Derks, Jean

Peters, Hans

English

Research Papers in Economics

Mathematical Social Sciences 36 (1998) 175–182
Orderings, excess functions, and the nucleolus
ab , * Jean Derks , Hans Peters
a
Department of Mathematics, University of Maastricht, P.O. Box 616, 6200 MD Maastricht, Netherlands
bDepartment of Economics, University of Maastricht, P.O. Box 616, 6200 MD Maastricht, Netherlands
Received 2 May 1997; received in revised form 6 January 1998; accepted 14 January 1998
Abstract
The nucleolus of a cooperative game can be described with the aid of the leximin ordering but
also on the basis of two other orderings. In this note the relation between these orderings is studied
in a more general framework. The results are applied to the nucleolus corresponding to so-called
normal excess functions. Also the Kohlberg criterion is extended to this more general case.
Ó 1998 Elsevier Science B.V. All rights reserved.
Keywords: Cooperative games; Nucleolus; Orderings; Excess functions
1. Introduction
The nucleolus of a cooperative game with transferable utility, introduced in Schmeid-
ler (1969), lexicographically minimizes the nonincreasingly ordered excesses of the
coalitions over the imputation set of the game. More generally, let M be a ﬁnite set of
M agents (e.g. coalitions) and let P #R be some set of feasible vectors (e.g. the excess
vectors corresponding to a collection of payoff vectors in a game, see Section 3 below).
Consider the ordering K (d from ‘desirable’) deﬁned by d
uB (a)u5uB (b)u for all t .t9 tt aK b :. there is a t9[R withH d uB (a)u,uB (b) t9 t9
or uB (a)u5uB (b) for all t [R tt
for all a, b[P, where B () : 5hj[M:a $tj denotes the set of agents for which the ta j
corresponding coordinates in a are at least t, and uB (a)u denotes the cardinality of this t
set. It is straightforward to verify that
*Corresponding author. Tel.: 131 43 3883834; fax: 131 43 3258535; e-mail: h.peters@ke.unimaas.nl
0165-4896/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.
PII: S0165-4896(98)00010-9176 J. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182
aK b Û a*K b*, d lexmin
where a* arises from a by rearranging the coordinates in nonincreasing order and
K denotes lexicographical minimization; that is, a*K b* if, and only if, lexmin lexmin
** ** a ,b for the smallest j with a ±b . Thus, the nucleolus may alternatively be deﬁned jj jj
using K . This ordering has the advantage that it is closer in formulation to the two d
orderings to be deﬁned next. For a comprehensive survey on the merits of the nucleolus
and related solutions see Maschler (1992).
In Justmann (1977), Justman considers the ordering
aK b : Û a 5b, or there is a j [M such that: J
a ,b and for all i [M:a .b implies a #a . jj ii ij
He shows that under certain conditions an iterative process based on this ordering
converges to the nucleolus, when applied to a game. Also Osborne and Rubinstein
propose an alternative ordering in their deﬁnition of the nucleolus (see Osborne and
Rubinstein, 1994). This is the following ordering:
aK : Û a 5b, or there is a j [M such that: OR
a ,b and for all i [M:a .b implies a #b . jj ii ij
The advantage of both these orderings over the desirability relation K or the lexmin d
ordering K is that they admit a more transparent interpretation in terms of lexmin
objections and counterobjections (cf. Osborne and Rubinstein, 1994, p. 286). To see this,
ﬁrst realize that in agreement with their interpretation as excesses, coordinates should be
seen as disutilities; so lower coordinates are better. Concerning the ordering K one can J
imagine some agent i objecting against some proposal a by referring to the alternative b;
then agent j may counterobject by stating, not only that a is better for him, j, than b, but
also that a is in fact better for agent i than for agent j himself; thus, by insisting on a
agent j actually accepts that he will end up less satisﬁed than agent i. Observe that such
an interpretation assumes that coordinates of different agents can be meaningfully
compared. The relation K can be given an interpretation in the same spirit, with the OR
difference that this time agent j, in counterobjecting against b, refers to the fact that b is
worse for him than a is for agent i.
