Coalitional Power Indices Applied to Voting Systems

Xavier Molinero

1

and Joan Blasco

2

1

Mathematics Department, Universitat Polit

`

ecnica de Catalunya·BarcelonTech,

Barcelona Graduate School of Mathematics, Spain

2

Barcelona School of Informatics, Universitat Polit

`

ecnica de Catalunya·BarcelonTech, Spain

xavier.molinero@upc.edu, joan.blasco@est.ﬁb.upc.edu

Keywords:

Simple Games, Coalitional Power Indices, Voting Systems.

Abstract:

We describe voting mechanisms to study voting systems. The classical power indices applied to simple games

just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions

among parties, players or voters. We measure the power of each party, player or voter when there are coali-

tions among them. In particular, we study real situations of voting systems using extended Shapley–Shubik

and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to

compute them.

1 INTRODUCTION

Classical cooperative games provide mathematical

tools useful to study situations of conﬂict of inter-

est and cooperation arising from the real world. A

ﬁeld where cooperative games have been frequently

applied is political science. By using simple games,

collective decision–making mechanisms ruled by vot-

ing have often been described and analyzed (Carreras,

2004). The value theory (one of the main streams of

the cooperative game theory) has given rise, by re-

stricting it to simple games, to a “power index the-

ory”. The most conspicuous representatives of this

line of research are the Shapley–Shubik (Shapley and

Shubik, 1954) and the Banzhaf-Coleman (Penrose,

1946; Banzhaf III, 1964; Coleman, 1971; Owen,

1975; Moshe’Machover Dan, 1998) power indices. A

survey about power indices can be found in (Freixas,

2010).

However, it seems that this framework does not

sufﬁce to analyze all aspects of voting. Following

early initial papers by Lucas and Thrall (Thrall and

Lucas, 1963) and Myerson (Myerson, 1977) who,

respectively, generalized cooperative games in char-

acteristic function form to “games in partition func-

tion form” and extended the Shapley value (Shap-

ley, 1953) to this new class of games, Bolger (Bol-

ger, 1983) used a Banzhaf–type power index for the

so–called “multicandidate voting games”. Bolger

(Bolger, 1993) also deﬁned and axiomatically char-

acterized a new extension of the Shapley value rela-

tively to each alternative. We agree that this relativity

is an essential feature of any evaluation of games with

alternatives.

In classical simple games it is implicitly assumed

that players can join them to form coalitions or strate-

gies (Holler and Owen, 2013; Carreras and Owen,

1988; Owen, 1977a; Fagen, 1963). The goal of this

paper is to show the wide possibilities of games with

coalitions among players to represent real voting sys-

tems. We consider real situations and we study the ex-

act role of each player. The analysis is done with the

classical Shapley–Shubik and Banzhaf power indices

with coalitions. Note that some deﬁnitions and con-

cepts have been gotten from previous works as (Car-

reras and Maga

˜

na, 2008; Nieto, 1996), among others.

2 CLASSICAL COALITIONAL

POWER INDICES FOR SIMPLE

GAMES

This section considers Shapley-Shubik and Banzhaf

power indices, as well as coalitional Shapley-Shubik

and Banzhaf power indices. Next we introduce some

references and deﬁnitions related with them.

Simple games can be viewed as models of voting

systems in which a single alternative, such as a bill or

an amendment, is pitted against the status quo.

Deﬁnition 1 ((Freixas and Molinero, 2009)). A sim-

ple game is a pair (N,v) in which N = {1,2,...,n} and

v(S) = 1 or 0, where S in a subset of N, that satisﬁes:

(1) v(N) = 1, (2) v(

/

0) = 0 and (3) the monotonicity

property: v(s) = 1 and S ⊆ T ⊆ N implies v(T ) = 1 .

So far, a coalition has represented a set of agents

that worked on its own. In a coalition structure (CS),

the different coalitions are intended to work indepen-

dently of each other. We can also interpret a coali-

tion to represent a group of agent that is more likely

to work together within a larger group of agents (be-

cause of personal or political afﬁnities).The members

of a coalition do not mind working with other agents,

but they want to be together and negotiate their payoff

together, which may improve their bargaining power.

This is the idea used in games with a priori unions.

Formally, a game with a priori unions is similar to a

game with CS: it consists of a triplet (N,v,B) where

(N,v) is a simple game

1

and B = {B

1

,B

2

,...,B

m

} is a

CS deﬁned over the set of players, N.

