computation of

the power indices of the manipulators in the original

game and the power index of the bloc formed by these

agents in a new game account for most of the compu-

tational time required by this algorithm. We also note

that it is unclear to the manipulators on how to deter-

mine a coalition that is beneﬁcial without having to

compute and compare the power indices of the ma-

nipulators in both games.

Since we seek to ﬁnd the coalition with the most

improved beneﬁt among these coalitions, it is not dif-

ﬁcult to see that the algorithm is prone to engaging in

irrelevant computation of power indices of agents in

the two games for coalitions whose merging are dis-

advantageous. We deﬁne evaluation criteria that we

use to prune away such coalitions without having to

compute the power indices of the blocs in the new

games, thus gaining ample computational time. The

basic idea of the evaluation criteria is to prune away

all coalitions havingtheir factor of increment less than

ICAART 2012 - International Conference on Agents and Artificial Intelligence

or equal to the estimated minimum possible factor of

increment in the altered WVGs. The criteria are de-

ﬁned for both Shapley-Shubik and Banzhaf indices.

First, consider merging using the Shapley-Shubik

index. Let G be a WVG of n agents. Let the Shapley-

Shubik power index of an agent i in a game G be

(G). Consider a manipulators’ coalition S ⊆ I with

k agents. Let agents i

, i

, . . . , i

be the distinguished k

manipulators in S that want to merge into a single bloc

&S. Let Π

n−k

be the set of all permutations of the re-

maining n−k agents in G (i.e., not including the k ma-

nipulators). Consider a certain permutation π ∈ Π

n−k

in which we insert all the k manipulators starting from

the r-th position in π (where r is any arbitrary position

in π), and such that one of the manipulators is pivotal

for π. There are k! permutations in G for π in which

the members of S occur together beginning at posi-

tion r. For example, consider a permutation π = 1, 2, 3

of other agents in G which excludes the three ma-

nipulators i

, i

, and i

. The 3! permutations in G

for π when all the manipulators appear together and

starting at the 3-rd position are: π

= 1, 2, i

, i

, 3,

= 1, 2, i

, i

, 3, π

=, ···,π

= 1, 2, i

, i

, 3.

Now, consider a permutation f(π) of agents in

the altered game G

′

obtained from π by inserting the

bloc &S (formed by agents i

∈ S) at the r-th posi-

tion in f(π). It is easy to see that the k! permutations

, · ·· , π

for π in G when the manipulators appear

together from the r-th position correspond to exactly

one permutation f(π) in G

′

. Also, since one of the

manipulators is pivotal for each of the permutations

, · ·· , π

for π in G, &S is also pivotal for the corre-

sponding f(π) in G

′

. Observe that counting the num-

ber of distinct permutations π ∈ Π

n−k

in G in which

we insert all the manipulators at certain positions and

such that one of them is pivotal for each π provides a

lower bound on the number of times the blocs formed

by these agents in G

′

will be pivotal. That it is a lower

bound is clear as the manipulators may also be pivotal

in other cases when they do not all appear together.

In our implementation, we ﬁrst count the num-

ber of times each of the manipulators is pivotal when

they all follow one another in G. We then pick the

smallest among these numbers denoted by |Π

∗

|. Now,

if S merges to form a bloc &S, then, the number of

agents in the new game G

′

is n − k + 1. We estimate

the Shapley-Shubik power index of the bloc in G

′

|Π

∗

(n−k+1)!

. We compare the estimated power index of

the bloc in G

′

to the sum of the Shapley-Shubik power

indices,

∑

i∈S

(G), of the manipulators in G. Specif-

ically, if

|Π

∗

(n−k+1)!

≤

∑

i∈S

(G), we eliminate the ma-

nipulators’ coalition S as the coalition cannot possibly

be a candidate coalition that provides the most im-

proved beneﬁt to the manipulators.

Second, consider merging using Banzhaf in-

dex. Let η

(G) be the number of coalitions for which

an agent i is critical in G. Also, denote by β

(G)

the Banzhaf power index of agent i in G. As before,

we consider a manipulators’ coalition S ⊆ I with k

agents. Let agents i

, i

, . . . , i

be the distinguished k

manipulators in S that want to merge into a single bloc

&S. Let Γ

n−k

be the set of all losing coalitions of the

remaining n − k agents in G (i.e., not including the k

manipulators). Consider a certain coalition C ∈ Γ

n−k

in which the inclusion of at least one of agents i

∈ S

makes C ∪ {i

} a winning coalition and such that at

least one of agents i

is critical for C∪ {i

}. There are

multiple such winning coalitions that can be formed

from the union of C and the subsets of S depending

on the quota of the game.

