Reconfirming the Prenucleolus

MATHEMATICS OF OPERATIONS RESEARCH Vol. 28, No. 2, May 2003, pp. 283–293 Printed in U.S.A. RECONFIRMING THE PRENUCLEOLUS GUNI ORSHAN and PETER SUDHÖLTER By means of an example, it is shown that the prenucleolus is not the only minimal solution that satisfies nonemptiness, Pareto optimality, covariance, the equal treatment property, and the reduced game property, even if the universe of players is infinite. This example also disproves a conjecture of Gurvich et al. (1994). As a second result, we prove that the prenucleolus is axiomatized by nonemptiness, covariance, the equal treatment property, and the reconfirmation property, provided the universe of players is infinite. 1. Introduction. The prenucleolus and the prekernel are widely accepted solutions for cooperative transferable utility games. Introduced as auxiliary solutions of the prebargaining set, they became important solutions in their own rights, heavily supported by the fact that they can be justified by simple and intuitive axioms. Both are closely related because they share many properties and because one, the prenucleolus, is a subsolution of the other. Indeed, both solutions are nonempty (NE), Pareto optimal (PO), covariant under strategic equivalence (COV), anonymous (AN), and satisfy the equal treatment property (ETP), and the reduced game property (RGP). One additional property for each of the solutions suffices to characterize it: The prenucleolus is single-valued (SIVA) and the prekernel satisfies the converse reduced game property (CRGP). In fact, the prenucleolus is axiomatized by means of SIVA, COV, AN, and RGP (see Sobolev 1975), whereas the prekernel is axiomatized by NE, PO, COV, ETP, RGP, and CRGP (see Peleg 1986). AN can be replaced by ETP in Sobolev’s axiomatization (see Orshan 1993) and CRGP in Peleg’s axiomatization can be replaced by a maximality principle: The prekernel is the maximum solution that satisfies the remaining axioms NE, PO, COV, ETP, and RGP. In this work we investigate the role of SIVA in the foregoing axiomatizations of the prenucleolus, and we provide two results. The first result may be motivated as follows. In view of the facts that SIVA may be considered as a “minimality principle” and that the prenucleolus of a game is a distinguished special point of the prekernel, the following question arises in a natural way: (1) Is it possible to replace SIVA by NE and a minimality principle, i.e., is the prenucleolus the minimum (or at least the unique minimal) solution that satisfies NE, PO, COV, ETP, and RGP? The first result, Theorem 3.7, shows that the answer to question (1) is negative. Indeed,
that is finite-valued and satisfies NE, PO, COV, AN, ETP, RGP we construct a solution