Based on Shapley’s value, Owen characterizes ax-

iomatically a new value, now known as Owen’s coali-

tional value or simply coalitional value, which reﬂects

the different possibilities of each player depending on

the subset of the coalition structure to which he be-

longs. In the development of this idea, the possibility

that the different blocks of the coalition structure co-

operate with each other is not excluded. On the con-

trary, it is considered that each block chooses a rep-

resentative, k for the B

k

block, and these are the ones

that negotiate (play). If M = {1,2,...,m} is the set

formed by the representatives of the blocks of B, then

the coalitional value of a player i ∈ B

k

is:

Deﬁnition 2 ((Owen, 1977b)). The Owen’s coali-

tional value is

φ

i

[v;B] =

∑

S⊆B

k

∑

T ⊆M

(t − 1)!(m −t)!

m!

·

(s − 1)!(b

k

− s)!

b

k

!

·

[v

S

(T ) − v

S

(T \ {i})]

(1)

where t = |T |, m = |M|, s = |S| and b

k

= |B

k

|.

Later, Owen also deﬁned the corresponding coali-

tional value for Banzhaf power index as follows.

Deﬁnition 3 ((Owen, 1981)). The Banzhaf-Owen’s

coalitional value is

ψ

i

[v;B] =

∑

S⊆B

k

∑

T ⊆M

1

2

m−1

·

1

2

b

k

−1

·

[v

S

(T ) − v

S

(T \ {i})]

(2)

where t = |T |, m = |M|, s = |S| and b

k

= |B

k

|.

Note that Owen’s coalitional and Banzhaf-Owen’s

coalitional power indices are also called coalitional

Shapley-Shubik and coalitional Banzhaf power in-

dices, respectively.

1

Note that it also works for cooperative games (see (Tay-

lor and Zwicker, 1999)).

3 THE IMPLEMENTATION OF

CLASSICAL COALITIONAL

POWER INDICES

Using all the information from the previous section,

Formulas 1 and 2, we have done a program that

computes coalitional Shapley-Shubik and coalitional

Banzhaf power indexes.

The basic case for this formula is when every

party is in a coalition by itself, i.e., B

k

= {k} for

1 ≤ k ≤ n = m. Then, the formula is the same as the

one for simple games without coalitions. The algo-

rithm uses backtracking to get the 2

b

k

+m−2

different

combinations. The function will have b

k

+ m − 2 lev-

els but this time the options are between getting that

player or ignoring it.

We divided our function in two parts.

The ﬁrst one chooses all combinations with b

k

players inside the coalition B

k

in which our player i

belongs to, i.e., i ∈ B

k

. See Figure 1.

Figure 1: Combinations of players inside of B

k

.

Afterwards, we call the 2

nd

function. This function

considers all combinations with m indices inside M

including the index j ∈ M which our player i belongs

to, i.e., i ∈ B

j

. See Figure 2.

Figure 2: Combinations of indices inside of M.

3.1 Dynamic Programming

In this section we just consider the so-called weighted

majority games, a subclass of simple games deﬁned as

follows.

Deﬁnition 4 ((Freixas and Molinero, 2009)).

A weighted majority game is a simple game

(N,v), where there exists a representation

(q,w = {w

1

,...,w

n

}) such that v(S) = 1 if and

only if w(S) ≥ q, where w(S) =

∑

i∈S

w

i

.

There is an algorithm that uses dynamic program-

ming to calculate the Shapley-Shubik and Banzhaf in-

dices, for weighted majority games, with a time com-

plexity of O(n

2

q) and space requirement of O(nq) for

a deﬁned quota q (Lucas, 1983; Brams and Affuso,

1976).

We extended the previous dynamic algorithm in-

troduced for simple games (Lucas, 1983; Brams and

Affuso, 1976) to one that can use coalitions.

Our ﬁrst step in our algorithm is to reshape our n

players divided in m coalitions into n

0

players that will

serve as our input.

Those n

0

players will be composed for the b

k

− 1

players in our coalition and the m − 1 coalitions that

we will transform into players c

i

with the weight of

the whole coalition.