For example, let G = [23, 20, 10, 11, 15;50] be a

WVG of ﬁve agents I = {1, 2, 3, 4, 5} in order. Let

S = {3, 4, 5} be a set of manipulators. Consider a los-

ing coalition C = {1, 2} which excludes the three ma-

nipulators. There are three winning coalitions: C

C∪ {3}, C

= C ∪ {4}, and C

= C ∪ {5} that can be

formed from the union of C and the subsets of S such

that at least one agent in S is critical in the resultant

set. Note that adding two members of S to C would

yield coalitions in which no agent is critical.

Now, consider a winning coalition f(C) of agents

in the altered game G

′

obtained from the union of the

losing coalition C in game G and the bloc &S (formed

by agents i

∈ S). It is easy to see that all the winning

coalitions C

, · ·· ,C

(where m ∈ N) obtained from C

in G correspond to exactly one winning coalition f (C)

in G

′

. Also, since at least one of the manipulators is

critical for each of the winning coalitions C

, · ·· ,C

the bloc &S is also critical for the corresponding win-

ning coalition f(C) in G

′

. Observe that counting the

number of distinct losing coalitionsC ∈ Γ

n−k

in which

the inclusion of agents i

∈ S makes C ∪ {i

} a win-

ning coalition and such that at least one of agents i

is critical for C ∪ {i

} gives the number of times the

bloc formed by the manipulators in G

′

is critical.

In our implementation, we compute the sum

∑

i∈S

(G) of the number of times all the manipula-

tors are critical in G, and then compute the number

′

) of times the bloc formed by the manipula-

tors will be critical in G

′

as described above. In or-

der to estimate the power of the bloc &S in G

′

need to know the number of times for which each of

the other agents in G

′

other than the bloc is also crit-

ical. These numbers are not available. Since we al-

ready know the number of times all agents in G are

critical, we estimate the number of times for which

each agent i (other than &S) in game G

′

would be crit-

ical as η

′

) =

(G)

, where τ is deﬁned as a measure

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES

to scale down the number of times an agent is criti-

cal in G to G

′

. This is required since there are more

agents in the original game G than G

′

. The number of

coalitions for which the non manipulating agents in

G is critical is always more than the number of coali-

tions for which they are critical in G

′

. More precisely,

we estimate the scaling factor between the two games

using the following ratio τ =

∑

i∈S

(G)

′

)

. We now com-

pute the estimated Banzhaf power index of the bloc in

′

as β

′

) =

′

)

′

∑

i∈I\S

′

)

. We compare the

estimated power index of the bloc in G

′

to the sum

of the Banzhaf power indices,

∑

i∈S

(G), of the ma-

nipulators in G. Speciﬁcally, if β

′

) ≤

∑

i∈S

(G),

we prune the manipulators’ coalition S as the coalition

cannot possibly be a candidate coalition that provides

the most improved beneﬁt to the manipulators.

5.2 Annexation Heuristic

We recall the deﬁnition of annexation in Section 2

and from (Machover and Felsenthal, 2002; Aziz et al.,

2011), the power of the assimilated bloc in an altered

WVG is compared to the power of the annexer in the

original game. By this deﬁnition, intuition suggests

that annexation should always be advantageous. This

intuition is indeed true using the Shapley-Shubik in-

dex to compute agents’ power. However, there exists

situations where annexation is disadvantageous for

the annexer using the Banzhaf index. See (Machover

and Felsenthal, 2002; Aziz and Paterson, 2009; Aziz

et al., 2011) for different examples of WVGs where

annexation is disadvantageous for the annexer using

the Banzhaf index. This case where annexation results

in power decrease for the annexer is refer to as the

bloc paradox (Machover and Felsenthal, 2002). Fur-

thermore, (Aziz et al., 2011) have also shown that de-

termining whether a player can beneﬁt from annexing

a given coalition is NP-hard for the Banzhaf index.

