Games and Economic Behavior 28, 73᎐104 Ž1999.
Article ID game.1999.0692, available online at http:rrwww.idealibrary.com on
The Banzhaf Index in Representative Systems with
Multiple Political PartiesU
Shigeo Muto†
Department of Value and Decision Science (VALDES), Graduate School of Decision
Science and Technology, Tokyo Institute of Technology, 2-12-1 Oh-okayama,
Meguro-ku, Tokyo 152-8552, Japan
Received January 6, 1994
The proposed modification of the Banzhaf index is used to evaluate voter power
in representative multiple-party Ži.e., more than two. systems. The modified index
shows that a voter’s ability to affect the outcome of legislative decisions increases
asymptotically as the inverse of the square root of the district’s population. The
square-root effect thus holds even in cases involving more than two parties. Journal
of Economic Literature Classification Numbers: C71, D72. 䊚 1999 Academic Press
1. INTRODUCTION
In Banzhaf Ž1966., J. F. Banzhaf evaluated citizen-voter power in representative systems with multiple electoral districts: he counted the probability that individual voters alter outcome by changing their votes assuming
that all coalitions on each side of the issue are equally likely. The index he
used, now called the Banzhaf index, showed that a voter’s ability to affect
the outcome of legislative decisions increases asymptotically as the inverse
of the square root of the district’s population, called the square-root effect.
Banzhaf’s model is, however, limited in the sense that each elected
legislator in effect polls the district on each issue and casts a vote based on
the majority of vote ŽBanzhaf, 1966, p. 1327.. This is equivalent to
U
This is a revised version of ‘‘Voters’ Power in Indirect Voting Systems with Political
Parties: the Square Root Effect,’’ CentER Discussion Paper No. 9372, the Center for
Economic Research ŽCentER., Tilburg University, The Netherlands, 1993. The author is
indebted to CentER for its support and excellent research environment. The Canon Foundation in Europe Visiting Research Fellowship which made his stay at CentER possible is
gratefully acknowledged. This research was supported in part also by the Ministry of
Education, Science and Culture in Japan ŽNo. 08453015, No. 08680448, 1996, No. 09680428,
1997.. The author wishes to acknowledge anonymous referees for their invaluable comments.
†
E-mail: [email protected].
73
0899-8256r99 $30.00
Copyright 䊚 1999 by Academic Press
All rights of reproduction in any form reserved.
74
SHIGEO MUTO
assuming that legislators are elected on each issue and that, each time,
there are two candidatesᎏone for and one against the issue.1
In most representative systems, however, constituents vote for a political
party or vote for a person fielded by such a party. Elected legislators serve
for a period, usually several years, during which they decide whether to
pass bills put to a vote in the legislature. They usually do not poll their
districts on each individual issue. In the decisions, legislators belonging to
the same party usually act as a block.
This paper proposes a modification of the Banzhaf index applicable to
the above representative systems, in particular to multiple-party systems
that are characterized by: Ž1. the existence of more than one political
party; Ž2. multiple electoral districts, each being allotted one seat; Ž3. each
constituent casts a vote for one candidate in the constituent’s district, and
the candidate with the largest number of votes is elected as a representative; and Ž4. elected legislators serve for a period during which they decide
whether to pass bills put to a vote based on majority rule; each legislator
casts one vote and members belonging to the same party act as a block.
With the proposed modification, the square-root effect is proved to hold in
this system. A typical example of the system I have in mind is the new
election system of the Japan’s Lower House of Representatives. There are
300 districts each of which has one seat 2 ; five major parties currently exist
in Japan; and members of each party usually act as a block.
Section 2 describes the model under discussion. Section 3 gives the
Banzhaf index modification. Section 4 presents the theorem showing that
the square-root effect holds even when more than two parties are involved.
Section 5 provides the proof: the Appendix gives mathematical details.
Section 6 ends the paper with short remarks.
2. MODEL
Let D s d 1, . . . , d n4 be the set of electoral districts. Each district d i,
i s 1, . . . , n, has q i constituents and one seat. Let P s p1 , . . . , pm 4 be the
set of political parties. In each district, there are m candidates: the jth
candidate stands for the jth party pj . Each constituent has one vote and
1
Thus, the model implicitly assumes a two-party system.
Strictly speaking, in addition to 300 legislators, 200 legislators are elected in 11 blocs: each
bloc consists of several districts and has multiple seats. Thus, the House has Ž300 q 200 s.500
legislators. Each constituent has two votes, one for electing the constituent’s district’s
representative and the other for the constituent’s bloc’s representative. The analysis developed here could, however, be applied to evaluating constituents’ influence through the
election of their representatives in districts if we suppose that constituents’ voting patterns in
blocs are similar to those in districts.
2
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
75
casts it for one of the candidates. The candidate who obtains the largest
number of votes wins the seat. Ties are resolved by a random equiprobable
choice.
For notational convenience, denote by the m-dimensional vector ¨ i k s
Ž ¨ i k1 , . . . , ¨ i km . the candidate for whom the kth constituent of the ith
district casts a vote: ¨ i k consists of one 1 and Ž m y 1. 0’s where ¨ i kj s 1
implies that the kth constituent votes for the jth candidate. Let ¨ ij be the
sum of ¨ i kj over all constituents of the ith district; hence ¨ ij is the number
of votes cast for the jth candidate in the ith district. Let jU s argmax ¨ ij :
j s 1, . . . , m4 ; then the jU th candidate wins the seat. If there are more
than one such jU , each wins with equal probability. The m-dimensional
vector r i s Ž r i1 , . . . , r im . denotes which candidate Žor which party. wins the
seat in the ith district: r i consists of one 1 and Ž m y 1. 0’s where r ij s 1
implies that the jth candidate Žor the jth party. wins the seat. Let
r j s Ý nis1 r ij , j s 1, . . . , m, and r s Ž r 1 , . . . , rm .. Number r j is the total
number of seats that the jth party holds in the House.
In the House, ‘‘Approval ŽA.’’ or ‘‘Disapproval ŽD.’’ of each bill is
decided by the elected legislators. Legislators belonging to the same party
are supposed to act as a block; hence all vote ‘‘Yes ŽY.’’ or all vote ‘‘No
ŽN.’’. Thus, for each bill submitted, if the set S of parties voting ‘‘Y’’
exceeds the quota, the bill is approved. The condition is written as
Ý j g S r j G q where q is the quota satisfying Ý mjs1 r j G q ) ŽÝ mjs1 r j .r2.
Our main concern is to measure each constituent’s power, i.e., to
evaluate the individual’s influence on decisions on bills in the House. The
power index below is similar to the Banzhaf index, at least in spirit. Each
constituent, however, influences decisions in the House only indirectly.
What the constituent does directly is to alter the party that wins the seat in
the constituent’s district. That alters the number of legislators belonging to
each party in the House, and thus may change final decisions on bills in
the House.
3. MODIFIED BANZHAF INDEX
The Banzhaf index in direct voting systems measures each voter’s power
by counting the number of ‘‘Y-N’’ combinations of votes in which a voter
may change final outcomes by changing the voter’s individual vote. More
precisely, consider all ‘‘Y-N’’ combinations of all voters. In each combination, a voter is called a swing if the voter can change the final outcome
from ‘‘A’’ to ‘‘D’’ Žor from ‘‘D’’ to ‘‘A’’. by changing a vote from ‘‘Y’’ to
‘‘N’’ Žor from ‘‘N’’ to ‘‘Y’’.. The Ž absolute. Banzhaf index of a voter is given
76
SHIGEO MUTO
by the probability that the voter is a swing provided that every combination
is equiprobable.3
In our model, however, each voter may only indirectly alter final outcomes through the change of the voter’s representative; and, further, the
voter has Ž m y 1. ways to change a vote: recall that there are m candidates. Based on the model’s characteristics, we define a power index for
our model as follows.
Following the original definition of the Banzhaf index, consider all
possible configurations of voters’ choices, and suppose each is equiprobable. Let ⌸ i be the set of all configurations of voters’ choices in the ith
district, i s 1, . . . , n. Since there are q i voters in the ith district and each
i
has m alternatives, ⌸ i has m q elements. Thus, each configuration of all
1
n
voters’ choices emerges with equal probability 1rŽ m q = ⭈⭈⭈ = m q ..