contains a minimal subsolution having and that does not contain the prenucleolus. Thus,
the desired properties (see Corollary 3.8). This result yields a new aspect of the impact of SIVA in Sobolev’s (1975) and in Orshan’s (1993) axiomatization of the prenucleolus. Hence it “reconfirms” the prenucleolus in the sense that it implicitly reveals a part of the special character of this solution. As a byproduct, the nonuniqueness result, Theorem 3.7, disproves the following conjecture raised by Gurvich et al. (1994): Any TU game is the reduced game of a “huge” game, Received September 12, 2001; revised March 18, 2002, and September 26, 2002. MSC 2000 subject classification. Primary: 91A12. OR/MS subject classification. Primary: Games/group decisions, cooperative. Key words. Prenucleolus, game theory. 283 0364-765X/03/2802/0283/$05.00 1526-5471 electronic ISSN, © 2003, INFORMS 284 G. ORSHAN AND P. SUDHÖLTER the prekernel of which consists of the prenucleolus only, with respect to the prenucleolus of the “huge” game. If this conjecture were true, then a proof of it would also yield a new proof of Sobolev’s or Orshan’s result, because any solution with the desired properties is a subsolution of the prekernel. Hence, the prenucleolus cannot be characterized by replacing the “maximality principle” CRGP in the aforementioned axiomatization of the prekernel by a suitable minimality principle. Nevertheless, one may ask the following question: (2) Is it possible to find an intuitive axiom that characterizes the prenucleolus, if the “maximality principle” CRGP is replaced by this axiom, i.e., is the prenucleolus characterized by NE, PO, COV, ETP, RGP, and some additional intuitive axiom? The second result, Theorem 4.3, shows that there is a suitable axiom that yields the desired characterization. This axiom requires from any member of the solution of every game that every element of the solution of the reduced game with respect to every coalition, when combined with the restriction of the initial element to the complement coalition, establishes a member of the solution of the initial game. In Balinski and Young (1982) this axiom is one condition of a property they called “uniformity”, and Shimomura (1992) used the term “flexibility”. Recently, Hwang and Sudhölter (2001) called this intuitive axiom the reconfirmation property (RCP), which reflects a natural interpretation of the property, and they employed RCP in an axiomatization of the core, which emphasizes the importance of this axiom. Due to private communication, the authors know since 1993 that the prenucleolus is characterized by NE, COV, ETP, RGP, and RCP. Note that PO can be deduced from the first four axioms. Surprisingly, it turns out that RGP is not needed in this characterization; that is, the prenucleolus is axiomatized by NE, COV, ETP, and RCP. We think this axiomatization gives a new insight into the character of the prenucleolus and that it reinforces RCP as a significant property that plays an important role in some characterizations of “classical” solutions like the core and the prenucleolus. Of course, this result also demonstrates the “power” of RCP when combined with ETP. Indeed, AN, even together with PO and RGP, does not replace ETP in Theorem 4.3 because, for example, the positive core (see Definition 3.1) satisfies NE, PO, AN, COV, RGP, and RCP. Also it should be remarked that, for single-valued solutions, RGP and RCP are equivalent. If singlevaluedness is not required, then there are important solutions that satisfy RCP but violate RGP. For example, the least core satisfies NE, PO, AN, COV, and RCP, but it violates RGP. Particularly this fact demonstrates that RCP stands for itself as an interesting property, independently of RGP. Though the main theorems, the nonuniqueness result, and the new axiomatization of the prenucleolus are mathematically independent, the above discussion shows their close relationship. We now briefly review the contents of the paper. Section 2 recalls definitions of some relevant solutions and contains a list of some properties of solutions. In §3, Theorem 3.7 is proved, and §4 contains the new characterization of the prenucleolus and examples that show that each of the employed axioms is logically independent of the remaining axioms. 2. Notation, solutions, and properties. In this section some relevant definitions and results from Maschler et al. (1972) and Peleg (1986) are recalled. Let U be a universe of players containing, without loss of generality, 1     k whenever
U
≥ k. A (cooperative TU) game is a pair N  v such that  = N ⊆ U is finite and v 2N →
 v = 0. For any game N  v let X ∗ N  v = x ∈
N
xN ≤ vN  and XN  v = x ∈
N
xN = vN  denote the set of feasible and Pareto optimal feasible payoffs (preimputations), respectively.  We use xS = i∈S xi x = 0 for every S ∈ 2N and every x ∈
N as a convention. 285 RECONFIRMING THE PRENUCLEOLUS Additionally, xS denotes the restriction of x to S, i.e., xS = xi i∈S . For disjoint coalitions S T ∈ 2N let xS  xT = xS∪T . For x ∈
N  S ⊆ N  and distinct players k l ∈ N let eS x v = vS − xS and skl x v = max S⊆N \ l k∈S eS x v denote the excess of S and the maximal surplus of k over l, respectively, at x with respect to N  v . The prekernel of N  v is given by    ℘N  v = x ∈ XN  v  skl x v = slk x v ∀k ∈ N  l ∈ N \ k  For X ⊆
N let  N  v  X denote the nucleolus of N  v with respect to X, i.e. the set of members of X that lexicographically minimize the nonincreasingly ordered vector of excesses of the coalitions (see Schmeidler 1969). It is well known that the nucleolus with respect to X ∗ N  v is a singleton, the unique element of which is called the prenucleolus of N  v and is denoted by N  v . In general, a solution
associates with each game N  v a subset of X ∗ N  v . Let
be a solution. Then
(1) is covariant under strategic equivalence (COV), if for all games N  v  N  w satisfying w = v + z for some  > 0 z ∈
N the equation
N  w = 
N  v + z holds. (Here we use the convention that identifies z ∈
N with the additive coalitional function,  again denoted by z, on the player set N defined by zS = i∈S zi for all S ∈ 2N . Also note that the games v and w are called strategically equivalent.); (2) is nonempty (NE), if
N  v =  for every game N  v ; (3) is Pareto optimal (PO), if
N  v ⊆ XN  v for every game N  v ; (4) is single-valued (SIVA), if