Recall again from Equation 4 that the annexer

needs to examine only a polynomial number of as-

similated coalitions of size at most k − 1 to ﬁnd the

most improved power gain. It is also known that in

computing agents’ power index in a WVG using both

the Shapley-Shubik and Banzhaf indices, the power

index of an agent with a higher weight cannot be

less than the power index of an agent with a smaller

weight (Bachrach et al., 2010). In our case, since we

are restricting the manipulators’ coalition size to k,

the assimilated coalitions with maximal weights are

those of size k − 1. Based on this observation and

the fact due to the bloc paradox as discussed above,

it is enough to check only the assimilated coalitions

of size exactly k − 1 in order for an annexer to ﬁnd

the coalition with the most improved beneﬁt using the

two power indices. There are only





n− 1

k− 1



such assimilated coalitions to be considered when the

n agents act as an annexer in turn.

6 EXPERIMENTAL RESULTS

We have studied the performance of the two ma-

nipulation algorithms. As noted in the introduction,

the simple random approach to manipulation via an-

nexation and merging seems unintelligent. Thus, it

is impractical that strategic agents would employ

such method. We make a simple modiﬁcation to this

method which provides manipulators with higher av-

erage factor of increment. The modiﬁcation involves

the selection of the best factor of increment from three

random choices (which we refer to as the best-of-three

method). We compare the results of our search-based

method with those of the simple random and best-of-

three methods. However, for clarity of presentation,

we show our results compared with only those of the

best-of-three method.

We randomly generate WVGs. The weights of

agents in each game are randomly chosen so that

all weights are integers and drawn from a uni-

form distribution over the range [1,W], where W ∈

{10, 20, 30, 40, 50}. We have chosen different weight

distributions in order to provide some generalization

of the performanceof the two methods under different

conditions. We run two different set of tests in which

the number of agents, n, in each of the original WVGs

is either 10 or 20 while the number of assimilated

agents, k, is chosen to be either 5 or 10. When creat-

ing a new game, the quota, q, of the game is randomly

generated such that

w(I) < q ≤ w(I), where w(I) is

the sum of the weights of all agents in the game.

Using the manipulation algorithms, the simple

random and the best-of-three methods, the power in-

dex of an assimilated bloc formed by annexation in

an altered game is compared to the power index of

the annexer in the original game. Similarly, the power

index of the assimilated bloc formed by merging is

compared to the sum of the original power indices of

the agents in the merged bloc. The factor of increment

(decrement) by which the annexer (or the merged

bloc) gains (loses) in the annexation (or merging) is

computed. We repeat each experiment 100 times and

compute the average factor of increment.

6.1 Results for Merging

Figure 1 shows the beneﬁts from merging for both the

ICAART 2012 - International Conference on Agents and Artificial Intelligence

Figure 1: The average factor of increment for merging for the search-based and best-of-three methods using different agents’

weights distributions. (a) n = 10 and k = 5 (b) n = 10 and k = 10 (c) n = 20 and k = 5 (d) n = 20 and k = 10.

best-of-three method and our search-based approach

for various values of n, k, and W using the two power

indices. The x-axis indicates the weight distributions

of agents while the y-axis is the average factor of in-

crement achieved by manipulating agents. The error

bars in this and the subsequent ﬁgure indicate 5% er-

ror amounts in the average factor of increment.

We found from the data of Figures 1(a)-(d) that

our search-based method achieves improvement in

beneﬁts of 28% to 45% more than those of the best-

of-three method in merging for various values of n, k,

and W, and for the two power indices. Since this per-

centage increment of the search-based approach over

the best-of-three method can be achieved with only a

polynomial amount of work, then, manipulators are

more likely to seek a much improved power gain in

merging using the search-based approach.

6.2 Results for Annexation

Figure 2 shows the average factor of increment from

annexation for both the best-of-three method and the

search-based approach for various values of n, k, and

W using the two power indices.

It is clear from the ﬁgure that the average factor

of increment found by the search-based approach for

the two power indices are higher than those of the

corresponding power indices using the best-of-three

method. Speciﬁcally, we found from the data of Fig-

ures 2(a)-(d) that the search-based method achieves

improvement in beneﬁts of over 300% more than

those of the best-of-three method in annexation for

various values of n, k, and W, and for the two power

indices. Again, this percentage increment of the

search-based approach over the best-of-three method

can be achieved with only a polynomial amount of

work. Thus, we conclude that manipulation via merg-

ing and annexation is more serious than was presented

in the simple random simulation of the previous work.