In our model, a configuration of voters’ choices determines which party
wins the seat in each district; and thus, determines the number of seats
that each party has in the House. Elected legislators serve for a period,
usually several years, and decide whether to pass bills put to a vote.
Legislators follow their parties’ decisions. We assume that the frequency of
each ‘‘Y-N’’ combination of parties’ votes can be estimated, for example,
from the past record of decisions in the House.4 Let ⌸ H be the set of all
‘‘Y-N’’ combinations; for each ␲ H g ⌸ H , PrŽ␲ H ., the probability that ␲ H
will arise, is assumed to be known. For each ␲ H , denote the set of parties
voting ‘‘Y’’ in ␲ H by P Y Ž␲ H ..
Take a configuration ␲ s Ž␲ 1 , . . . , ␲ n, ␲ H ., ␲ i g ⌸ i, i s 1, . . . , m,
H
␲ g ⌸ H. Assume the number of legislators in parties belonging to
P Y Ž␲ H . exceeds the quota; hence, the decision in the House is ‘‘A.’’ Pick
voter k in the ith district, and assume that this voter votes for the jth
candidate in configuration ␲ i. Since there are Ž m y 1. candidates other
than the jth, the voter has Ž m y 1. different ways of changing a vote. We
say that the voter is a swing in ␲ if, by altering the choice from the jth to
one of the other candidates, for example, to the jX th, the voter also alters
the representative from the jth to the jX th and alters the final outcome
from ‘‘A’’ to ‘‘D’’.5 The voter is also called a swing if the change of the
3
Refer to Dubey and Shapley Ž1979. for a more precise definition and properties of the
Banzhaf index.
4
More precisely, one may determine parties’ ideological positions from the record of past
decisions, etc.: see, for example, Owen Ž1971., Shapley Ž1977., Rapoport and Golan Ž1985.,
Rabinowitz and Macdonald Ž1986., and Ono and Muto Ž1997.. Then assuming that all types
of bills are equally likely to be introduced, one may calculate, for each ‘‘Y-N’’ combination of
parties’ votes, the probability that it will arise. See Shenoy Ž1982. and Rapoport and Golan
Ž1985..
5
It is possible that the change of the voter’s vote induces the new final outcome ‘‘D’’ even
X
though the voter’s representative changes from jth to a candidate different from the j th, but
in this case the voter does not qualify as a swing since the change of the representative was
not the one that the voter intended. We use the term ‘‘swing’’ in a somewhat strict sense.
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
77
voter’s vote alters the final outcome from ‘‘D’’ to ‘‘A’’. Let s k Ž␲ . be the
number of ways that k can be a swing in ␲ . Thus 0 F s k Ž␲ . F m y 1.
Assuming that configurations Ž␲ 1, . . . , ␲ n . are equiprobable, the probabilities PrŽ␲ H . are given, and ties in elections of representatives are resolved
by a random equiprobable choice, we define a modified Banzhaf index ␤ k
of voter k by the expected number of times that the voter is a swing. Since
voters in the same district are symmetric, their modified Banzhaf indices
are identical. Thus, in what follows, we denote the modified Banzhaf index
of a voter in the ith district by ␤ i.
4. PASSAGE TO THE LIMIT AND THE
SQUARE-ROOT EFFECT
Our concern is to evaluate each voter’s relative power measured by the
modified Banzhaf index when every district’s population is large. For this
purpose, we increase all districts’ populations, keeping their proportions
fixed, and study the asymptotic behavior of ratios of voters’ indices in
different districts. Let ␣ 1, . . . , ␣ n be rational numbers representing proportions of districts’ populations, and K be their least common denominator. Let M be a positive integer and q i Ž M . s ␣ i KM, i s 1, . . . , n. Let
␤ i Ž M . be the modified Banzhaf index of a voter in the ith district when
., . . . , q n Ž M . voters. We examine the asymptotic behavdistricts have q 1 Ž M
X
iŽ
i
ior of ␤ M .r␤ Ž M . as M ª ⬁ for each pair of i, iX s 1, . . . , n, i / iX .
The following theorem shows that the ‘‘square-root effect,’’ shown by
Banzhaf Ž1966. 6 for the ‘‘two-parties’’X case, holds even if there exist more
than two parties. That is, ␤ i Ž M .r␤ i Ž M . asymptotically converges to the
inverse of the square root of the districts’ populations.
THEOREM.
X
For each two districts d i and d i , i, iX s 1, . . . , n, i / iX ,
lim
Mª⬁
␤ iŽ M .
X
␤i Ž M.
X
'␣ i
s
'␣ i
.
5. PROOF OF THE THEOREM
5.1. Decomposition of ␤ i Ž M .
Recall that parties are denoted by p1 , . . . , pm and districts are by
d 1, . . . , d n. Each district d i, i s 1, . . . , n, has ␣ i KM voters. Pick the ith
district. We first present a mathematical formula of the modified Banzhaf
6
See also Lucas Ž1983..
78
SHIGEO MUTO
index ␤ i Ž M . for a voter in the ith district. Pick one voter in the district. In
what follows we use the following notation.
V Ž j ., j s 1, . . . , m: the event that the voter votes for the jth candidate;
recall that the jth candidate belongs to the jth party;
V Ž j ª k ., j, k s 1, . . . , m, j / k: the event that the voter changes his
vote from the jth to the kth candidate;
R i Ž j ., j s 1, . . . , m: the event that the jth party wins the seat in the ith
district;
R i Ž j ª k ., j, k s 1, . . . , m, j / k: the event that the seat changes from
the jth to the kth party in the ith district; and
A l D: the event that the final outcome in the House changes from A
to D or from D to A.
Since all configurations Ž␲ 1 , . . . , ␲ n . of voters’ choices are assumed
equiprobable and ties in elections of representatives are resolved by a
random equiprobable choice,
Pr Ž V Ž j . . s
1
m
j s 1, . . . , m
and
Pr Ž R i Ž j . . s
1
m
j s 1, . . . , m
hold.
Now since the modified Banzhaf index ␤ i Ž M . is defined as the expected
number of times that a voter in the ith district is a swing, ␤ i Ž M . is written
as
m
␤ iŽ M . s
Ý
X
j s1
m
ž
Y
Ý
Y
j s1, j /j
X
Pr Ž A l D N V Ž jX ª jY . . Pr Ž V Ž jX . . . Ž 5.1.
/
Since the voter can change the final outcome through changing the
winning party of the district, Ž5.1. is rewritten as
m
␤ iŽ M . s
Ý
X
j s1
m
ž
Y
Ý
Y
j s1, j /j
X
Pr Ž A l D N R i Ž jX ª jY . .
=Pr Ž R i Ž jX ª jY . N V Ž jX ª jY . . Pr Ž V Ž jX . . . Ž 5.2.
/
79
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
5.2. E¨ aluation of PrŽ A l D N R i Ž jX ª jY .., jX , jY s 1, . . . , m, jX / jY
Denote PrŽ A l D N R i Ž jX ª jY .. by aŽ jX , jY . and denote the set of parties voting Y by P Y. Then
a Ž jX , jY . s
Ý Pr Ž Ž A l D, P Y s S .
R i Ž jX ª jY . .
S:P
s
Ý Pr Ž A l D N R i Ž jX ª jY . , P Y s S . = Pr Ž P Y s S . .
S:P
Denote PrŽ A l D N R i Ž jX ª jY ., P Y s S . = PrŽ P Y s S . by aŽ jX , jY , S ..
Then aŽ jX , jY , S . s 0 if both jX and jY vote ‘‘Y’’ Žthus in S . or both vote
‘‘N’’ Žthus not in S .; because even if the representative of the ith district
changes from the jX th to the jY th party, the number of legislators belonging to parties in S remains unchanged. Suppose jX g S and jY g P y S.
Then aŽ jX , jY , S . is positive only when Ý j g S r j s q; i.e., when among
the representatives elected in the districts other than i, exactly q y 1 of
them vote Y. If jX g P y S and jY g S, aŽ jX , jY , S . is positive only when
Ý j g S r j s q y 1; or when exactly q y 1 of the representatives elected in
the district other than i vote Y. Since all configurations of voters’ choices
are equiprobable and ties are resolved by a random equiprobable choice,
each party wins the seat with equal probability 1rm in each district as we
noticed in the previous subsection. Thus letting
m
½
⌫ s Ž r 1 , . . . , rm .
