New Issues
in Voting Power Analysis
1
František Turnovec2
Charles University, Prague, Center for Economic Research and Graduate Education,
& Economics Institute of the Academy of Sciences of the Czech Republic,
Draft August 12, 1999
Abstract: The paper introduces concept of power indices with constraints on
coalition formation, so called constrained power indices. It is shown in the
paper that some of most frequently used power indices based on extremely
simple model of weighted voting game can be extended for more
sophisticated voting models of voting situations.
Keywords: A priori unions, committee, conflicting configuration structure,
constrained power indices, spatial power indices
This research was undertaken with support from the Grant Agency of the Academy of
Sciences of the Czech Republic, project No. A8085901. Presented at the 15th Triennial
Conference IFORS '99, August 16-20, 1999, Beijing. Author's participation at the conference
was cosponsored by World Bank's Development Grant Facility as a part of the FY98
program on Regional Research Consortia.
1
2
Address: CERGE UK, Politických vězňů 7, 111 21 Prague 1, Czech Republic, tel.: (+420 2) 24
00 52 11, fax: (+420 2) 242 27 143, e-mail: [email protected].
1
1. INTRODUCTION
Measuring of voting power in committee systems was introduced by Penrose
[1946] and Shapley and Shubik [1954] more than 50 years ago. Recently we
can observe growing interest to different aspects of analysis of distribution
and concentration of power in committees, related to development of new
democratic structures in countries in transition as well as to institutional
reform in European Union. Both methodological and applied problems of
analysis of power are being intensively discussed during last few years (see
e.g. recent discussion in Freixas and Gambarelli [1997], Holler [1997],
Mercik [1997], Nurmi [1997], Sosnowska [1997], Turnovec [1997]).
This paper presents an attempt to clarify some problems of interpretation of
different concepts of power indices introduced and used in analysis of
different committee systems.
A model of a committee in terms of quotas and allocations of weights of
committee members is used instead of traditional model of cooperative
simple games. A subset of the set of all committee members is called a
winning voting configuration if the total weight of all its members is at least
equal to the quota. The power index is defined as a mapping from the space
of all committees into the unit simplex, representing a reasonable
expectation of the share of voting power given by ability to contribute to
formation of winning voting configurations.
Considering the fact that in different real situations not all formally possible
voting configurations matter, a set of feasible voting configurations is
introduced and definition of basic power indices reconsidered with respect
to constraints on voting configuration formation. Constrained power indices
are defined as power indices defined over the feasible sets of voting
configurations. Three alternative ways of introducing constraints (reflecting
somehow preferences of committee members) into voting power are
considered: an a priori union approach (proposed originally for two
particular concepts of power indices by Owen, 1977, 1982), spatial power
indices (introduced by Edelman, 1997), and conflicting configuration
structures (first time analysed in a very special case of a "paradox of
quarrelling members" by Kilgour, 1974). Extension of different power
indices concepts on models with a priori union structures is discussed in
more details.
2
2. POWER IN COMMITTES
Let N = {1, ..., n} be the set of members (players, parties) and ωi (i = 1, ..., n)
be the (real, non-negative) weight of the i-th member such that
  i = 1,  i  0
iN
(e.g. the share of votes of party i, or the ownership of i as a proportion of the
total number of shares, etc.). Let γ be a real number such that 0  γ  1.
The (n+1)-tuple
[  , ] = [ , 1 ,  2 ,...,  n ]
such that
n
  i = 1, i  0, 0    1
i=1
we shall call a committee of the size n = card N with quota γ and allocation
of weights
 = (1 ,  2 , ...,  n )
(by card S we denote the cardinality of the set S, for empty set card  = 0).
Any non-empty subset S  N we shall call a voting configuration. Given an
allocation ω and a quota γ, we shall say that S  N is a winning voting
configuration, if
 i  
iS
and a losing voting configuration, if
 i <
iS
(i.e. the configuration S is winning, if it has a required majority, otherwise it
is losing).
3
Let 2N be the power set of N, the set of all subsets of N. In the committee
framework 2N is the set of all formally possible voting configurations. Let us
define   2N as a set of feasible voting configurations, such voting
configurations that express some preferences of committee members. The set
 can be given by a list of configuration, by a rule of configuration
formation etc. Now we can extend our model of a committee as an (n+2)tuple
[  , ,] = [ , 1 ,  2 ,...,  n ,]
such that
n
  i = 1, i  0, 0    1, _ 2 N
i=1
A member i  N of the committee [γ, ω, ] is said to be dummy if he cannot
benefit any feasible voting configuration by joining it, i.e. the player i is
dummy if
 i   
kS
 i  
kS -{i}
for any winning configuration S   such that i  S.
Two distinct members i and j of a committee [γ, ω, ] are called symmetric
if their benefit to any feasible voting configuration is the same, that is, for
any S   such that i, j  S