The ﬁrst objective of this note is to clarify the relations between the three orderings
deﬁned above. This is done in Section 2. In Section 3 the result is applied to so called
normal excess vectors in cooperative games with transferable utility, which leads to a
generalization of existing results. A corresponding generalization of the Kohlberg
criterion involving balanced collections completes the paper.
1.1. Notation
# denotes set inclusion, , denotes strict set inclusion.
2. Comparison of K , K , and K Jd O R
M Let P #R , as in the Introduction. The following lemma is immediate from the
deﬁnitions.J. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182 177
Lemma 1. Let a, b[P. Then:
(i) aK b ÞaK b JO R
(ii) aK b, a ±b Þnot bK a, not bK a . JJ O R
As to the relation with K we have the following result. d
Theorem 1. Let a, b[P. Then: aK bÞaK ÞaK b. Jd O R
Proof. Assume aK b.I fa5b then uB (a)u5uB (b)u for all t[R,s oaK b. Jt t J
Now assume a±b, and let j[M as in the deﬁnition of aK b. Deﬁne t9:5maxhb :b ± Jk k
a j. Take t$t9 and suppose that b ,t#a for some i[M. Then a #a ,b #t9,a ki i i j j
contradiction. This shows that B (a)#B (b) and hence uB (a)u#uB (b)u for all t#t9. tt t t
For t5t9 this inequality is strict: for k[M with b 5t9, a .b would imply kk k
b ,a #a ,b contradicting the deﬁnition of t9, hence a ,b 5t9. For t.t9 the kkjj kk
inequality is in fact an equality, which can be seen as follows. Suppose, to the contrary,
that there is an i[M with i[B (b)\B (a). Then b $t.a ,s ob #t9,t by deﬁnition of t9, tt i i i
a contradiction.
It follows that also in this case aK b. d
Next, assume aK b.I fa5b then a± b. Now assume a±b. Let t9 be such that dO R
B (a)5B (b) for all t.t9, and B (a)±B (b). tt t 9 t9
Suppose B (b),B (a), then obviously bK a. Together with aK b this implies uB (a)u5 t9 t9 dd t
uB (b)u for all t, and in particular for t5t9, a contradiction. It follows that B (b)\B (a)± t t9 t9
[.
So, take j[B (b)\B (a). Then b .a . For i with a .b one must have a #t9 since t9 t9 jj ii i
otherwise i[B (a)5B (b) for t5a , and thus, b #a , a contradiction. Therefore, a #t# tt i i i i
b , hence aK b. h jO R
Remark. In the proof of Theorem 1 the following has in fact been shown as well.
B (a)5B (b) for all t .t9 tt aK b Þ there is a t9[R withH J B (a),B (b) t9 t9
or a 5b ,
and
B (a)5B (b) for all t .t9 tt aK b Ü there is a t9[R withH OR B (b)/B (a)±[ t9 t9
or a 5b ,
It can be shown that the implication in the K -case is strict, and in the K -case it is an jO R
equivalence.
In general – that is, without speciﬁc conditions on the feasible set P – the
implications in Theorem 1 cannot be reversed. For instance, let M5h1, 2j, a5(2, 0), and
b5(1, 2). Then neither aK b nor bK a; aK b but not bK a; and aK b as well as JJ d d O R
bK a. OR
Under the following condition the reversal of the implications in Theorem 1, at least178 J. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182
M for maximal elements of the three relations, will hold. Call a set P # ⁄ R weakly convex
if for all a, b[P and every e.0 there is a c[P with (Euclidean) distance to a smaller
than e and with for all i[M:
a .b Þa .c .b ii iii
a ,b Þa ,c ,b ii iii
a 5b Þa 5c 5b (1) ii iii
Thus, the condition of weak convexity of a set means that between any two points of the
set there exists another point of the set as close to one of the two points as desired.