Ý r j s q y 1, Ý r j s n y 1
jgS
js1
5
,
we obtain that
Ž n y 1. !
a Ž jX , jY , S . s
Ý
Ž r 1 , . . . , r m .g⌫
r 1 != ⭈⭈⭈ = rm !
1
ny 1
ž /
m
= Pr Ž P Y s S .
X
holds when Ž j g S and jY g P y S . or Ž jX g P y S and jY g S .. Let
⌰ s S : P N Ž jX g S, jY g P y S . or Ž jX g P y S, jY g S . 4 .
Then the desired probability is given by
a Ž jX , jY . s
Ý
Sg⌰
ž
Ž n y 1. !
Ý
Ž r 1 , . . . , r m .g⌫
r 1 != ⭈⭈⭈ = rm !
1
ž /
m
ny 1
/
= Pr Ž P Y s S . .
Ž 5.3.
For each S : P, PrŽ P Y s S . is equal to PrŽ␲ H . where ␲ H is a configuration in which parties in S vote ‘‘Y’’ and other parties vote ‘‘N.’’ PrŽ␲ H . is
given exogenously. Recall Section 3.
80
SHIGEO MUTO
5.3. E¨ aluation of PrŽ R i Ž jX ª jY . N V Ž jX ª jY .., jX , jY s 1, . . . , m, jX / jY
To save space, denote PrŽ R i Ž jX ª jY . N V Ž jX ª jY .. by bŽ jX , jY .. It suffices
to examine the following three cases. Recall ¨ ij denotes the number of
votes that the jth party obtains in the ith district.
1
"
¨ jiX s ¨ jiY q 1,
¨ jiX G ¨ ji
2
"
¨ jiX s ¨ jiY q 2,
¨ jiY q 1 G ¨ ji
j / jX , jY ;
for all
for all
j / jX , jY ;
and
¨ jiX s ¨ jiY G ¨ ji
3
"
j / jX , jY .
for all
Thus we have
1 N V Ž jX ª jY .
b Ž jX , jY . s Pr R i Ž jX ª jY . & "
ž
/
2 N V Ž jX ª jY .
q Pr R i Ž jX ª jY . & "
ž
/
3 N VŽ j ª j ./.
q Pr ž R Ž j ª j . & "
i
X
X
Y
Y
Ž 5.4.
Now the following proposition holds. The proof is given in the Appendix.
PROPOSITION.
Ž1. PrŽ R i Ž jX ª jY . & "
1 N V Ž jX ª jY ..
f
M
1
=
2'␲
(
m
␣ KM
i
= ⌽ Ž Ž y, y 3 , . . . , ymy1 . N
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . .
Ž2. PrŽ R i Ž jX ª jY . & "
2 N V Ž jX ª jY ..
3 N V Ž jX ª jY .
s Pr R i Ž jX ª jY . & "
ž
f
M
1
2
=
1
2'␲
=
(
m
␣ KM
i
/
= ⌽ Ž Ž y, y 3 , . . . , ymy1 . N
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . .
81
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Here ⌽ Ž y, y 3 , . . . , ymy1 . is the Ž m y 2.-¨ ariate normal distribution with
mean ¨ ector Ž0, . . . , 0. and the ¨ ariance-co¨ ariance matrix
¡2
m
ž
y
¢
2
1y
y
^
y
/
m
2
1
m2
..
.
m
2
m
ž
2
m
1y
..
.
y
2
2
1
m
/
1
m
2
⭈⭈⭈
y
⭈⭈⭈
y
..
.
⭈⭈⭈
`
1
m
ž
¦¦
2
m
1
2
¥m y 2 ;
m2
..
.
1y
1
m
/§§_
my 2
and the symbol f denotes that the ratio of the both sides con¨ erges to 1 as
M
M ª ⬁.
From the proposition and Ž5.4. we obtain
b Ž jX , jY . f
M
1
=
'␲
(
m
␣ KM
i
= ⌽ Ž Ž y, y 3 , . . . , ymy1 .
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . . . Ž 5.5.
Note that bŽ jX , jY . depends neither on jX nor on jY .
X
5.4. E¨ aluation of ␤ i Ž M . and of the ratio ␤ i Ž M .r␤ i Ž M .
It follows from Ž5.2., Ž5.3., and Ž5.5. that the inner sum of Ž5.2. is given
by
m
Y
Ý
Y
j s1, j /j
f
M
X
a Ž jX , jY . = b Ž jX , jY .
1
'␲
=
(
m
␣ KM
i
= ⌽ Ž Ž y, y 3 , . . . , ymy1 .
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . .
m
=
Y
Ý
Y
j s1, j /j
X
a Ž jX , jY . .
82
SHIGEO MUTO
Recall bŽ jX , jY . depends neither on jX nor on jY . Since PrŽ V Ž jX .. s 1rm,
jX s 1, . . . , m, we obtain from Ž5.2.
␤ iŽ M . f
M
1
'␲
=
1
'm ␣ i KM
= ⌽ Ž Ž y, y 3 , . . . , ymy1 .
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . .
=
m
m
Ý
Ý
X
Y
Y
j s1 j s1, j /j
X
a Ž jX , jY . .
From Ž5.3., Ý mjXs 1 Ý mjY s 1, jY / jX aŽ jX , jY . is independent of i. Thus
Ž1r 'm ␣ i KM . is the sole term that depends on i, and all ␤ i Ž M ., i s
1, . . . , n, have other terms in common. Therefore we obtain the desired
relation
␤ iŽ M .
X
␤i Ž M.
X
'm ␣ i KM
s
'm ␣ i KM
X
'␣ i
s
'␣ i
.
Thus the proof of the theorem is completed.
6. CONCLUDING REMARKS
In this paper, we have studied the power of constituents in indirect
voting systems with more than two political parties. We first proposed a
modification of the Banzhaf index. Then, using the modified index, we
showed that the square-root effect holds even when more than two parties
exist. We conclude the paper with remarks concerning possible future
research directions.
The first is to study systems in which elections in districts and elections
in blocs are conducted jointly. As stated in Footnote 2, the study is
necessary to truly evaluate voters’ influences on decisions made in the
Japan’s Lower House of Representatives.
The second is to extend the model to districts with unequal numbers of
seats.7 In the case of two parties, studies were done by Owen Ž1975. and
Muto Ž1989..8 Owen assumed that the party receiving a majority of votes
takes all seats, and analyzed voters’ power in the United States Presiden7
In the election of the Japan’s Upper House of Representatives there are 47 electoral
districts to which 2᎐8 seats are allotted.
8
Banzhaf Ž1966. also analyzed the case with unequal number of seats; but his analysis was
limited to voters’ influences within districts. Voters’ influences on final decisions in the House
were not examined.
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
83
tial election. The latter assumed that seats are given to each party in
proportion to the votes received, and demonstrated that: Ž1. voters in
districts with even seats are powerless and Ž2. voters’ power in districts
with odd seats depends only on districts’ populations and does not depend
on the number of seats. It would be interesting to study their counterparts
in a case with more than two parties. Owen’s work on United States
Presidential election was later extended by Bolger Ž1983. to a case with
more than two candidates. Bolger defined a modification of the Banzhaf
index: his definition is essentially the same as ours, although the underlying voting systems are somewhat different. He showed the square-root
effect for the cases with three and four candidates, but the square-root
effect for a case with arbitrary number of candidates was not proved, but
remainded as conjecture. His conjecture can be proved by the use of the
idea developed in this paper.
The third and more conceptually difficult question is on the application
of the Shapley-Shubik index, another well-known and widely used index.