N

0

= {p

1

,..., p

b

k

−1

} ∪ {c

1

,...,c

m−1

} (3)

In which:

p(S) = |V|,V ⊆ S, ∀x ∈ V, x ∈ B

k

(4)

Then we will make a partition of N

0

players into sub-

sets N

0

1

,N

0

2

,...,N

0

z

satisfying that:

1. N

0

1

∪ N

0

2

∪ ...∪ N

0

z

= N

0

,N

0

x

∩ N

0

y

=

/

0(x 6= y),

2. For all x,y : 1 ≤ x < y ≤ z, ∀i ∈ N

0

x

, ∀ j ∈ N

0

y

, w

i

>

w

j

,

3. For all x : 1 ≤ x ≤ z, ∀i, j ∈ N

0

x

,w

i

= w

j

.

For every player i, ˆc

i

(w,t, s, x) denotes the number

#{S ⊆ N

0

− {i} : w(S) = w, |S| = t, p(S) = s,

S ∩ N

0

x

6=

/

0,S ∩ N

0

x+1

= ... = S ∩ N

0

z

=

/

0}.

and, for every player i, c

i

(w,t, x) denotes the number

#{S ⊆ N

0

− {i} : w(S) = w, |S| = t, S ∩ N

0

x

6=

/

0,

S ∩ N

0

x+1

= ... = S ∩ N

0

z

=

/

0}.

Then the Owen (φ) and Banzhaf-Owen (ψ) coalitional

power indices of the player i ∈ N

y

are described as

follows,

φ

i

[v;B] =

n

0

−1

∑

t=1

q−1

∑

w=q−w

i

z

∑

x=1

b

k

∑

s=1

(t−s)!(m−(t−s)−1)!

m!

·

s!(b

k

−s−1)

b

k

!

·

ˆc

i

(w,t, s, x),

and

ψ

i

[v;B] =

1

2

m−1

·

1

2

b

k

−1

·

n−1

∑

t=1

q−1

∑

w=q−w

i

z

∑

x=1

c

i

(w,t, x),

respectively.

Note that v( j) is the number of combinations get-

ting j elements from the s

0

elements, times, the com-

binations getting y − j elements from y

0

− s

0

.

Then we can get an algorithm with time complex-

ity of O((n

0

b

k

)

2

q) and space requirement of O(n

0

q).

This algorithm improves the time to calculate our

index when the number of players grows but it be-

comes slower when the quota is too big. To reduce

the quota we can try to apply Greatest Common Divi-

sor to the weight of our players and the quota.

Nevertheless the biggest parliament in the world is

China’s National People’s Congress with 2890 seats

in which the biggest party controls the 71.1% of the

seats. This parliament is followed by the UK Upper

House with 793 and the European Parliament with

751 seats. So, in the worst case scenario where China

needs a quota of 75% and there will be n players in

B

k

our algorithm will have a time cost of O(1987n

3

).

That means it will start to be faster than the exponen-

tial version for n > 24. See Figure 3.

Figure 3: Complexity of the standard (O(2

n

)) and the dy-

namic programming (O(1987n

3

)) algorithms.

For the parliament of the UK Upper House, the aver-

age parliament has 300 seats, with a quota of 75% and

half of the players in the coalition, our algorithm will

have a time cost of O(100n

3

). So our algorithm will

act faster than the exponential version for n > 19 .See

Figure 4.

Figure 4: Complexity of the standard (O(2

n

)) and the dy-

namic programming (O(100n

3

)) algorithms.

4 REAL VOTING SYSTEMS

To success a game (the approval or defeat of a mo-

tion), a number of players need to agree in the same

decision to reach a certain quorum amount of votes.

Here we mention three different scenarios.

• Voting a motion that requires a qualiﬁed majority

of 2/3 of the players. This kind of motion is nec-

essary to reform the constitution in the Spanish

parliament.

• Voting a motion that requires an absolute majority

of 1/2 of the players. This motion is used for a

censure motion in the Spanish parliament.

• Voting a motion that requires a relative majority,

i..e, more votes for the motion than against it. This

is the most common form of voting and the one

in which abstention plays a different role than in

the other two cases because we do not take into

account abstention votes, i.e., we just compare the

votes in favour with the votes against.

We study now two different voting rules: Catalan Par-

liament 2019 and German Parliament 2019,

4.0.1 Catalan Parliament 2019

φ

i

[v

q

,B] is the coalitional Owen’s power index for the

player i, with quota q and structure of coalitions

B = {{JUNT SxCAT,ERC}, {C

0

s},{PSC},

{EnComu},{CUP},{PP}} ,

see Table 1, and

B = {{JUNT SxCAT,ERC,CUP},{C

0

s},{PSC},

{EnComu},{PP}} ,

see Table 2.