Obviously, convex sets are weakly convex. Further, one easily shows that a closed set
is weakly convex whenever for each a,b[P there exists a c[P with property (1).
M Theorem 2. Let P #R be weakly convex, and let K, K9 be any of the three orderings
K , K , K . Let a[P. If aKb for all b[P, then aK9b for all b[P. JdO R
Proof. In view of Theorem 1 it is sufﬁcient to prove the implication for K5K and OR
K95K . So let aK b for all b[P.I fa5b then by deﬁnition aK b. Now suppose JO R J
a±b, then in particular a$b. Choose a point c satisfying (1) and so close to a that
c ,a for all i, j[M with a .a . Because aK c there is a coordinate j with a ,c ji ij O R jj
and a .c Þa #c for all i[M.B y( 1 ) ,a ,b . Take any i[M with a .b . Then, by ii ij jj ii
(1), a .c , hence a #c . By the choice of c this implies a #a . It follows that a Kb. h ii ij ij j
Theorem 2 applies in particular to a convex set P (the usual case for the nucleolus of
a cooperative game). Convex sets are connected but, clearly, connectedness of P is not a
necessary condition for the conclusion of the theorem to hold. It is also not a sufﬁcient
condition, as the next example shows.
Example 1. Let M5h1, 2j, let
M M P 5h(x,1 )[R :0#x #1j<h(1, x)[R :0#x #1j
and let a5(0, 1). Then it is straightforward to verify that aK b for all b[P, but not OR
(e.g.) aK (1, 0). Observe that the set P is connected, but clearly does not satisfy (1). J
The proof of Theorem 2 uses the assumption that c may be chosen sufﬁciently close
to a. The following example shows that this assumption cannot be dropped.
Example 2. Let M5h1, 2j, and
M P 5h(0, 1)j<h(x,12x/2)[R :1,x #2j ,
and further, let a5(0,1). Again, it is straightforward to verify that aK b for all b, but OR
not (e.g.) aK (2, 0). P satisﬁes the property that for each a, b there exists a c[P with JJ. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182 179
property (1), but for a5(0, 1) we cannot choose c arbitrarily close to a. Observe that P
is not closed.
3. Normal excess functions and the nucleolus
A (cooperative) game (with transferable utility) is a pair (N, v), or brieﬂy v, where
N N:5h1,...,nj is the set of players and the characteristic function v:2 ®R assigns to each
N coalition S[2 the worth v(S), with the convention that v([)50. Feasible allocations
N are vectors in a given set F(v)# ⁄ R , to be interpreted as possible payoff vectors for the
players in the game v. One way to evaluate feasible payoff vectors is to consider excess
or complaint functions e 5(e ) , where the excess e (v,x) measures the N SS [2 ¥h[j S
dissatisfaction of coalition S in case the allocation x[F(v) is chosen as the outcome of
the game v. (The best known example is e (v, x):5v(S)2o x for every coalition S.) Si [Si
Thus, for a given collection of excess functions, any feasible allocation x gives rise to a
M corresponding vector of excesses e(x)5e(v, x)5(e (v, x)) in R , where the indices SS ±[
in M correspond to the nonempty coalitions.
For a given collection of excess functions e the nucleolus of a game v with feasible
allocation set F(v) consists of those feasible allocations of which the corresponding
excess vectors are most desirable:
n(v, F(v), e)5hx [F(v): e(x)K e(y) for all y [F(v)j . d
Most nucleoli considered in the literature (Schmeidler, 1969; Kohlberg, 1971; Grotte,
1971; Sobolev, 1975; Owen, 1977; Wallmeier, 1980, 1983; Potters and Tijs, 1992) are
deﬁned with respect to excess functions that share the following property. A collection
of excess functions e is called normal if for every game v with feasible allocation set
F(v) every e is continuous on F(v) and satisﬁes: S
e (x),e (y)Ûx(S).y(S) for all x, y [F(v), SS
where x(S):5o x . Thus, normality implies that the excess functions depend only on i[Si
the sums of the individual coordinates.