The difficulty arises from the two-stage structure of the model; i.e.,
elections of legislators in districts and decisions by the elected legislators
in the House. Recently, the Shapley-Shubik index Žor the Shapley value. in
multiple choice games has been extensively studied, for example by Bolger
Ž1993., Hsiao and Raghavan Ž1993., etc. We may combine these studies
with the nonsymmetric Shapley-Shubik index proposed by Owen Ž1971.,
Shapley Ž1977., and Owen and Shapley Ž1989.; the latter is necessary for
evaluating parties’ influences in the House. In combining these two streams
of research, the probabilistic approach to the Shapley-Shubik index by
Straffin Ž1983. would be helpful. These problems will be studied in future
work.
APPENDIX
Proof of Proposition. We will show only Ž1. of the proposition; Ž2. can
be shown in a similar manner. Let s be the number of candidates other
than jX who obtained ¨ Ž jX . votes. Then the jX th candidate wins with
probability 1rŽ s q 1. when the voter votes for the candidate; while the
jY th candidate wins with probability 1rŽ s q 1. when the voter changes the
vote. Thus the representative changes from the jX th to the jY th with
probability 1rŽ s q 1. 2 . Let
⍀ Ž s, x . s Ž x sq3 , . . . , x m . s Ž x q 1 . q 2 x q x sq3 q ⭈⭈⭈ qx m s
␣ i KM y 1,
x G x sq3 , . . . , x m G 0 4 .
84
SHIGEO MUTO
Then the first term of Ž5.4. is rewritten as
1 N V Ž jX ª jY .
Pr R i Ž jX ª jY . & "
ž
?Ž ␣ i KMy1ys .r Ž sq2 .@
1
s
/
Ž s q 1.
Ý
2
Ž ␣ i KM y 1. ! 4
Ý
xs uŽ ␣ i KMy1ys .rm v Ž x sq3 , . . . , x m .g⍀ Ž s, x .
= Ž x q 1 . ! ⭈⭈⭈ Ž x q 1 . ! x! x! x sq 3 ! ⭈⭈⭈ x m !
½^
`
_
5
y1
␣ i K My1
1
ž /
Ž A.1.
m
s
where the symbols ? @ and u v denote the largest integer less than or equal
to and the smallest integer greater than or equal to , respectively.
Denote the r.h.s Žright-hand side. of ŽA.1. by AŽ s .. For simplifying
discussion, we only show that
䢇
䢇
䢇
䢇
AŽ 0. f
M
1
2'␲
=
(
m
␣ KM
i
= ⌽ Ž Ž y, y 3 , . . . , ymy1 .
y G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . . Ž A.2.
where f implies that the ratio of both sides converges to 1 when M goes
M
to infinity. Through similar but much more lengthy discussion, one can
show that AŽ s ., s s 1, . . . , m y 2, are approximated by multivariate normal distributions with dimensions lesser than the above. Thus AŽ s .,
s s 1, . . . , m y 2, are negligibly small compared to AŽ0., and hence, the
desired probability is approximated by the r.h.s of ŽA.2..
Substituting s s 0 into the r.h.s. of ŽA.1., we obtain
?Ž ␣ i KMy1 .r2 @
AŽ 0. s
Ž ␣ i KM y 1 . !
Ý
Ý
xs uŽ ␣ i KMy1 .rm v Ž x 3 , . . . , x m .g⍀ Ž0, x .
x! x! x 3 ! ⭈⭈⭈ x m !
␣ i K My1
1
ž /
m
.
For simplifying notation, we denote ␣ i KM y 1 by L, ⍀ Ž0, x . by ⍀ Ž x ., and
? Lr2@, u Lrm v by Lr2, Lrm, respectively. Then
Lr2
AŽ 0. s
L!
Ý
Ý
xsLrm Ž x 3 , . . . , x m .g⍀ Ž x .
L
1
x! x! x 3 ! ⭈⭈⭈ x m !
ž /
m
.
For each integer x in the interval between Lrm and Lr2, let
L!
f Ž x. s
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
s
L!
x! x!
1
ž /
m
1
x! x! x 3 ! ⭈⭈⭈ x m !
L
L
ž /
m
1
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
x 3 ! ⭈⭈⭈ x m !
.
Ž A.3.
85
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
For integers a, b, a O b, in the interval between Lrm and Lr2, let
b
f Ž a, b . s
Ý f Ž x. .
xsa
Then
Lr2
AŽ 0. s
f Ž x . s f Ž Lrm, Lr2. .
Ý
xsLrm
By taking Lrm q L2r3 between Lrm and Lr2,9 AŽ0. is rewritten as
A Ž 0 . s f Ž Lrm, Lrm q L2r3 . q f Ž Lrm q L2r3 q 1, Lr2 .
ž
s f Ž Lrm, Lrm q L2r3 . 1 q
f Ž Lrm q L2r3 q 1, Lr2 .
f Ž Lrm, Lrm q L2r3 .
/
.
Ž A.4.
To prove ŽA.2., we will show the following two facts.
Ž a.
Ž b.
f Ž Lrm q L2r3 q 1, Lr2 .
f Ž Lrm, Lrm q L2r3 .
f Ž Lrm, Lrm q L2r3 . f
L
ª0
Ž A.5.
L
r.h.s. of
Ž A.2.
Ž A.6.
where A ª B implies that A converges to B as L ª ⬁ while A f B
L
L
implies that the ratio of A to B converges to 1 as L ª ⬁. If both Ža. and
Žb. hold, ŽA.2. follows from ŽA.4..
Proof of Ža.. We first examine the denominator of ŽA.5.; and then the
numerator.
Ž1. DENOMINATOR OF ŽA.5.. Since each f Ž x ., Lrm F x F Lrm q L2r3 ,
is positive and L is a large number,
f Ž Lrm, Lrm q L2r3 . ) f Ž Lrm . .
Ž A.7.
Let mX and mY be the quotient and the leftover when we divide L by m;
i.e., mX , mY are integers satisfying L s mmX q mY , 0 O mY O m y 1. Then
9
Strictly speaking, L2r3 is u L2r 3 v.
86
SHIGEO MUTO
from ŽA.3.
¡
s
f
ž /~
L
m
1
X
Ž mmX . !
mm
ž /
mX !mX !^mX !`
⭈⭈⭈ mX_
!
m
my 2
if mY s 0
X
m m q1
1
Ž mmX q 1 . !
)
m
⭈⭈⭈ mX_
! Ž mX y 1 . !
Ž mX q 1 . ! Ž mX q 1 .^mX !`
ž /
if mY s 1X Y
m m qm
1
)
m
my 3
ž /
Ž mmX q mY . !
=
⭈⭈⭈ mX_
!
Ž mX q 1 . ! Ž mX q 1! . Ž mX q 1 . ! ⭈⭈⭈ Ž mX q 1 . !^mX !`
¢
^
if mY G 2.
`
Y
_
m y2
Y
my m
Therefore from ŽA.7.
f
ž
L
,
L
m m
q L2r3
¡
1
ž /
m
)
~
/
Ž mmX . !
L
mX !mX !^mX !`
⭈⭈⭈ mX_
!
if mY s 0
1 L
ž /
m
my 2
Ž mmX q 1 . !
⭈⭈⭈ mX_
! Ž mX y 1 . !
Ž mX q 1 . ! Ž mX q 1 . !^mX !`
if mY s 1
1 L
ž /
¢
m
my 3
Ž mmX q mY . !
⭈⭈⭈ mX_
!
Ž mX q 1 . ! Ž mX q 1! . Ž mX q 1 . ! ⭈⭈⭈ Ž mX q 1 . !^mX !`
if mY G 2.
^
`
Y
m y2
_
Y
my m
Ž A.8.
For simplifying presentation, we give a proof of Ža. only to the case of
mY s 0. A similar proof applies to the other cases of mY s 1 and mY G 2.
Therefore we assume Lrm s mX in the following.
Ž2. NUMERATOR OF ŽA.5.. To evaluate the numerator of ŽA.5., for each
integer x in the interval between mX q L2r3 q 1 and Lr2, let mU Ž x . and
87
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
mUU Ž x . be the quotient and leftover when we divide L y 2 x by m y 2;
i.e., mU Ž x ., mUU Ž x . are integers satisfying L y 2 x s Ž m y 2. mU Ž x . q
mUU Ž x ., 0 O mUU Ž x . O m y 3. Then it is easily shown that for each combination of x 3 , . . . , x m satisfying x 3 q ⭈⭈⭈ qx m s L y 2 x and x P x 3 , . . . , x m
P 0,
1
F
x 3 ! ⭈⭈⭈ x m !
1
U
Ž m Ž x . q 1 . ! ⭈⭈⭈ Ž m Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
^
`
U
_^
mUU Ž x .
`
m y 2 y mUU Ž x .
_
.
Since x 3 , . . . , x m are integers satisfying x 3 q ⭈⭈⭈ qx m s L y 2 x and x P
x 3 , . . . , x m P 0, each x 3 , . . . , x m takes an integer value between 0 and
minŽ x, L y 2 x .. Since minŽ x, L y 2 x . O Lr3, each x 3 , . . . , x m takes at
most Lr3 q 1 values. Hence
1
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
x 3 ! ⭈⭈⭈ x m !
1
-
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
-
ž
L
3
^Ž
mU Ž x . q 1 . ! ⭈⭈⭈ Ž mU Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
_^
`
mUU Ž x .
my 2
q1
/
`
_
m y 2 y mUU Ž x .
1
^Ž
mU Ž x . q 1 . ! ⭈⭈⭈ Ž mU Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
_^
`
mUU Ž x .
`
m y 2 y mUU Ž x .
_
Thus from ŽA.3.,
f Ž x. -
L!
1
x! x!
ž /ž
m
L
3
my2
q1
/
1
=
^Ž
s L!
L
U
m Ž x . q 1 . ! ⭈⭈⭈ Ž m Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
`
_^
mUU Ž x .
1
L
ž /ž
=
U
m
L
3
`
m y 2 y mUU Ž x .
_
my2
q1
/
1
x! x! Ž m Ž x . q 1 . ! ⭈⭈⭈ Ž m Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
^
U
`
mUU Ž x .
U
_^
`
m y 2 y mUU Ž x .
_
.
Ž A.9.
88
SHIGEO MUTO
Denote the last term of ŽA.9. by g Ž x ., i.e.,
gŽ x. s
1
x! x! Ž m Ž x . q 1 . ! ⭈⭈⭈ Ž m Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
^
U
U
`
_^
mUU Ž x .
`
m y 2 y mUU Ž x .
_
.
Ž A.10.
Then g Ž x . is monotone decreasing in x if x G mU q L2r3 q 1. In fact,
g Ž x q 1.
gŽ x.
!
#
"!
mUU Ž x .
#
m y 2 y mUU Ž x .
"
s x! x! Ž mU Ž x . q 1 . ! ⭈⭈⭈ Ž mU Ž x . q 1 . ! mU Ž x . ! ⭈⭈⭈ mU Ž x . !
½
5
= Ž x q 1. ! Ž x q 1. !
½
= Ž mU Ž x q 1 . q 1 . ! ⭈⭈⭈ Ž mU Ž x q 1 . q 1 . !
^
`
_
mUU Ž x q 1 .
= mU Ž x q 1 . ! ⭈⭈⭈ mU Ž x q 1 . !
^
`
UU
my 2y m
5
y1
_
Ž x q 1.
where mU Ž x ., mUU Ž x . are, as defined above, integers satisfying L y 2 x s
Ž m y 2. mU Ž x . q mUU Ž x ., 0 O mUU Ž x . O m y 3, and mU Ž x q 1. and
mUU Ž x q 1. are integers satisfying L y 2Ž x q 1. s Ž m y 2. mU Ž x q 1. q
mUU Ž x q 1., 0 O mUU Ž x q 1. O m y 3. Through simple calculations, we
obtain
g Ž x q 1.
gŽ x.
Ž mU Ž x . q 1 . Ž mU Ž x . q 1 .
F
.
Ž x q 1. Ž x q 1.
We now show that, for all x G mX q L2r3 q 1, mU Ž x . - x holds. First from
the equation L y 2 x s Ž m y 2. mU Ž x . q mUU Ž x ., 0 O mUU Ž x . O m y 3,
we have mU Ž x . O Ž L y 2 x .rŽ m y 2.. Since x G mX q L2r3 q 1, we have
x ) mX s Lrm. Thus
Ly2x
my2
yxs
L y mx
my2
- 0.
Therefore we obtain mU Ž x . O Ž L y 2 x .rŽ m y 2. - x for all x G mX q
L2r3 q 1. This implies that for all x G mX q L2r3 q 1
g Ž x q 1.
gŽ x.
F
Ž mU Ž x . q 1 . Ž mU Ž x . q 1 .
- 1.
Ž x q 1. Ž x q 1.
89
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Hence g Ž x . is decreasing in x when x G mX q L2r3 q 1. This implies that
g Ž x . - g Ž mX q L2r3 q 1 .
x G mX q L2r3 q 1.
for all
From ŽA.9. we obtain
f Ž x . - L!
L
1
L
ž /ž
m
3
my2
q1
g Ž mX q L2r3 q 1 .
/
for all
x G mX q L2r3 q 1;
hence
f Ž mX q L2r3 q 1, Lr2 .
ž /ž
. ž / ž
/
/ ž /ž /
ž
my2
2m
L L!
L
1
- Ž Lr2 y mX L!
-
L
1
- Ž Lr2 y Ž mX q L2r3 . . L!
L
m
1
m
m
3
L
L
3
L
3
my2
q1
g Ž mX q L2r3 q 1 .
/
my2
q1
g Ž mX q L2r3 q 1 .
my2
q1
Ž3. NUMERATOR r DENOMINATOR
AND
g Ž mX q L2r3 q 1 . .
PROOF
OF
Ž A.11.
ŽA.5..
1 NUMERATOR r DENOMINATOR. From ŽA.8., ŽA.10., and ŽA.11. we
"
obtain
f Ž mX q L2r3 q 1, Lr2 .
f Ž mX , mrL q L2r3 .
-
ž
my2
2m
L
/ž
L
3
y2
! m#
"
my 2
q1
/ ½
mX !mX ! mX ! ⭈⭈⭈ mX !
5
= Ž mX q 1 q L2r3 . ! Ž mX q 1 q L2r3 . !
½
= Ž mU Ž 噛 . q 1 . ! ⭈⭈⭈ Ž mU Ž 噛 . q 1 . ! mU Ž 噛 . ! ⭈⭈⭈ mU Ž 噛 . !
^
_^
`
mUU Ž 噛 .
X
`
m y 2 y mUU Ž 噛 .
_5
y1
. Ž A.12.
For saving space, 噛 stands for m q L q 1. Symbols mU Ž噛. and mUU Ž噛.
are integers satisfying L y 2噛 s Ž m y 2. mU Ž噛. q mUU Ž噛., 0 O mUU Ž噛.
O m y 3.
We will show that
2r3
the r.h.s. of Ž A.12. ª 0
L
90
SHIGEO MUTO
Denote the fraction of the r.h.s. of ŽA.12. by X, i.e.,
X
y2
! X m#
"
X
X
X s m !m ! m ! ⭈⭈⭈ m !
½
5
= Ž mX q 1 q L2r3 . ! Ž mX q 1 q L2r3 . !
½
= Ž mU Ž 噛 . q 1 . ! ⭈⭈⭈ Ž mU Ž 噛 . q 1 . ! mU Ž 噛 . ! ⭈⭈⭈ mU Ž 噛 . !
^
_^
`
mUU Ž 噛 .
`
m y 2 y mUU Ž 噛 .
_5
y1
.
Then from ŽA.12.
f Ž mX q L2r3 q 1, Lr2 .
f Ž mX , mX q L2r3 .
F Ž term of a polynomial order of L . = X .
Ž A.13.
2 EVALUATION
"
Xf
L
X
OF
X. By Stirling’s formula, we obtain
Xm
m
X
½ Ž'2␲ m m expŽ ym . . 5
= ½ ž '2␲ Ž m q 1 q L . Ž m q 1 q L
X
X
2r3
2r3 m q1qL
X
2r3
.
=exp Ž y Ž mX q 1 q L2r3 . .
=
U
U
½ ž '2␲ Ž m Ž 噛. q 1. Ž m Ž 噛. q 1.
mU Ž 噛 .q1
=exp Ž y Ž mU Ž 噛 . q 1 . .
=
U
U
½ ž '2␲ m Ž 噛. m Ž 噛.
mU Ž 噛 .
=mU Ž 噛 .
X
= mX m m Ž mX q 1 q L2r3 .
½
/
mUU Ž 噛 . y1
exp Ž ymU Ž 噛 . .
s mX m r2 Ž mX q 1 q L2r3 . Ž mU Ž 噛 . q 1 .
X
2 y1
/5
5
my2ymUU Ž 噛 .
5
y1
mUU Ž 噛 .r2
Ž my2ymUU Ž 噛 .. r2 y1
2 Ž m q1qL2r3 .
4
UU
Ž mU Ž 噛. q 1 . m
=mU Ž 噛 .
Ž 噛 .Ž mU Ž 噛 .q1 .
Ž my2ymUU Ž 噛 .. mU Ž 噛 . y1
5
. Ž A.14.
91
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Since
L
X
ms
U
m
, m Ž 噛. s
L y 2 Ž mX q 1 q L2r3 . y mUU Ž 噛 .
my2
,
0 F mUU Ž 噛 . F m y 3,
the first term of ŽA.14., i.e.,
mX m r2
UU
Ž mX q 1 q L2r3 . Ž mU Ž 噛 . q 1 . m
Ž 噛 .r2
mU Ž 噛 .
Ž my2ymUU Ž 噛 .. r2
is of a polynomial order of L. Denote the second term of ŽA.14. by X X .
Then
X f Ž term of a polynomial order of L . = X X .
L
Ž A.15.
X X . By replacing mU Ž噛. q 1 by mU Ž噛. in the
denominator of X , we obtain
3 EVALUATION
"
XX F
X
OF
mX L
Ž mX q 1 q L2r3 .
X
2 Ž m q1qL2r3 .
mU Ž 噛 . Ly2
Ž mXq1qL2r3 .
Since mX s Lrm, we have
Ž mX q 1 q L2r3 .
X
2 Ž m q1qL2r3 .
s Ž mX q 1 q L2r3 .
X
X
2 m q2 L2r3
2r 3
2r3 2 m q2 L
X
s Žm qL
s U 2U 1 q
ž
.
1
U
ž
Ž mX q 1 q L2r3 .
1q
1
mX q L2r3
2
X
2 m q2 L2r 3
/
Ž mX q 1 q L2r3 .
2U
/
Ž U q 1.
2
where U s mX q L2r3 s Ž Lrm. q L2r3. Note that U ª ⬁.
L
2
92
SHIGEO MUTO
Moreover
mU Ž 噛 . Ly2
s
)
s
s
Ž mXq1qL2r3 .
L y 2 Ž mX q 1 q L2r3 . y mUU Ž 噛 .
ž
ž
my2
X
Ly2 Ž m q1qL2r3 .
/
L y 2 Ž mX q 1 q L2r3 q 1 . y Ž m y 3 .
my2
L
ž
ž
m
L
m
=
sV
ž
y
y
L
m
2
2r3
L
my2
2
2r3
L
my2
y
Ž my2.V
ž
2
y
my2
mq1
y
my2
L
/
/
my 2 Lrmy2 L2r 3
my2
Ž my2 . V
mq1
Ž m y 2. V
/
Ly 2 Lrmy2 L2r 3 y4
mq1
y
2r3
my2
1y
mq1
X
Ly2 Ž m q1qL2r3 q1 .
/
y4
/
ž
Vy
mq1
my2
y4
/
where V s Lrm y 2 L2r3rŽ m y 2.. Note that V ª ⬁.
L
Therefore we obtain
X X - mX L U 2U 1 q
1
2U
½ ž /
= 1y
ž
2
Ž U q 1 . V Ž my2.V
U
Ž my2 .V
mq1
Ž m y 2. V
/
ž
Vy
mq1
my2
y4 y1
/
5
.
In the last expression,
ž
1q
1
U
2U
/
ªe
2
and
L
ž
1y
Ž my2 .V
mq1
Ž m y 2. V
/
ª eyŽ mq1. .
L
since U, V ª ⬁ and 1 q Ž krx . x ª e k . Each of ŽU q 1. 2 and Ž V y
x
L
Ž m q 1.rŽ m y 2..y4 is of a polynomial order of L. Let
Y
X s
mX L
U 2U V Ž my2.V
.
93
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Then when L is large, we have
X X - Ž term of a polynomial order of L . = X Y .
X Y . Dividing the denominator and the numerator
by m s Ž Lrm. L , we obtain
4 EVALUATION
"
of X
Y
XL
Y
X s1
ž
1q
OF
m
L1r3
2 Lrmq2 L2r 3
ž
/
1y
2 mr Ž m y 2 .
L1r3
Ž my 2 . Lrmy2 L2r 3
/
Ž A.17.
Thus
Y
ln X s yln 1 q
ž
s yL
1r3
=
žž
ln
1q
2 Lrmq2 L2r 3
m
žž
ž
/
L1r3
1q
L1r3
2
m
L1r3
L1r 3
m
1y
1y
/ž
2
/ / žž
1y
2 mr Ž m y 2 .
L1r3
2 mr Ž m y 2 .
L1r3
2 mr Ž m y 2
L1r3
Ž my 2 . Lrmy2 L2r 3
/
L2r 3 rm
my2
/ /
.
/ /
L1r 3
y2
Ž A.18.
where ln is the natural logarithm log e .
Now the first term in the logarithm of ŽA.18. is rewritten as
žž
1q
s
2
m
L1r3
žž
/ž
1q
1y
2m
1r3
L
2 mr Ž m y 2 .
L1r3
q
m2
/ž ž
L
q
s 1y
ž
2r 3
sWL
m3r Ž m y 2 .
L2r3
rm
ž
1qO
/ /
1y
2r3
ž
my2
2
1
qO
m y 2 2 mr Ž m y 2 .
1
L1r3
/
/ž
L
=
2 mr Ž m y 2 .
L1r3
L2r 3 rm
ž //
ž / /
1
L2r 3 rm
my2
L
1
W
L2r 3 rm
2
1
/ ž ///
qO
L
L2r 3 rm
94
SHIGEO MUTO
where W s 1 y Ž m3rŽ m y 2. L2r3 . and O Ž1rL. denotes the term which is
at most of the order 1rL. Since
W
L2r 3 r m
s 1y
ž
m3r Ž m y 2 .
L2r3
ž
and
1qO
1
ž /
L
L2r 3 rm
ª eym
/
2
rŽ my2.
L
=
1
W
L2r 3 rm
ª1
/
L
the first term of ŽA.18. converges to eym rŽ my2. as L ª ⬁.
The second term of ŽA.18. is rewritten as
2
žž
L1r 3
m
1q
L1r3
2
/ / žž
1y
ª Ž e m . Ž ey2 m rŽ my2. .
2
/ /
L1r3
y2
L
s e2 m
2
y2
L1r 3
2 mr Ž m y 2 .
rŽ my2.
.
Therefore the logarithm in the last expression of ŽA.18. converges to
2
ln e m rŽ my2. s m2rŽ m y 2. as L ª ⬁. Thus ln X Y f y L1r3 Ž m2rŽ m y 2..;
L
and hence X Y f expŽyL1r3 Ž m2rŽ m y 2....
L
5 PROOF OF ŽA.5.. From ŽA.13., ŽA.15., ŽA.16., and the fact
"
X Y f expŽyL1r3 Ž m2rŽ m y 2..., we obtain
L
f Ž Lrm q L2r3 q 1, Lr2 .
f Ž Lrm, Lrm q L2r3 .
ª 0.
L
Thus the proof of Ža. is completed.
Proof of Žb.. We next show Žb., i.e.,
f Ž Lrm, Lrm q L2r3 . f r.h.s of Ž A.2. .
L
Ž1. TRANSFORMATION OF f Ž Lrm, Lrm q L2r3 .. Recall, for each integer x in the interval between Lrm and Lr2, f Ž x . is given by
L!
f Ž x. s
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
s
L!
x! x!
1
ž /
m
1
x! x! x 3 ! ⭈⭈⭈ x m !
L
L
ž /
m
1
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
x 3 ! ⭈⭈⭈ x m !
95
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
and
LrmqL2r3
f Ž Lrm, Lrm q L2r3 . s
f Ž x.