kS {i}
k   

kS { j}
k  
Let [γ, ω, ] be a committee with the set of members N and
 :N N
be a permutation mapping. Then the committee
[  , , ]
we shall call a permutation of the committee [γ, ω,] and σ(i) is the new
number of the member with original number i.
4
Let
n


() = (  , )  R n +1 :   i = 1,  i  0, 0    1 
i =1


be the space of all committees [, ω, ] of the size n with the set   2N of
feasible voting configurations and


E =  e  R n : ei = 1, ei  0 (i = 1,..., n) 
iN


be the unit simplex.
A power index is a vector valued function
 : ()  
that maps the space Γ of all committees into the unit simplex E, such that:
a) for dummy member i
 i (  , ,) = 0
(dummy axiom)
b) if [γ; ω, ] is a committee and [γ; σω, σ] its permutation, then
  (i) (  , , ) =  i ( ,  ,)
(permutation axiom)
c) if i and j (i  j) are symmetric, then
 i (  , ,) =  j ( ,  ,)
(symmetry axiom)
A power index represents a reasonable expectation of the share of decisional
power among the various members of a committee, given by ability to
contribute to formation of winning voting configurations. By πi(γ, ω, ) we
denote the share of power that the index π grants to the i-th member of a
committee with weight allocation ω, quota γ and the set  of feasible voting
configurations. Such a share is called a power index of the i-th member.
If  = 2N, we shall speak about unconstrained power indices, if   2N, we
shall speak about constrained power indices (Turnovec, 1997).
5
3. MARGINALITY
We shall term a member i of a committee [γ, ω, ] to be marginal with
respect to a configuration S  , i  S, if
  k   and
 k <
kS
kS {i}
A voting configuration S   such that at least one member of the
committee is marginal with respect to S will be called a critical winning
configuration (CWC).
Let us denote by C(γ, ω, ) the set of all CWC in the committee with the
quota γ, allocation ω and the set of feasible configurations . By Ci(γ, ω, )
we shall denote the set of all CWC the member i  N is marginal with
respect to, and by Cis(γ, ω, ) the set of all CWC of the size s (by size we
mean cardinality of CWC, 1  s  n) the member i  N is marginal with
respect to. Analogically by Pis(γ, ω) we shall denote the set of all CWC
having exactly s marginal members (1  s  n) the member i  N is marginal
with respect to. Then
n
C i (  , ,) =  Cis ( ,  ,)
s =1
C( , ,) =  Ci ( , ,)
iN
A voting configuration S   is said to be a minimal critical winning
configuration (MCWC), if
  i   and
iS
 k <
kT
for any T _ S
Let us denote by M(γ, ω, ) the set of all MCWC in the committee [γ, ω, ].
By Mis(γ, ω, ) we shall denote the set of all minimal critical winning
configurations S of the size s such that i  S, and by Mi(γ, ω, ) the set of all
minimal winning configurations containing member i.
6
Lemma 1
A member i  N of a committee [γ, ω, ] is dummy if and only if for
any s = 1, 2, ..., n
card C is (  , ,) = 0
Remark
Since for all s
M is (  , ,)  Cis ( ,  ,)
the member i  N is dummy if and only if for any s = 1, 2, ..., n
card Mis ( ,  ,) = 0
Lemma 2
Let [γ; ω, ] is a committee and [γ; σω, σ] its permutation, then for
any s = 1, 2, ..., n
card Cis ( ,  ,) = card C (i) s ( ,  , )
Remark
The same statement is true for the sets Mis(γ, ω, ); for any s = 1, 2, ...,
n
card Mis ( ,  ,) = card M (i) s ( ,  , )
Lemma 3
If the members k, r  N (k  r) of a committee [γ, ω, ] are
symmetric, then for any s = 1, 2, ..., n
card Cks ( ,  ,) = card Crs ( ,  ,)
Remark:
Using the same arguments we can show that for k, r symmetric and
for any s
card Mks ( ,  ,) = card Mrs ( ,  ,)
(Proofs Turnovec 1998)
7
4. UNCONSTRAINED POWER INDICES
Traditional power analysis is an unconstrained one, assuming  = 2N. The
five most widely known power indices were proposed by Shapley and
Shubik (1954), Penrose, Banzhaf and Coleman (1946, 1965, 1971), Johnston
(1978), Holler and Packel (1978, 1983), and Deegan and Packel (1978). All of
them measure the power of each member of a committee as a weighted
average of the number of his marginalities in CWC or MCWC.
SHAPLEY-SHUBIK
The Shapley-Shubik (SS) power index assigns to each member of a
committee the share of power proportional to the number of permutations of
members in which he is pivotal. A permutation is an ordered list of all
members. A member is in a pivotal position in a permutation if in the
process of forming this permutation by equi-probable additions of single
members he provides the critical weight to convert the losing configuration
of the preceding members to the winning one. It is assumed that all winning
configurations are possible and all permutations are equally likely. Thus SS
power index assigns to the i-th member of a committee with quota γ and
weight allocation ω the value of his share of power
 iSS (  , ) =
(card S- 1)!(card N- card S)!
(card N)!
S N