It is not hard to prove that if F(v) is convex and e is a normal collection of excess
functions, then the image e(F(v)) is a weakly convex set. This observation implies the
following immediate corollary of Theorem 2.
Theorem 3. Let v be a game and let F(v) be a convex set. Let e be a normal collection
of excess functions. Then
n(v, F(v),e)5hx [F(v): e(x)K e(y) for all y [F(v)j J
5hx [F(v): e(x)K e(y) for all y [F(v)j OR
The rest of this section and of the paper deals with the Kohlberg criterion. As is well
known, the nucleolus can be characterized in terms of so called balanced collections of180 J. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182
coalitions (Kohlberg, 1971). This Kohlberg criterion will be extended now to normal
excess functions.
N A collection C#2 \h[j of coalitions is balanced if it is empty or there exist positive
ˆ weights l .0, S[C, such that for each player i[N the sum o l equals 1. S Słi,S[CS
In the following lemma balancedness is characterized in terms of sidepayments (a
N y[R is a side-payment if y±0 and y(N)50). The lemma states that a non-empty
collection of coalitions is balanced if and only if it cannot perform a reallocation,
beneﬁcial for at least one coalition in the given collection without hurting others (see
also Zumsteg (1995) for an application in the context of the computation of the
nucleolus).
N Lemma 2. A non-empty collection C#2o f coalitions is balanced if and only if for
N each side-payment y[R either y(S)50 for all S[C, or there are two coalitions
S,T[C with y(S).0 and y(T),0.
Instead of this lemma a slightly different version will be formulated and proved, using
N Farkas’ Lemma. Let 7 be a collection of coalitions. A collection C#2 \h[j is called
7-balanced if there is a subset 79 of 7 such that C<79 is balanced. So, the standard
notion of balancedness is incorporated in this deﬁnition by taking 7 equal to the empty
set. Further, observe that each collection is hhij: i[Nj-balanced.
NN Lemma 3. Let T#2 \h[j. A non-empty collection C# ⁄ 2 \h[j of coalitions is 7-
N balanced if and only if for each side-payment y[R with y(S)$0 for all S[C<7 there
is no coalition S[C with y(S).0.
SS N S Proof. Let e denote the indicator vector of coalition S, i.e. e [R and e 51i fi[S, i
S e 50i fi[ ⁄ S. i
NN S Proof of ‘if’: The inequalities y?e $0, y?2e $0, y?e $0, for S[C<7, imply
S S y?2e $0, for S[C; according to Farkas’ Lemma this implies that for each S[C, 2e
has to be a nonnegative weighted sum of the indicator vectors in C<7, and the vectors
NN e and 2e . Therefore,
SS NS N S T 2e 5l e 2m e 1 O l e , S [C , NN T
T[C<7
SS with all weights nonnegative. Observe that l ,m for each S[C and, thus, there exist NN
SS nonnegative weights g , T[C<7, with g .0, such that TS
NS T e 5 O g e . T
T[C<7
S Now deﬁne the nonnegative weights l 5(o g )/(uCu), T[C<7. Deﬁne 79:5 TS [CT
hT[7: l .0j. Observe that l .0 for every T[C. Then the collection C<79 is TT
balanced, as follows fromJ. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182 181
ST OO g e S[CT [C<7 T NT T ]]]]]] e 55 O l e 5 O l e . TT uCu T[C<7 T[C<7 9
Hence, C is 7-balanced.
Proof of ‘only if’: Let C<79 be balanced for a subset 79 of 7, and let y be a
side-payment with y(S)$0 for all S[C<7. Then, with l .0 for S[C<79 such that S
NS e 5o l e , S[C<7 9 S
NS 05y ?e 5y ? O l e 5 O l y(S). SS SD
S[C<7 9 S[C<7 9
This is only possible if y(S)50 for all S[C<79. h
The following theorem gives a characterization of the nucleolus in terms of balanced
collections for a speciﬁc case of normal excess functions. It is an extension of a result in
Potters and Tijs (1992).