N  v
= 1 for every game N  v ; (5) is anonymous (AN), if the following condition is satisfied for all games N  v and M w . If  N → M is a bijection such that v = w, then
M w = 
N  v . (In this case the games N  v and M w are isomorphic.); (6) satisfies the equal treatment property (ETP), if for every game N  v , for every x ∈
N  v , xk = xl for all substitutes k l ∈ N . (Here k and l are substitutes, if vS ∪ k = vS ∪ l ∀S ⊆ N \ k l.); (7) satisfies the reduced game property (RGP), if for every game N  v   = S ⊆ N  and every x ∈
N  v , xS ∈
S vSx . (The reduced game S vSx is defined by vSx  = 0, vSx S = vN − xN \ S , and vSx T = maxQ⊆N \S vT ∪ Q − xQ for  = T
S); (8) satisfies the converse reduced game property (CRGP), if for every game N  v with
N
≥ 2 the following condition is satisfied for every x ∈ XN  v : If, for every S ⊆ N with
S
= 2, xS ∈
S vSx , then x ∈
N  v ; (9) satisfies the reconfirmation property (RCP), if for every game N  v   = S ⊆ N  for every x ∈
N  v and y ∈
S vSx , y xN \S ∈
N  v . For interpretations and discussions, in particular of the variants 7, 8, and 9 of the reduced game property, see Peleg (1986) and Sudhölter (2001). We conclude this section by stating the known characterizations of the prenucleolus. Theorem 2.2 will be used in §4. Theorem 2.1 (Sobolev 1975). The unique solution that satisfies SIVA, COV, AN, and RGP is the prenucleolus, provided
U
= . Theorem 2.2 (Orshan 1993). The unique solution that satisfies SIVA, COV, ETP, and RGP is the prenucleolus, provided
U
= . 286 G. ORSHAN AND P. SUDHÖLTER 3. Nonuniqueness of a minimal solution. This section is devoted to the construction
that satisfies NE, COV, PO, ETP, RGP and that does not contain the prenuof a solution
cleolus as a subsolution. Moreover, as it is finite valued, the solution contains a minimal subsolution satisfying the axioms. As a byproduct, the constructive proof provides an example that disproves the conjecture of Gurvich et al. (1994) mentioned in §1. Throughout this section we shall assume that
U
≥ 4. First, a specific coalition structure, i.e., a partition of the set of players, is defined. Let N  v be a game and x = N  v . Define the binary relation ∼v on N by k ∼v l ⇔ k = l or k = l and skl x v ≤ 0 and note that ∼v is reflexive and transitive. It is also symmetric (an equivalence relation), because x ∈ ℘N  v . Let  N  v denote the set of equivalence classes of ∼v . In Section 3 of Maschler et al. (1972), the collection  N  v of coalitions is the partition, which corresponds to the smallest excess strictly greater than 0, of the profile generated by the prekernel element x = N  v . (Of course,  N  v = N  in the case that there is no coalition of positive excess.) Definition 3.1. Let N  v be a game and x = N  v . Let t + = max t 0 denote the positive part of the real number t. The positive core of N  v is the set  + N  v = y ∈ X ∗ N  v
eS x v + = eS y v + ∀S ⊆ N  Note that the positive core was first mentioned by Orshan (1994), who fully described this solution in the three-person case. Though this solution may be regarded as an interesting core extension, in the present paper it serves as an auxiliary solution only. Lemma 3.2. The positive core satisfies RGP and RCP. Though this lemma is contained in Sudhölter (1993, Lemma 2.2), we shall present a proof that proceeds analogously to the proof of RGP of the prenucleolus due to Peleg (1988). Proof. Let N  v be a game, x ∈ XN  v and  = S ⊆ N . For ! ∈
define ! x v = S ⊆ N
eS x v ≥ ! ∪  N  Then Definition 3.1 can be formulated as (3.1) x ∈  + N  v ⇔ ! x v = ! N  v  v ∀! > 0 According to Kohlberg (1971), the prenucleolus is characterized by the equivalence   yN = 0 yT ≥ 0 ∀T ∈ ! x v x = N  v ⇔ ∀! ∈
∀y ∈
N (3.2)  ⇒ yT = 0 ∀T ∈ ! x v By Equations (3.1) and (3.2), the positive core is characterized by the equivalence   yN = 0 yT ≥ 0 ∀T ∈ ! x v + N (3.3)  x ∈  N  v ⇔ ∀! > 0 ∀y ∈
⇒ yT = 0 ∀T ∈ ! x v Note that xS ∈ XN  vSx and (3.4) ! xS  vSx = S ∩ T
T ∈ ! x v  ∀! ∈
 By Equation (3.4), for any ! ∈
, the set yS ∈
S
yS S = 0 yS Q ≥ 0 ∀Q ∈ ! xS  vSx  RECONFIRMING THE PRENUCLEOLUS 287 is the projection of y ∈
N
yN = 0 yi = 0 ∀i ∈ N \ S yT ≥ 0 ∀T ∈ ! x v  Thus RGP is implied by Equation (3.3). To show RCP let x ∈  + N  v and z ∈  + S vSx . By RGP, xS ∈  + S vSx , thus ! xS  vSx = ! z vSx ∀! > 0 by (3.1). Hence, by Equation (3.4) applied to z xN \S , ! z xN \S  v = ! x v for every ! > 0. Thus Equation (3.3) finishes the proof.  Remark 3.3. Let N  v be a game and x = N  v . By the definition of the positive core we obtain   skl x v > 0 ⇒ skl x v = skl y v and ∀k l ∈ N  k = l ∀y ∈  + N  v  skl x v ≤ 0 ⇔ skl y v ≤ 0 Let N  v be a game. For any total order  of  =  N  v , let us say  = T1      Tt  with T1 ≺ · · · ≺ Tt , recursively define !v ∈
 (we abbreviate !v by ! if there is no danger of confusion) by    !T1 = min yT1  y ∈  + N  v (3.5) and (3.6)    !Ti = min yTi  y ∈  + N  v  yTj = !Tj ∀j = 1     i − 1 for all i = 2     t and put (3.7) X N  v = z ∈
N
zT = !v T ∀T ∈  N  v  Remark 3.4. Let N  v be a game. Note that !v is well defined because  + N  v is nonempty and compact. Therefore X N  v is a nonempty convex set of preimputations, which, by Equations (3.5), (3.6), and (3.7), intersects the positive core; i.e., we have (3.8) + N  v =  + N  v ∩ X N  v =  This subset of the positive core can also be expressed as  + N  v = y ∈  + N  v
yT1      yTt (3.9)  ≤lex zT1      zTt ∀z ∈  + N  v  Lemma 3.5. Let N  v be a game and  be a total order of  N  v . Then the nucleolus of N  v with respect to X N  v is a singleton, which belongs to  + N  v . Proof. By Equation (3.8), + N  v = . Hence,   (3.10)  N  v  X N  v =  N  v  + N  v by the definition of the positive core. By Equation (3.9) the latter set of preimputations is a nonempty, convex, and compact set. Thus
 N  v   + N  v
= 1 (see Schmeidler 1969).  Definition 3.6. For any game N  v and any total order  of  N  v let  N  v
is defined by be the unique member of  N  v  X N  v . The solution
 