Ý
xsLrm
LrmqL2r3
s
Ý
L!
Ý
L
1
x! x! x 3 ! ⭈⭈⭈ x m !
Ž x 3 ⭈⭈⭈ x m .g⍀ Ž x .
xsLrm
1
ž /
m
.
Let
r Ž x, x 3 , . . . , x m . s
L!
L
1
x! x! x 3 ! ⭈⭈⭈ x m !
ž /
.
m
Then this is rewritten as
r Ž x, x 3 , . . . , x m .
s
L!
2
2x
x3
1
xm
1
⭈⭈⭈
ž / ž / ž / ž
Ž 2 x . ! x 3 ! ⭈⭈⭈ x m ! m
m
m
Ž2 x. !
x! x!
=
1
22 x
/
.
Ž A.19.
Let xX s 2 x and replace x m by x m s L y Ž xX q x 3 q ⭈⭈⭈ qx my1 . in the
r.h.s. of ŽA.19.; then noting that Ž2rm. q Ž 1rm . q ⭈⭈⭈ q Ž 1rm . s 1,
^
ŽA.19. is further rewritten as
`
_
my 2
L!
X
Ž x . ! x 3 ! ⭈⭈⭈ x my1 ! Ž L y Ž xX q x 3 q ⭈⭈⭈ qx my1 . . !
=
x
2
X
1
x3
x my 1
1
⭈⭈⭈
ž /ž / ž /
m
m
m
X
ž ž
= 1y
2
m
q
1
q ⭈⭈⭈ q
^m `
1
Ly Ž x qx 3 q ⭈⭈⭈ qx my 1 .
//
_
m
my 3
=
Ž xX . !
1
ž / ž / 0
x
X
2
!
x
=
X
2
!
2x
X
.
96
SHIGEO MUTO
Let
s Ž xX , x 3 , . . . , x my1 .
s
L!
X
Ž x . ! x 3 ! ⭈⭈⭈ x my1 ! Ž L y Ž xX q x 3 q ⭈⭈⭈ qx my1 . . !
=
x
2
X
x3
1
x my 1
1
⭈⭈⭈
ž /ž / ž /
m
m
m
X
ž ž
= 1y
2
m
q
1
q ⭈⭈⭈ q
^m `
Ly Ž x qx 3 q ⭈⭈⭈ qx my 1 .
1
//
Ž A.20.
_
m
my 3
and
Ž xX . !
X
tŽ x . s
xX
xX
ž /ž /
2
!
2
=
1
2x
X
.
Ž A.21.
!
Then
f Ž Lrm, Lrm q L2r3 .
2 Ž LrmqL2r3 .
s
Ý
X
Ý
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
s Ž xX , x 3 , . . . , x m . t Ž xX . .
L r mqL X .
where Ý xX2Ž
s2 L r m, x is even means that the summation is over only even
numbers of xX , i.e., xX s 2 Lrm, 2 Lrm q 2, 2 Lrm q 4, . . . , 2Ž Lrm q
L2r3 . y 2, 2Ž Lrm q L2r3 ..
2r 3
L r mqL .
Ž2. EVALUATION OF Ý2Žx ⬘s2
Ž X
L r m, x ⬘ is even ÝŽ x 3 , . . . , x m .g ⍀ Ž x . s x , x 3 , . . . ,
X
X
X
x m . t Ž x .. Since t Ž x . is monotone decreasing in x , the following inequalities follow:
2r 3
2 Ž LrmqL2r3 .
t Ž 2 Lrm .
Ý
X
Ý
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
s Ž xX , x 3 , . . . , x m .
) f Ž Lrm, Lrm q L2r3 .
) t Ž 2 Ž Lrm q L2r3 . .
2 Ž LrmqL2r3 .
=
X
Ý
X
Ý
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
s Ž xX , x 3 , . . . , x m . .
97
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Dividing three terms by the first term, i.e.,
2 Ž LrmqL2r3 .
t Ž 2 Lrm .
Ý
X
Ý
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
s Ž xX , x 3 , . . . , x m . ,
we get
f Ž Lrm, Lrm q L2r3 .
1)
2 Ž LrmqL2r3 .
t Ž 2 Lrm .
Ý
X
s Ž xX , x 3 , . . . , x m .
Ý
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
)
t Ž 2 Ž Lrm q L2r3 . .
t Ž 2 Lrm .
.
Ž A.22.
We will show that
t Ž 2 Ž Lrm q L2r3 . .
t Ž 2 Lrm .
ª 1.
L
Since
Ž xX . !
X
tŽ x . s
xX
=
xX
ž /ž /
!
2
2
1
2x
X
,
!
by using Stirling’s formula we obtain
t Ž xX . f
L
(
2
1
␲
'xX
(
2
.
Ž A.23.
Therefore
1G
s
2 Ž 2 Ž Lrm q L2r3 . .
t Ž 2 Lrm .
(
␲
2r3
s
(
'2 Ž Lrm q L
2r3
(
L
Lrm
Lrm q L
f
1
2
1
␲
'2 Lrm
1
1 q mrL1r3
Since
(
1
1 q mrL1r3
ª 1,
L
.
.
98
SHIGEO MUTO
we obtain
t Ž 2 Ž Lrm q L2r3 . .
t Ž 2 Lrm .
ª 1.
L
Thus from ŽA.22.
f Ž Lrm, Lrm q L2r3 .
ª 1. Ž A.24.
2 Ž LrmqL2r3 .
t Ž 2 Lrm .
Ý
X
s Ž x , x3 , . . . , xm .
Ý
X
L
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
Ž3. EVALUATION OF DENOMINATOR OF ŽA.24.. As to the denominator of
ŽA.24., we first obtain from ŽA.23. that
t Ž 2 Lrm . f
L
(
2
1
␲
'2 Lrm
s
1
'␲
(
m
L
.
Ž A.25.
We next evaluate
2 Ž LrmqL2r3 .
X
Ý
Ý
X
x s2 Lrm , x is even Ž x 3 , . . . , x m .g⍀ Ž x .
s Ž xX , x 3 , . . . , x m .
when L ª ⬁.
L r mqL X .
Recall that Ý x2ŽXs2
L r m, x is even denotes the summation taken over only even
X
numbers of x , i.e., xX s 2 Lrm, 2 Lrm q 2, 2 Lrm q 4, . . . , 2Ž Lrm q
L2r3 . y 2, 2Ž Lrm q L2r3 .. Thus when L is large, the summation is
approximated by
2r 3
2 Ž LrmqL2r3 .
1
2
X
Ý
x s2 Lrm
s Ž xX , x 3 , . . . , x m .
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
L r mqL .
where Ý2ŽxXs2
is an ordinary summation, i.e., summation over all
Lr m
X
integers x between 2 Lrm and 2Ž Lrm q L2r3 ..
Since the r.h.s of ŽA.20. is a multinomial density function, we will show
that this converges to a multivariate normal density function by using the
convergence of a moment generating function. Let Ž X X , X 3 , . . . , X my1 . be
a vector of random variables whose joint density function is given as
ŽA.20.. Define random variables Y X , Y3 , . . . , Ymy1 by
2r 3
YX s
XX y
'L
2L
m
,
Y3 s
X3 y
'L
L
m
, . . . , Ymy1 s
X my1 y
'L
L
m
Ž A.26.
99
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
and let ␮ Ž tX , t 3 , . . . , t my1 . be a moment generating function of
Ž Y X , Y3 , . . . , Ymy1 .. Let
⌬ s Ž xX , x 3 , . . . , x my1 . N xXr2 G x 3 , . . . , x my1 G 0,
L y xX G x 3 q ⭈⭈⭈ qx my1 G L y 3 xXr2 4 .
Then we obtain
␮ Ž tX , t 3 , . . . , t my1 .
s E exp Ž tX Y X q t 3 Y3 q ⭈⭈⭈ t my1Ymy1 .
s E exp tX
s
2L
XX y
m
q t3
'L
L
X3 y
m
'L
q ⭈⭈⭈ qt my1
X my1 y
L
m
'L
0
L!