NN Theorem 4. Let 7#2 , let v be a game with F(v)5hx[R : x(S)$v(S) for all S[7,
x(N)5v(N). Let e be a normal collection of excess functions, and let x[F(v). Then the
following two assertions are equivalent:
1. for each t[R there is a subset 79 of hS[7: x(S)5v(S)j such that B (e(v, x))<79 is t
balanced;
2. x belongs to the nucleolus.
Proof. In this proof the notation B (x) is used instead of B (e(v, x)). tt
Proof of ‘(i)Þ(ii)’: Let z[F(v), z±x be arbitrary. Take t9 such that B (x)5B (z) for all tt
9 t.t9, and B (x)±B (z). Such a t9 exists because otherwise x(S)5z(S) for all coalitions, tt 9
implying x5z.
Claim: x(S)#z(S) for all S with e (x)$t9.To prove this claim, let S be a coalition with S
s:5e (x)$t9. Consider the following three cases: S
· s.t9: Then S[B (x)5B (z) and S[ ⁄ B (x)5B (z) for all t.s, implying s5e (z). ss tt S
· s5t9 and S[B (z): If e (z)5t.s then S[B (x) implying s$t.s. Therefore, s5 t9 St
e (z). S
· s5t9 and S[ ⁄ B (z): Obviously, e (z),s. t9 S
Hence, in all cases, e (x)$e (z) or, equivalently, x(S)$z(S). This proves the Claim. SS
Deﬁne y5z2x, then y(S)$0 for all S[B (x). Obviously, y(S)$0 for all S[7 with t9
x(S)5v(S). By Lemma 3 it follows that y(S)50 for all S[B (x). Hence, x(S)5z(S), and t9
9 therefore e (x)5e (z) for all S[B (x), implying B (x)#B (z). Because B (x)±B (z) SS t 9 t9 t9 t9 t
there is a coalition T with e (z)5t9.e (x). Also, e (x).e (z) implies e (x)#t95e (z). TT S S ST
So e(x)K e(z). By Theorem 3, x belongs to the nucleolus. OR182 J. Derks, H. Peters / Mathematical Social Sciences 36 (1998) 175–182
Proof of ‘(ii)Þ(i)’: Let x[F(v) and suppose there is a t[R such that B (x)<79 is not t
balanced for each subset 79 of hS[T: x(S)5v(S)j. According to Lemma 3 there is a
side-payment y and coalition S9[B (x) such that y(S9).0 and y(S)$0 for each coalition t
in B (x)<hS[7: x(S)5v(S)j. Obviously, y may be chosen to have Euclidean length of t
1. Choose d .0 such that B (x)5B (x). This is possible since by taking the largest tt 2d
excess t9 of a coalition outside B (x) one has t9,t, and any 0,d ,t2t9 can be taken. t
By continuity of the excess functions there is an e.0 so that for each z[P within
Euclidean distance from x one has e (z)$t2d for S[B (x), and e (z),t2d for St S
S[ ⁄ B (x), i.e., B (z)5B (x). tt 2d t2d
For z:5x1ey it holds that z(S)$x(S) for all S[B (z), with strict inequality for t2d
S5S9. This implies e (z)#e (x) for all S[B (z) and e (z),e (x). In particular, SS t 2d S9 S9
e (z).e (x) implies S[ ⁄ B (z), so e (z),t2d#e (z). Consequently, e(z)K e(x) which SS t 2d SS 9 J
yields, by Lemma 1, that e(x)K e(z) does not hold. By Theorem 3, x is not in the OR
nucleolus. h
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The Nucleolus of a characteristic function Game.

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Orderings, excess functions, and the nucleolus.

Orderings, excess functions, and the nucleolus.

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