where the sum is extended to all winning configurations S for which the i-th
player is marginal. We can use an alternative formula as an explicit function
of marginalities
n
(s - 1)!(n - s)!
card Cis ( , )
n!
s =1
 iSS (q, ) = 
8
PENROSE-BANZHAF
The Penrose-Banzhaf (PB) power index assigns to each member of a
committee the share of power proportional to the number of critical winning
configurations for which the member is marginal. It is assumed that all
critical winning configurations are possible and equally likely. Banzhaf
suggested the following measure of power distribution in a committee:
n
 iPB (  , ) =
card Ci ( ,  )
=
 card Ck ( ,  )
kN
 card Cis ( ,  )
s =1
n
  card Ckr ( ,  )
r =1 kN
(so called normalized PB-power index)
We can meet different names for the index (Banzhaf, Banzhaf-Coleman,
Penrose-Banzhaf) reflecting the fact that slightly different versions were
proposed independently by different authors. Some authors prefer the
following version of PB index (based on Dubey and Shapley (1979)
axiomatization:
1
2
n
card Cis ( ,  )
n-1 
s=1
But this version does not sum up to one, does not provide information on
relative power.
JOHNSTON
The Johnston (J) power index measures the power of a member of a
committee as a normalized weighted average of the number of cases when he
is marginal with respect to a critical winning configuration, using as weights
the reciprocals of the number of marginalities in each CWC:
n 1
 card Pis ( ,  )
s
 iJ (  , ) = n s=1
1
  card Pkv ( ,  )
v =1 v kN
9
HOLLER-PACKEL
The Holler-Packel (HP) power index assigns to each member of a
committee the share of power proportional to the number of minimal critical
winning configurations he is a member of. It is assumed that all winning
configurations are possible but only minimal critical winning configurations
are being formed to exclude free-riding of the members that cannot
influence the bargaining process. "Public good" interpretation of the power
of MCWC (the power of each member is identical with the power of the
MCWC as a whole, power is indivisible) is used to justify HP index. Holler
[1978] suggested and Holler and Packel [1983] axiomatized the following
measure of power distribution in a committee:
n
 iHP (  , ) =
card Mi ( ,  )
=
 card Mk ( ,  )
kN
 card Mis ( ,  )
s =1
n n
  card Mkr ( ,  )
k =1 r =1
DEEGAN-PACKEL
The Deegan-Packel (DP) power index measures the power of a member of a
committee as a normalized weighted average of the number of minimal
critical winning coalitions he is a member of, using as weights the
reciprocals of the size of each MCWC:
n 1
 card Mis ( ,  )
s
 iDP (  , ) = n s=1
1
  card M ks ( ,  )
v =1 v kN
10
5. CONSTRAINED POWER INDICES
All original definitions of voting power measures are based on an extremely
simple model of a weighted voting game, described by quota and
distribution of votes, or, by a super-additive simple game. Then, assuming
nothing about preferences, restrictions on voting configuration formation
etc., we ask the following question: considering all voting configurations (or
all ways of forming voting configurations) equally probable and having no
additional information that is not described by this model, then assuming
that an infinite (or very large) number of voting events will take place, what
is a probability (or, a reasonable expectation) that an actor's YES will be
decisive?
Some of political scientists criticize power index methodology for not taking
into account preferences of decision making actors (see e.g. Garett and
Tsebelis, 1999, and discussion of the problem in Lane and Berg, 1999). Not
going into philosophical arguments this paper tries to outline possibilities
how to relax equal probability assumption.
There are two ways how to approach the problem.
a) One way is to assume that the underlying simple model is complete:
there are no a priori preferences and no aversions and restrictions on
coalition formation (a sort of perfect voting competition).
b) The model is incomplete, and we just don't know additional
information.
In the first case we are measuring and abstract voting power in an abstract
model of a committee. In the second case we have to add something to the
model and classify possible cases.
Constrained power indices introduce a new element into the model: some
additional behavioural assumptions about voting configuration formation.
11
6. RELAXING EQUAL PROBABILITY
A PRIORI UNIONS
The first relaxation of equal probability assumption was introduced by
Owen, 1977, 1982. Owen modifies Shapley-Shubik and Penrose-Banzhaf
power measure in models with a priori union structure. A priori union
structure is a partition of the set of committee members, a collection of
disjoint voting configurations whose union is the set of all members. A priori
unions are configurations which in some sense made a priori commitment to
cooperate. We shall generalize a priori union structure power indices
approach to implement it for any power index satisfying conditions
formulated in section 2.
SPATIAL MODELS
Another innovation relaxing equal probability assumption was proposed by
Edelman, 1997. Edelman develops a simple model applying to a situation in
which the committee members have positions on a one dimensional
(ideology) space and only connected voting configurations are allowed. The
Shapley-Shubik power index is extended to this situation in Edelman paper.
Again, it is possible to show that spatial power indices approach can be
implemented for any power measure satisfying conditions of section 2, and
extended for more than one ideological dimension.
CONFLICTING CONFIGURATION STRUCTURE
The roots of the third approach can be traced in Kilgour, 1974. He is
analysing so called "paradox of quarrelling members". An extension of
Shapley-Shubik power index is considered for situations in which there is a
set of members, each of whom refuses to join any configuration already
containing another member from the set. Generalization of Kilgour's
approach assumes that there exists a list of infeasible voting configurations
and leads to constrained power indices in conflicting configuration structure.
In some sense spatial power indices can be treated as a special case of this
approach.
12
7. A PRIORI UNIONS
Let us consider a committee [γ, ω] with the set of members N, n = card N.
For the member set N let as have a partition
 = {T 1 , T 2 , ..., T m}
such that
Tk Tr=
for any r, k = 1, 2, ..., m, r  k, and
m
 Ti= N
i=1
Following Owen's terminology we shall call this partition an a priori union
structure in the committee [γ, ω].
Let us denote
t j = card T j
the number of committee members in the union Tj and
j
n j =  tk
k=1
(j = 1, 2, ...,m). For simplicity we shall assume that n0 = 0 and
T j = { n j -1 + 1, n j -1 + 2, ..., n j -1 + t j = n j} _
(j = 1, 2, ..., m). Then
 j = ( n
j-1
+1 ,  n j-1+ 2 , ...,  n j )
is the internal distribution of weights in the union Tj and
j=
nj