7 RELATED WORK

Weighted voting games and power indices are widely

studied (Brams, 1975; Felsenthal and Machover,

1998; Laruelle, 1999). Prominent real-life situations

where WVGs have found applications include the

United Nations Security Council, the International

Monetary Fund (Leech, 2002; Alonso-Meijide and

Bowles, 2005), the Council of Ministers, and the Eu-

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES

Figure 2: The average factor of increment for annexation for the search-based and best-of-three methods using different

agents’ weights distributions. (a) n = 10 and k = 5 (b) n = 10 and k = 10 (c) n = 20 and k = 5 (d) n = 20 and k = 10.

ropean Community (Felsenthal and Machover, 1998).

The need to compensate agents from jointly de-

rived payoff in WVGs has also necessitated the as-

signment of power to players. A widely accepted

method for measuring power of agents in WVGs

uses power indices. Fairness in the assignment of

power to players in a game is also a concern of most

of the power indices. The two most prominent and

widely used power indices are Shapley-Shubik(Shap-

ley and Shubik, 1954) and Banzhaf (Banzhaf, 1965)

power indices. Other power indices found in the lit-

erature include Deegan-Packel (Deegan and Packel,

1978), Johnsoton (Johnston, 1978), and Holler-Packel

(Holler and Packel, 1983) power indices.

Computing the Shapley-Shubik and Banzhaf

power indices of players in WVGs is NP-hard (Mat-

sui and Matsui, 2001). The power indices of voters

using any of Shapley-Shubik and Banzhaf power in-

dices can be computed in pseudo-polynomial time

using dynamic programming (Matsui and Matsui,

2000). Efﬁcient exact algorithms using generating

functions (Brams and Affuso, 1976; Bilbao et al.,

2000) also exist for both the Shapley-Shubik and

Banzhaf power indices for WVGs where the weights

of all agents are restricted to integers. There are

also approximation algorithms (Fatima et al., 2007;

Bachrach et al., 2010) for computing the Shapley-

Shubik and Banzhaf power indices in WVGs.

We now consider false-name manipulation via

annexation and merging in WVGs. (Machover and

Felsenthal, 2002) originally studied annexation and

alliance (or merging) in WVGs. They consider when

the blocs formed by annexation or merging are ad-

vantageous or disadvantageous. They show that using

the Shapley-Shubikpower index, it is always advanta-

geous for a player to annex some other players in the

game. However, this is not true for Banzhaf power

index. Furthermore, they show that merging can be

advantageous or disadvantageous for the two power

indices. In contrast to our work, they do not consider

the extent to which the agents involved in annexation

or merging may gain, which we study in this paper.

(Aziz et al., 2011) have also considered the com-

putational aspects of the problem of annexation and

merging in WVGs. They show that determining if

there exists a beneﬁcial merge in a WVG is NP-hard

using both Shapley-Shubik and Banzhaf indices. The

same is also true for determining the existence of ben-

eﬁcial annexation using the Banzhaf index. Our work

differ from that of these authors as we provide a com-

ICAART 2012 - International Conference on Agents and Artificial Intelligence

parison of the extent of power gain or beneﬁts that are

possible for manipulating agents in a restricted ver-

sion of this problem using the two indices.

8 CONCLUSIONS

We extend the simple random simulation of (Lasisi

and Allan, 2011) on susceptibility of power indices

to annexation and merging in WVGs to consider a

much improved beneﬁt achievable by manipulating

agents. Using similar assumptions for annexation and

meging as employed in the simulation of (Lasisi and

Allan, 2011), we show that manipulators need to do

only a polynomial amount of work to ﬁnd a much

improved beneﬁt and then present two search-based

pseudo-polynomial manipulation algorithms that ma-

nipulators can use.

We provide a modiﬁed version of the simple ran-

dom approach that considers the best beneﬁt from

three random choices (which we refer to as the

best-of-three method) that we compare results of

our search-based approach with. Experimental results

show that our search-based method achievesimprove-

ment in beneﬁts of over 300% more than those of the

best-of-three approach in annexation, while the im-

provement in beneﬁts is 28% to 45% more than the

best-of-three method in merging for all the WVGs

we considered. We conclude that since this percent-

age increment of the search-based approach over the

best-of-three method for both annexation and merg-

ing can be achievedwith only a polynomial amount of

work, and using pseudo-polynomial algorithms, then,

manipulators are more likely to seek for a much im-

proved power gain when faced with annexation and

merging in WVGs. Thus, we advance the state of the

art by showing that annexation and merging can be

more serious than as presented in the previous work.

ACKNOWLEDGEMENTS

This work is supported by NSF research grant

#0812039 entitled “Coalition Formation with Agent

Leadership”.

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A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES