Discussion Paper Series

Graduate School of Economics and School of Economics

Meisei University

Discussion Paper Series, No.43
March 22, 2019

A Generalization of Peleg’s Representation Theorem
on Constant-Sum Weighted Majority Games

Takayuki Oishi(Meisei University)

Hodokubo 2-1-1, Hino, Tokyo 191-8506
School of Economics, Meisei University
Phone:+81-(0)42-591-9479

Fax: +81-(0)42-599-3024

URL: https://keizai.meisei-u.ac.jp/econ/
E-mail: keizai@econ.meisei-u.ac.jp

A Generalization of Peleg’s Representation
Theorem on Constant-Sum Weighted Majority
Games
Takayuki Oishi
March 22, 2019

Corresponding author. Faculty of Economics, Meisei University, 2-1-1, Hodokubo, Hinocity, Tokyo 191-8506, Japan. (E-mail: takayuki1q80@gmail.com Tel: +81-42-591-5921)

Abstract We propose a variant of the nucleolus associated with distorted
satisfaction of each coalition in TU games. This solution is referred to as
the -nucleolus in which
is a pro…le of distortion rates of satisfaction of
all the coalitions. We apply the -nucleolus to constant-sum weighted majority games. We show that under assumptions of distortions of satisfaction of
winning coalitions the -nucleolus is the unique normalized homogeneous representation of constant-sum weighted majority games which assigns a zero toeach null player. As corollary of this result, we derive the well-known Peleg’s
representation theorem.
Keywords: Constant-sum weighted majority games; Homogeneous representation; -Nucleolus; Distorted satisfaction; Peleg’s representation theorem
JEL Classi…cation Number: C71
Acknowledgements: I would like to thank Tamás Solymosi for helpful discussions
and kind hospitality for my visit at Corvinus University of Budapest. For usefulcomments, I am grateful to Shin Sakaue. I am …nancially supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B), Project #17K13751. I am responsible
for any remaining errors.

1

Introduction

Constant-sum weighted majority games have been known as the most classicalgames applied to voting systems, e.g., von Neumann and Morgenstern (1944).
1

These games are simple games in which a coalition wins i¤ the sum of theweights of its members is larger than half the sum of weights of all the players. Since these games are derived from a pro…le of weights of players and aquota, a pair of the weight pro…le and the quota is called a representation. In
particular, if a quota is the minimum of the sum of the weights attainable bya minimal winning coalition, then a pro…le of weights is called a normalized
representation. In addition, if each minimal winning coalition carries the same
weight, then it is called a normalized homogeneous representation. If such ahomogeneous representation exists, a constant-sum weighted majority game iscalled homogeneous. Since the seminal work of von Neumann and Morgenstern(1944), whether a unique normalized homogeneous representation in constantsum weighted majority games exists has been an important open question
until Peleg (1968) solved this problem. Peleg (1968) showed that in constantsum weighted majority games the nucleolus (Schmeidler 1969) is always theunique normalized homogeneous representation which assigns a zero to eachnull player. The nucleolus is an established solution for coalitional games withtransferable utility (for short, TU games), and it is a game-theoretic expressionof the ‘di¤erence principle of social justice’ à la Rawls (1971). As Maschler(1992) pointed out, Peleg’s representation theorem shows one of the nicest
results of solutions for TU games applied to voting systems.
In this note, we generalize Peleg’s representation theorem on constant-sum
weighted majority games (Peleg 1968). In constant-sum weighted majority
games, the nucleolus is a single-weight that lexicographically maximizes eachcoalition’s satisfaction over the set of weights. Given a pro…le of weights,
satisfaction of each coalition is the di¤erence between the sum of weights of its
members and its coalitional worth. Thus satisfaction of each coalition plays
a role in the nucleolus. We consider another scenario of satisfaction of eachcoalition by using the notion of utopia payo¤s. The utopia payo¤ of each player
is her marginal contribution to the grand coalition. Reasonable hope of eachcoalition may be regarded as the sum of utopia payo¤s of its members. The
sum of utopia payo¤s might be the most that each coalition could reasonably
hope for in the sense of Milnor (1952). We employ the notion of aspirationlevels in computing satisfaction instead of coalitional worth. The aspirationlevel of each coalition is a value that lies somewhere between its coalitional
worth and its reasonable hope (i.e. the sum of utopia payo¤s of its members).
For each coalition S, we refer to the di¤erence between its reasonable hopeand its coalitional worth as the utopia gap of S. For each coalition S, thenumber S is a weight ratio on its utopia gap. The aspiration level of eachcoalition S is the sum of its coalitional worth and its weighted utopia gap
associated with S . Given a pro…le of weights, distorted satisfaction of eachcoalition is the di¤erence between the sum of weights of its members andits aspiration level. In this sense, for each coalition S the number S is a2

distortion rate of satisfaction. Using these notions, we introduce a variant of
the nucleolus, referred to as the -nucleolus, where is a pro…le of distortions ofsatisfaction of all the coalitions. We show that under assumptions of distortionsof satisfaction of winning coalitions the -nucleolus is the unique normalized
homogeneous representation of constant-sum weighted majority games whichassigns a zero to each null player. As a corollary of the representation theorem
in the present study, we derive that if no coalition has distortion of satisfaction,then the -nucleolus is the nucleolus. This is Peleg representation theorem.
In the representation theorem in the present study, we put two assumptions
on distortions of satisfaction of winning coalitions. Firstly, in constant-sum
weighted majority games that are homogeneous, it seems natural for us toconsider that each minimal winning coalition that carries the same weight has
the same distortion of satisfaction. The …rst assumption says that a distortion
rate of satisfaction of each minimal winning coalition is homogeneous, and it
is minimal among all the winning coalitions. Such a homogeneous distortion
rate is assumed to be at most one minus the maximal quota derived fromnormalized representations. Secondly, it also seems natural for us to consider
that a weight ratio on the utopia gap of each winning coalition that does notinclude a null player is invariant under a situation where the null player joins.
This is because a null player does not make any change of the utopia gapof each winning coalition that does not include the null player. The second
assumption says that a distortion rate of each winning coalition that does notinclude a null player is invariant under a situation where the null player joins.
This note is organized as follows. In Section 2, we de…ne the -nucleolus.
Using the notion of the -nucleolus, in Section 3, we present the unique representation of constant-sum weighted majority games. In Section 4, we remark
on the representation theorem in the present study.

2

The -nucleolus

Let N be a non-empty and …nite set of agents. A coalitional game withtransferable utility for N (a TU game for N , for short) is a function v : 2N !
R with v(;) = 0. For all S 2 2N , v(S) represents what coalition S canachieve on its own. Let V be the class of all TU games. Let V be aPgenericsubclass of V, that is, V
V. For x 2 RN and S
N , let x(S)i2S xi .
For all v 2 V, the following notations are introduced: For all i 2 N , let
Miv v(N ) v(N nfig) be the marginal contribution of agent i to the grandcoalition N . The number Miv P
is also called agent i’s utopia payo¤ of v.
vv
For each S
N , let M (S)i2S Mi be the sum of utopia payo¤s of themembers of S.
The aspiration level of each coalition S
N is a value that lies somewhere between v(S) and M v (S). For each coalition S
N , we refer to the3di¤erence between M v (S) and v(S) as the utopia gap of S. For each S N ,let g v (S) be the utopia gap of S, that is, g v (S)M v (S) v(S). For eachcoalition S
N , the number S 2 [0; 1] is a weight ratio on the utopia gap
of S. Let
( S )S22N , where S 2 [0; 1]. Since this pro…le of weight ratios
makes distorted satisfaction mentioned below, we call it a pro…le of distortions of satisfaction. Given = ( S )S22N , where S 2 [0; 1], the aspirationlevel of S, denoted v (S), is de…ned by setting for each S 2 2Nv (S)v(S) +v
S g (S):The -aspiration game of v is the mapping that associates with each coalition S N its aspiration level v (S). By the de…nition of v and g v , v (;) = 0.
Let Ef (v) be the set of vectors x 2 RN such that x(N ) = v(N ). Let IP(v)be the set of imputations x 2 RN such that x(N ) = v(N ) and for all i 2 N
N
xi v(fig). On the domain of V, given 2 [0; 1]2 , distorted satisfactionof each coalition S 2 2N with respect to x 2 Ef (v) is de…ned by setting foreach S 2 2N
f (S; x; v) x(S) v (S):N

Let e(x) (f (S; x; v))S22N 2 R2 , givenNgraphic ordering of R2 .1N

2 [0; 1]2 . Letlex be the lexico-N

De…nition 1 On the domain of V, given 2 [0; 1]2 , the -prenucleolus,
denoted PN (v), is de…ned as follows:on
PN (v)
x 2 Ef (v) e(x) lex e(y) for all y 2 Ef (v) :Proposition 1 On the domain of V, given
point.

N

2 [0; 1]2 , PN (v) is a single

The proof is identical to the uniqueness argument of the prenucleolus in
Theorem 5.1.14 in Peleg and Sudhölter (2003).2
Remark 1 PN (v) coincides with the following single-valued solutions.
(i) On the domain of V, if
cleolus (Schmeidler 1969).3
1

N= 0, then PN (v) coincides with the prenu-N

For all z 2 R2 , (z) 2 R2 is de…ned by rearranging the coordinates of z in nonNdecreasing order. For all z; z 0 2 R2 , z is lexicographically larger than z 0 if 1 (z) >
0
0
0
0
0
1 (z ) or [ 1 (z) = 1 (z ) and 2 (z) > 2 (z )] or [ 1 (z) = 1 (z ) and 2 (z) = 2 (z ) and00
3 (z) > 3 (z )], and so on. Then, we write z
lex z .
2
Theorem 5.1.14 is itself a consequence of Theorems 5.1.6 and Cololary 5.1.10 in Pelegand Sudhölter (2003).
3
An 2N -dimensional vector 0 = (0; 0;
; 0).

4

(ii) On the domain of V such that v(N ) M v (N ), if = 1, then PN (v)coincides with the ENSC value (Hou et al. 2018).4 Notice that the ENSC
value5 is de…ned by setting for each i 2 N ,ENSCi (v)

Miv +

v(N )

M v (N ):jN j

Let V be such that IP(v) 6= ; for all v 2 V. On the domain of V, givenN
2 [0; 1]2 , distorted satisfaction of each coalition S 2 2N with respect tox 2 IP(v) is de…ned by setting for each S 2 2N f (S; x; v) x(S) v (S). LetNNe(x) (f (S; x; v))S22N 2 R2 , given 2 [0; 1]2 .
De…nition 2 On the domain of V such that IP(v) 6= ; for all v 2 V, givenN
2 [0; 1]2 , the -nucleolus, denoted N (v), is de…ned as follows:
n
o
N (v)
x 2 IP(v) e(x) lex e(y) for all y 2 IP(v) :By the argument appearing in Schmeidler (1969) together with Proposition
1, the -nucleolus is a single point since IP(v) is nonempty, compact, andconvex.
The following example shows that thecoincide with the nucleolus.

-nucleolus does not necessarily

Example 1 Let N = f1; 2; 3g. Let v : 2N ! R such that for all i 2 Nv(fig) = 0, v(f1; 2g) = 30, v(f1; 3g) = 40, v(f2; 3g) = 80, v(N ) = 120, andv(;) = 0. For all S 2 2N , S = 1=2. By simple calculation, the nucleolus
is given by N(v) = (20; 45; 55), and the -nucleolus is given by N (v) =(10; 50; 60). Therefore, N(v) 6= N (v).
In this note, we will not proceed further investigation into game-theoretic
properties of the -nucleolus. Our target is to derive a unique homogeneous
representation of constant-sum weighted majority games by using the -nucleolus.
We will focus on this topic in the next section.
4
5

An 2N -dimensional vector 1 = (1; 1;; 1).
“ENSC” means “Egalitarian Non-Separable Contribution”.

5

3

The unique representation of constant-sum
weighted majority games

Let G(N; W) be a simple game, where W is the set of winning coalitions,i.e.,
N 2 W; ; 2= W;(S
T N and S 2 W) =)T 2 W.
Let vG be the corresponding TU game of G(N; W), where vG (S) = 1 if S 2 W;
and vG (S) = 0 otherwise. Notice that vG is monotonic. Let W m be the set ofminimal winning coalitions, that is,noWm
S2W T (S)T 2=W .

N and a pro…le of weights w = (wi )i2N 2 RN
+ , let w(S)P For each S
i2S wi . A simple game G(N; W) is a weighted majority game if there
exists a quota q > 0 and a pro…le of weights w = (wi )i2N 2 RN
+ such that
S 2 W () w(S)q:

If a simple game G(N; W) is a weighted majority game, then (q; w) is called a
representation of G(N; W). The simple game is strong if
S2= W () N nS 2 W:
A player i 2 N is a veto player if i 2 \S2W S. A player i 2 N is a nullplayer if for all S
N nfig vG (S) = vG (S [ fig). Let D be the set of nullplayers.
For w 2 RN
+ , letq(w)minm w(S):S2W

We call x 2 Ef (vG ) a normalized representation of G if (q(x); x) is a
representation of G.
A weighted majority game G is constant-sum if vG (S) + vG (N nS) = 1
for all S
N . Let G be a constant-sum weighted majority game forwhich there exists w 2 RN
+ such that
1
S 2 W () w(S) > w(N ):2
Since G is strong, vG is superadditive. Therefore, IP(vG ) 6= ;, which implies
that N (vG ) 6= ;. We call x 2 Ef (vG ) a normalized homogeneous representation of G if x is a normalized representation of G and x(S) = q(x)6for all S 2 W m .
Claim 1 Let G be a constant-sum weighted majority game. If a veto player
exists in G , then the number of veto players is one.
Proof. Suppose that the number of veto players is more than one. It su¢ ces
to consider the case where the number of veto players is two. Let k; k 0 be vetoplayers, i.e. k; k 0 2 \S2W S. Take S 2= W such that k 2 S and k 0 2= S. By thede…nition of G , N nS 2 W such that k 2= N nS and k 0 2 N nS, a contradiction.

Claim 2 Let G be a constant-sum weighted majority game. If no veto player
exists in G , then for all i 2 N fig2W.
=
Proof. Suppose that there exists { 2 N such that f{g 2 W. Since f{g 2 W m ,w{ > 12 w(N ), which implies that { must be a veto player, a contradiction.
Lemma 1 (Peleg 1968, Lemma 3.1) Let G be a constant-sum weighted majority game. An imputation x 2 IP(vG ) is a normalized representation of G
if and only if q(x) > 21 .
We introduce the following property of distortion, referred to as a minimal
homogeneous distortion with respect to G . This property states that in aconstant-sum weighted majority game a distortion rate of satisfaction of eachminimal winning coalition is homogeneous, and it is minimal among all thewinning coalitions. Such a homogeneous distortion rate is assumed to be at
most one minus the maximal quota derived from normalized representationsof G . Notice that there exists the maximal quota derived from normalized
representations of G .6
6

Since the nucleolus is a normalized representation (Peleg 1968, Theorem 3.4), the set of
normalized representations of G , denoted X, is nonempty. Let r q(x) for all x 2 X 6= ;.
We consider two cases. Case 1: A unique veto player exists in G . By Claim 1, let i be aunique veto player. Let us consider the following problem: max r subject to xi
r, xi 0for all i 2 N nfi g, and x(N ) = 1. The optimal solution is r = 1, which is attainable byx 2 RN such that xi = 1 and xi = 0 for all i 2 N nfi g. Since x is an imputation andr > 1=2, there exists max q(x) in this case. Case 2: No veto player exists in G . By Claim2, a normalized representation is an imputation. Let us consider the following problem:
max r subject to x(S)r > 1=2 for all S 2 W m , xi
0 for all i 2 N , and x(N ) = 1.
Since the nucleolus is feasible for the problem and the objective function is bounded above,
there exists max q(x) in this case. By the argument of the two cases mentioned above,
max q(x) 2 (1=2; 1].

7

De…nition 3 (Minimal homogeneous distortion) Let G be a constantsum weighted majority game. Let X be the nonempty set of normalized representations of G . A pro…le of distortions of satisfaction
is a minimal0
homogeneous distortion with respect to G if (i) for all S; S 2 W m such that
S 6= S 0 , S = S 0 1 maxx2X q(x), and (ii) for all T 2 W m and all T 0 2 W
TT0.
Lemma 2 Let G be a constant-sum weighted majority game. Assume thata pro…le of distortions of satisfaction is a minimal homogeneous distortion
with respect to G . Then if an imputation x 2 IP(vG ) is a normalized representation of G , q(N (vG )) q(x).
Proof. We consider two cases.
Case 1: A veto player exists in G .
By Claim 1, there exists a unique veto player i . For all x 2 IP(vG ),min f (S; x; vG ) = min [x(S)(1= minfq(x)

1;

S22N

S22N

= q(x)
since q(x)S )vG (S)

SM

(S)]

min xi g

i2N nfi g

1;

1 2 ( 1=2; 0] by Lemma 1, and mini2N nfi g xi

min f (S; N (vG ); vG ) = q(N (vG ))

1

S22N

which implies q(N (vG ))vG

0. Then,min f (S; x; vG ) = q(x)

S22N

1;

q(x).

Case 2: No veto player exists in G .
By Claim 2, for all i 2 N fig2W.
=
For all i 2 N , since N nfig 2 W,vGm
Mi = 0. Let = S for all S 2 W . For all x 2 IP(vG ),min f (S; x; vG ) = min [x(S)

S22N

(1

S22N

SM

vG

(S)]

1 + ; min xi g

= minfq(x)= q(x)S )vG (S)

i2N

1+ ,

since q(x) 1 +0 by the assumption of the minimal homogeneous distortion, and mini2N xi 0. Then,min f (S; N (vG ); vG ) = q(N (vG ))

S22N

1+

min f (S; x; vG ) = q(x)S22N

which implies q(N (vG ))

q(x).
8

1+ ;

Proposition 2 Let G be a constant-sum weighted majority game. Assume
that a pro…le of distortions of satisfaction is a minimal homogeneous distortion with respect to G . Then the -nucleolus of G is a normalized representation of G .
Proof. Let x 2 IP(vG ) be a normalized representation of G . By Lemma 1,
q(x) > 21 , and by Lemma 2, q(N (vG )) q(x), which implies that q(N (vG )) >1
. Again, by Lemma 1, N (vG ) is a normalized representation of G .
2
Next, we introduce the following notations. For all players i; j 2 N suchthat i 6= j , letnoTij (N )S N i 2 S and j 2=S :For all x 2 IP(vG ), let sij (x) = minS2Tij (N ) f (S; x; vG ). The -kernel of vG
is de…ned as follows:nfor all i; j 2 N with i 6= j sij (x) sji (x) o
K (vG )
x 2 IP(vG ):or for all k 2 N xk = vG (fkg)

The -kernel is a variant of the kernel (Davis and Maschler 1965) associatedwith distorted satisfaction of each coalition.
For all x 2 IP(vG ), letnoNNF(x; vG )S 2 2 nfN; ;g f (S; x; vG ) f (T; x; vG ) 8T 2 2 nfN; ;g :The collection F is separating if i; j 2 N such that i 6= j and F(x; vG ) \Tij (N ) 6= ;, then F(x; vG ) \ Tji (N ) 6= ;.

Claim 3 Let G be a constant-sum weighted majority game in which no vetoplayer exists. F(N (vG ); vG ) is separating.
N

Proof. Fix an arbitrary 2 [0; 1]2 . On the domain of V such that for allv 2 V IP(v) 6= ;, it is well known that the nucleolus is included in the kernel.
By the same argument as in the proof of this inclusion (e.g., Theorem 5.1.17
in Peleg and Sudhölter (2003)), it follows that N (vG )K (vG ), which
implies that K (vG ) 6= ;. Since no veto player exists in G , K (vG ) is theset of x 2 IP(vG ) such that for all i; j 2 N with i 6= j sij (x) = sji (x). For allx 2 K (vG ) F(x; vG ) is separating. Therefore F(N (vG ); vG ) is separating.
Next, we introduce the following property of distortion, referred to as thenull player property of distortion with respect to G . This property statesthat in a constant-sum weighted majority game, given an arbitrary null playerk, for each winning coalition S that does not include k, a distortion rate of S
is the same as that of S [ fkg.
9

De…nition 4 (Null player property of distortion) Let G be a constantsum weighted majority game. Let D be the set of null players. Fix an arbitrarynull player k 2 D in G . A pro…le of distortions of satisfaction satis…es thenull player property of distortion with respect to G if for each S 2 W withk2
= S, S = S[fkg .
We are in the position to present the main result.
Theorem 1 Let G be a constant-sum weighted majority game. Assume that(i) a pro…le of distortions of satisfaction is a minimal homogeneous distortion with respect to G , and (ii) it satis…es the null player property of distortion
with respect to G . Then the -nucleolus of G is the unique normalized homogeneous representation of G which assigns a zero to each null player ofG.
Proof. Let D be the set of null players of G . Let y be a normalized homogeneous representation of G which satis…es yi = 0 for all i 2 D. Since y ishomogeneous,y(S) = q(y) for all S 2 W m :Let R be the set of imputations r 2 IP(G ) such that r(S)q(y) for allm
S 2 W and ri = 0 for all i 2 D. Let x = N (vG ). We consider three steps.
Step 1: For all k 2 D, xk = 0.
By Claims 1 and 2, we consider two cases of Step 1.
Case 1 of Step 1: A unique veto player exists in G .
It is clear that for all k 2 D, xk = 0. This is because x = (0;; 0; 1; 0;; 0),
where 1 is assigned to a unique veto player and 0 is assigned to each player
except for the veto player.
Case 2 of Step 1: No veto player exists in G .
By Claim 2, it is clear that for all i 2 N , xi vG (fig) = 0. It su¢ ces toshow that for all k 2 D, xk
0. Suppose that there exists k 2 D such thatxk > 0. Fix such a k. By the assumption (ii), for k 2 D, and each S 2 W
with k 2= S, S = S[fkg . Fix an arbitrarily …xed S 2 W such that k 2=S .
Since S = S [fkg and xk > 0,
f (S ; x; vG ) = x(S )< x(S )

1+
1+

S
S [fkg + xk

= f (S [ fkg; x; vG );10

which implies that f (S ; x; vG ) < f (S [ fkg; x; vG ). For an arbitrarily …xed
S 2
= W such that k 2= S , f (S ; x; vG ) = x(S ) < x(S [ fkg) = f (S [fkg; x; vG ). Therefore, for k 2 D and each S N such that k 2= S,
f (S; x; vG ) < f (S [ fkg; x; vG );which implies that k 2= [S2F (x;vG ) S, a contradiction to Claim 3.
Step 2: x 2 R.
By the assumption (i), Lemma 2 and Proposition 2 hold. By the factthat q(x)q(y) by Lemma 2 together with the fact that for all S 2 W mx(S) q(x) by Proposition 2, x(S) q(y) for all S 2 W m . By this observation
together with Step 1, x 2 R.
Step 3: R = fxg.
Suppose not. Since R 6= ; by Step 2, R has an extreme point z such thatz 6= x. Since y(S) = q(y) for all S 2 W m , there exists j 2= D such thatmzj = 0. Fix such a j. Since j 2= D, there exists S 2 W such that j 2 S.
Fix such an S. Since Snfjg 2= W, R(N nS) [ fjg 2 W. Since z 2 R,z(S) q(y) = y(S). By Lemma 1,
1< z(R) = z(N nS) = 1
2

z(S)

1

1
y(S) < ;2which is impossible. Therefore, y = x.
As a corollary of Theorem 1, we derive the well-known representation theorem on constant-sum weighted majority games (Peleg 1968).
Corollary 1 If
= 0, then the nucleolus of G is the unique normalized
homogeneous representation of G which assigns a zero to each null player ofG (see Peleg 1968, Theorem 3.5).

4

Concluding remarks

Finally, we remark on Theorem 1. Let G be a constant-sum weighted majority game. Let f ;g be any pair of pro…les of distortions of satisfaction.
Assume that (1)andare minimal homogeneous distortions with respect to G , and (2)andsatisfy the null player property of distortion
with respect to G . According to Theorem 1, by uniqueness, the -nucleolus
and the-nucleolus must coincide for G . By this observation together with
Corollary 1, the -nucleolus and the nucleolus must coincide for G if is aminimal homogeneous distortion and it satis…es the null player property. As
a consequence, even if coalitions have distortions of satisfaction satisfying the11two assumptions proposed in the present study, the nucleolus is the uniquenormalized homogeneous representation of constant-sum weighted majority
games. In this respect, Theorem 1 is a generalization of Peleg’s representation
theorem. We close this note with the following example that shows the fact
mentioned above.
Example 2 Let G (N; W) be a constant-sum weighted majority game, where
the set of players is given by N = f1; 2; 3; 4; 5; 6g, the set of minimal winningcoalitions is given by W m = ff1; 2g; f1; 3g; f2; 3; 4g; f2; 3; 5g; f1; 4; 5gg, andthe set of null players is given by D = f6g. Consider an arbitrary pro…leof distortions of satisfaction
satisfying that (i) the minimal homogeneous
distortion assumption: for all S; S 0 2 W m such that S 6= S 0 , S = S 0 4=9,
and for all T 2 W m and all T 0 2 W T
T 0 , and (ii) the null player propertyof distortion: for each S 2 W with 6 2= S, S = S[f6g . Then the uniquenormalized homogeneous representation of G which assigns a zero to the nullplayer is the -nucleolus of G , that is, N (vG ) = (3=9; 2=9; 2=9; 1=9; 1=9; 0) =N(vG ), where N(vG ) is the nucleolus of G .

12

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[7] Rawls, J.: A Theory of Justice, Oxford University Press, 1971.
[8] Schmeidler, D.: The Nucleolus of a characteristic function game. SIAM J. Appl.
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[9] von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior,Princeton University Press, 1944.

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