wk
k = n j-1+1
is the total weight of the union Tj.
Now let us consider two level process: a first level inter-union committee
game
[  , 1 , 2 , ..., m ]
13
and m internal union second level subgames
[  , 1 , ..., j -1 , j ,  j+1 , ...,  m ]
(j = 1, 2, ..., m). We can use the following interpretation:
In the first level game each union Tj selects a representative to act for the
union in bargaining with other unions. In that case we have a standard
voting committee situation with m members (union) using their total
weights. First level game provides a total power evaluation of the union.
In the second level subgame the total power of the union has to be
decomposed among individual members. The power of individual members
should be proportional to the power they can gain by forming
subconfigurations within the union and by defection of individuals or
subconfigurations containg particular individuals from the union and
forming configurations with other unions taken as entities.
Let
 j [  , ]
be the power index of the j-th union calculated from the first level game, and
 ij [  , ,  j ]
(j= 1, 2, ..., m, i  Tj) be the power index of the i-th member of the committee
in the j-th subgame. Then
 ij [ ,  ,  j ]
 i [  , , T ] =  j [ ,  ]
  kj [  ,  ,  j ]
kT j
defines the power index of the member i of the union Tj  T in the
committee with an a priori union structure T.
In 1977 Owen proposed the modification of Shapley-Shubik power index for
committees with a priori union in terms of the characteristic function
cooperative game. In 1982 he repeated this exercise for non-normalized
Penrose-Banzhaf.
14
8. ANY POWER INDEX CAN BE MODIFIED
FOR A PRIORI UNIONS
Based on marginality concept, we can obtain equivalent explicit formulas (in
terms of weights and quotas) modifying any power index for evaluation of
power in committees with a priori union structures.
SHAPLEY-SHUBIK
 iSS [  , , T] =
m-1+ t j
m
 (s - 1)!(m - s)! card C js ( , )  (l - 1)! (m - 1 + t j - l)! card Cil ( , ,  j)
s
l
nj
m!