Ý
Ž x X , x 3 , . . . , x my1 .g⌬
=
x
2
X
X
Ž x . ! x 3 ! ⭈⭈⭈ x my1 ! Ž L y Ž xX q x 3 q ⭈⭈⭈ x my1 . . !
x3
1
x my 1
1
⭈⭈⭈
ž /ž / ž /
m
m
m
X
ž ž
= 1y
2
m
1
q
q ⭈⭈⭈ q
^m `
1
Ly Ž x qx 3 q ⭈⭈⭈ qx my 1 .
//
_
m
my 3
=exp tX
s
2L
xX y
m
'L
q t3
x3 y
L
m
'L
q ⭈⭈⭈ qt my1
x my1 y
'L
L
m
0
L!
Ý
Ž x X , x 3 , . . . , x my1 .g⌬
=
ž
2
m
⭈⭈⭈ =
tX
X
Ž x . ! x 3 ! ⭈⭈⭈ x my1 ! Ž L y Ž xX q x 3 q ⭈⭈⭈ x my1 . . !
x
X
=
ž ' // ž
ž ž ' //
exp
1
m
L
exp
t my 1
L
1
m
x my 1
exp
t3
ž ' //
L
x3
= ⭈⭈⭈
100
SHIGEO MUTO
X
ž ž
= 1y
2
1
q
q ⭈⭈⭈ q
^m `
m
Ly Ž x qx 3 q ⭈⭈⭈ qx my 1 .
1
//
_
m
my 3
=exp ytX
ž
s
ž
2
m
exp
2'L
y t3
m
tX
q
ž' /
L
1
m
'L
'L
y ⭈⭈⭈ yt my1
m
exp
t3
ž' /
L
/
m
q ⭈⭈⭈ q
1
m
exp
t my1
ž' /
L
L
q1 y
ž
2
m
1
q
1
q ⭈⭈⭈ q
^m `
//
_
m
my 3
=exp ytX
ž
2'L
y t3
m
'L
m
'L
y ⭈⭈⭈ yt my1
/
m
Hence
ln ␮ Ž tX , t 3 , . . . , t my1 .
s L ln 1 q
ž
2
m
tX
y1 q
exp
L
q ⭈⭈⭈ q
y tX
2'L
m
1
t3
y1
ž ž' / / ž ž' / /
ž ž ' / //
y t3
'L
m
y ⭈⭈⭈ yt my1
exp
m
1
exp
m
'L
m
L
t my 1
L
y1
.
By Taylor’s theorem, we obtain
exp
exp
tX
ž' /
ž' /
L
t3
L
y1s0q
y1s0q
1
'L
1
'L
tX q
t3 q
1
2L
1
2L
tX 2 q
t 32 q
1
3!L'L
1
3!L'L
tX 3 q ⭈⭈⭈
t 33 q ⭈⭈⭈
⭈⭈⭈
exp
t my 1
ž' /
L
y1s0q
1
'L
t my 1 q
1
2L
t my12 q
1
3!L'L
t my13 q ⭈⭈⭈ .
101
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
Let
us
2
tX
y1
ž ž' / /
ž ž' / /
exp
m
1
q
L
t3
exp
m
y 1 q ⭈⭈⭈ q
L
1
m
t3
y1 .
ž ž' / /
exp
L
Then
2
us
tX q
m'L
q
2
2 mL
6 mL'L
q ⭈⭈⭈ .
t 3 q ⭈⭈⭈ q
m'L
tX 2 q
2
q
1
1
2 mL
t my1
m'L
t 32 q ⭈⭈⭈ q
1
tX 3 q
1
1
2 mL
t my12
t 33 q ⭈⭈⭈ q
6 mL'L
1
6 mL'L
t my13
By Taylor’s theorem
ln Ž 1 q u . s 0 q u y 12 u 2 q 13 u 3 y ⭈⭈⭈ .
Therefore we obtain
ln ␮ Ž tX , t 3 , . . . , t my1 .
s L ln Ž 1 q u . y tX
s
2
2m
y
tX 2 q
2
m
2
1
m
y
ž
1y
2
m
2
2m
2
m
2
2
m
2m
m
t
2 3
2
tX 2 q
tX t 3 y ⭈⭈⭈ y
'L
2m
1
m
1
2m
2
m
2
ž
t t
2 3 4
1y
'L
y ⭈⭈⭈ yt my1
m
1
y ⭈⭈⭈ y
tX t my 1 y
/
y t3
t 32 q ⭈⭈⭈ q
1
tX 2 y
y ⭈⭈⭈ y
s
1
2'L
m
t my12
1
2m
t
2 my1
y ⭈⭈⭈ y
1
m
tX t my1 y
/
2
y
1
m2
2
m2
t my2 t my1 q o Ž 1 .
t 32 q ⭈⭈⭈ q
1
m
t t
2 3 4
tX t 3
1
2m
y ⭈⭈⭈ y
ž
1y
1
m2
1
m
/
t my12
t my2 t my1 q o Ž 1 . ,
102
SHIGEO MUTO
where oŽ1. is a function of tX , t 3 , . . . , t my1 tending to zero as L ª ⬁.
Hence
␮ Ž tX , t 3 , . . . , t my1 . ª exp
L
1
ž ž
m
q
y
2
m
2
tX t 3 y ⭈⭈⭈ y
2
m
2
2
1y
1
2m
ž
m
/
1y
tX t my1 y
1
m
tX 2
1
m
t t
2 3 4
/
t 32 q ⭈⭈⭈ q
y ⭈⭈⭈ y
1
m2
1
2m
ž
1
1y
m
/
t my12
/
t my2 t my1 . Ž A.27.
The r.h.s of ŽA.27. is a moment generating function of Ž m y 2.-variate
normal distribution with mean vector Ž0, 0, . . . , 0. and variance᎐covariance
matrix
¡2
m
ž
1y
y
¢
y
2
m
y
/
2
1
m2
..
.
m
2
m
2
ž
2
m
2
1y
..
.
y
1
m
/
y
⭈⭈⭈
y
..
1
m
⭈⭈⭈
.
⭈⭈⭈
2
1
m
ž
¦
2
m
1
2
m2
..
.
1y
1
m
/§
Thus when L is very large, a distribution function of random variables
Ž Y X ,Y3 , . . . , Ymy1 . is approximated by this Ž m y 2.-variate normal distribution. Denote this Ž m y 2.-variate normal distribution by ⌽.
In the expression
2 Ž LrmqL2r3 .
Ý
X
x s2 Lrm
s Ž xX , x 3 , . . . , x m . ,
Ý
Ž x 3 , . . . , x m .g⍀ Ž x .
it follows from ŽA.26. that if xX s 2 Lrm, then yX ª 0 as L ª ⬁10 ; and if
xX s 2Ž Lrm q L2r3 ., then yX ª ⬁ as L ª ⬁. Further the conditions on
xX , x 3 , . . . , x m are rewritten as
yX G 2 y 3 , . . . , yX G 2 ymy1 , yX G y
10
X
X
2
3
Ž y 3 q ⭈⭈⭈ qymy1 . .
Strictly speaking, x s u Lrm v; thus, we have y ª 0 as L ª ⬁.
BANZHAF INDEX IN REPRESENTATIVE SYSTEMS
103
Therefore
2 Ž LrmqL2r3 .
X
Ý
x s2 Lrm
Ý
s Ž xX , x 3 , . . . , x m .
Ž x 3 , . . . , x m .g⍀ Ž x .
ª ⌽ Ž Ž yX , y 3 , . . . , ymy1 .
L
yX G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . . Ž A.28.
Ž4. PROOF
OF
ŽA.6.. From ŽA.24., ŽA.25., and ŽA.28., we obtain
f Ž Lrm, Lrm q L2r3 .
f
L
1
2'␲
(
m
L
⌽ Ž Ž yX , y 3 , . . . , ymy1 .
=yX G max Ž 2 y 3 , . . . , 2 ymy1 , y 23 Ž y 3 q ⭈⭈⭈ qymy1 . , 0 . . .
Since L s ␣ i KM, the proof of Žb. is completed.
We have thus shown that ŽA.2. holds.
Q.E.D.
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Bolger, E. M. Ž1993.. ‘‘A Value for Games with n Players and r Alternatives,’’ Internat. J.
Game Theory 22, 319᎐334.
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