Table 1: Power distribution in the Catalan Parliament 2019

using coalitions B

1

= {JUNT SxCAT,ERC}.

party weight %weight φ

i

[v

2/3

,B ] φ

i

[v

1/2

,B ]

C’s 36 26.66% 21.6667 6.6667

JUNTSxCAT 34 25.19% 31.6667 33.3333

ERC 32 23.70% 38.3333 33.3333

PSC 17 12.59% 10 6.6667

EnComu 8 5.92% 5 6.6667

CUP 4 2.96% 1.6667 6.6667

PP 4 2.96% 1.6667 6.6667

In the government in catalonia 2019 it was a coalition

between JUNTSxCAT and ERC (Table 1), but if

CUP, which had similar political interest, was added

to the coalition they will have a complete majority.

Table 2: Power distribution in the Catalan Parliament 2019

using coalitions B

1

= {JUNT SxCAT,ERC,CUP}.

party weight %weight φ

i

[v

2/3

,B ] φ

i

[v

1/2

,B ]

C’s 36 26.66% 20 0

JUNTSxCAT 34 25.19% 31.6667 43.3333

ERC 32 23.70% 38.3333 43.3333

PSC 17 12.59% 11.6667 0

EnComu 8 5.92% 3.3333 0

CUP 4 2.96% 2.5 13.3333

PP 4 2.96% 3.3333 0

4.0.2 German Parliament 2019

ψ

i

[v

q

,B] denotes the coalitional Banzhaf-Owen’s

power index for the player i, with quota q and struc-

ture of coalitions B . Tables 3 and 4 analyze the Ger-

man parliament 2019 using the Banzhaf-Owen coali-

tional value for B without coalitions and

B = {{CDU, SPD,CSU },{A f D},{FDP},

{DIELINKE},{GR

¨

UNE}} ,

which is the current coalition in the government, re-

spectively.

Table 3: Power distribution in the German Parliament (Bun-

destag) 2019 without coalitions.

party weight %weight ψ

i

[v

2/3

,B ] ψ

i

[v

1/2

,B ]

CDU 200 28.21% 30.4348 29.0323

SPD 153 21.58% 21.7391 19.3548

AfD 94 13.26% 13.0435 16.129

FDP 80 11.28% 10.7696 9.6774

DIE LINKE 69 9.73% 8.6956 9.6774

GR

¨

UNE 67 9.45% 8.6956 9.6774

CSU 46 6.49% 6.5217 6.4516

Table 4: Power distribution in the German Parliament (Bun-

destag) 2019 using coalitions B

1

= {CDU,SPD,CSU}.

party weight %weight ψ

i

[v

2/3

,B ] ψ

i

[v

1/2

,B ]

CDU 200 28.21% 32.5581 52.9412

SPD 153 21.58% 23.2558 35.2941

AfD 94 13.26% 13.9535 0

FDP 80 11.28% 13.9535 0

DIE LINKE 69 9.73% 4.6512 0

GR

¨

UNE 67 9.45% 4.6512 0

CSU 46 6.49% 6.9767 11.7647

Note that if we take into account the coalition that

forms the government, the power of each party

changes a lot, specially if that coalition has absolute

majority.

5 CONCLUSIONS AND FUTURE

WORK

We have computed some speciﬁc measures to es-

tablish the power that each party in a parliament

holds, taking special attention when coalitions ap-

pears. In this vein, we have implemented the exten-

sion of Shapley-Shubik and Banzhaf power indices

with coalitions. We have also studied more real vot-

ing systems in deep.

For our future work we are also planning on ex-

tending the considered deﬁnitions and algorithms. It

could express multiple alternatives (as abstention) be-

tween input options, voting likeness (two parties with

opposed ideologies are less likely to vote the same),

and try to put both concepts together.

We also plan to improve the complexity of our

algorithm (by the way, all of them are NP-Hard) us-

ing approximation methods for computing power in-

dices (Fatima et al., 2012; Bachrach et al., 2010) and

another methods (Alonso-Meijide and Bowles, 2005).

ACKNOWLEDGEMENTS

X. Molinero has been partially supported by funds

from the Spanish Ministry of Economy and Com-

petitiveness (MINECO) and the European Union

(FEDER funds) under grants MTM2015-66818-P

(VOTA-COOP) and MDM-2014-044 (BGSMath).

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