m-1+ t j
 (r - 1)! (m - 1 + t j - r)! card Ckr ( , ,  j)
k = n j-1+1 r =1
(i  Tj)
PENROSE-BANZHAF
m
 iPB [  , , T] =
m-1+ t j
 card C js ( , )  card Cil ( , ,  j)
s
m m
  card C jv ( , )
j=1 v =1
l
nj

m-1+ t j
 card Ckr ( , ,  j)
k = n j-1+1 r =1
(i  Tj)
15
JOHNSTON
m-1+ t j1
m1
 card P js ( , )  card Pil ( , ,  j)
s
l l
 iJ [  , , T] = m ms
m
1
+
t j1 n j
1
  card P jv ( , ) 
 card Pkr ( , ,  j)
v
r
v =1 j=1
r =1 k = n j-1+1
(i  Tj)
HOLLER-PACKEL
m
 iHP [  , , T] =
m-1+ t j
 card M js ( , )  card Mil ( , ,  j)
s
m m
  card M jv ( , )
j=1 v =1
l
nj

m-1+ t j
 card M kr ( , ,  j)
k = n j-1+1 r =1
(i  Tj)
DEEGAN-PACKEL
m-1+ t j1
1
card
(

,

)

 card Mil ( , ,  j)
M js
s
l l
 iDP [  , , T] = m ms
m
1
+
t j1 n j
1
  card M jv ( , ) 
 card M kr ( , ,  j)
v =1 v j=1
r =1 r k = n j-1+1
m
(i  Tj)
If the generic power index satisfies dummy axiom, permutation axiom and
symmetry axiom, then the corresponding modification for a priori unions
also satisfies these axioms.
16
9. CONCLUDING REMARK
Using a priori union approach we tried to demonstrate that introducing
constraints on voting configuration formation into voting committee models
allows to extend unconstrained power measures and modify results of
analysis, reflecting different behavioural conjectures. The same exercises
can be made for other two approaches, extending simple committee model:
spatial models and conflicting configuration structures. Therefore, criticism
of power index methodology as completely ignoring preferences of decision
makers and providing too simplified tool for political sciences in explaining
voting outcomes, seems not to be fully justified.
17