N  v =  N  v
 is a total order of  N  v 

satisfies NE, PO, COV, ETP, RGP, and AN. Moreover, Theorem 3.7. The solution

. the prenucleolus is not a subsolution of
288 G. ORSHAN AND P. SUDHÖLTER Proof. Let N  v be a game and  be a total order of  =  N  v .
satisfies NE, AN, PO, and COV. The vector x =  N  v is a (1) We show that

N  v , thus

satisfies NE. A bijection  N → M maps  bijectively to member of

satisfies AN as well. Moreover,  M v , because the prenucleolus satisfies AN. Thus

satisfies PO, because it is a subsolution of  + . Let  > 0 z ∈
N  w = v + z, and
y = x + z. Then  N  w =  , because the prenucleolus satisfies COV. Moreover, by COV of the positive core (which can be shown in a straightforward way), the equation + N  w = + N  v + z
satisfies COV. is valid, thus  N  w =  N  v + z = y by Equation (3.10). Hence
(2) To show ETP and RGP a derived game N  v is defined by v S = vS  if S ∈ 2N \   !v T  if S = T for some T ∈   Then (3.11)   N  v  X N  v  =  N  v  X N  v  because v differs from v at most on the partition  , the elements T of which receive a fixed amount zT = !v T by every preimputation z of X N  v . The expression on the right-hand side of Equation (3.11) is the prenucleolus N  v   of the game N  v with the coalition structure  . Thus, (3.12) x =  N  v = N  v    It is well known that the prenucleolus of a game with coalition structure is a member of the prekernel of the game with coalition structure. For every pair k l of distinct players the equation (3.13) skl x v = if k l ∈ T for some T ∈   skl x v  skl N  v  v  otherwise is a consequence of the definition of the derived game and of Lemma 3.2 and Remark 3.3,
is a subsolution of respectively. Equations (3.12) and (3.13) imply x ∈ ℘N  v . Thus
the prekernel which satisfies ETP.
satisfies RGP, let  = S ⊆ N and w = vSx . Using the well-known fact To show that
skl xS  w = skl x v ∀k l ∈ S with k = l we obtain (3.14)  S w = T ∩ S
T ∈  and T ∩ S =  Let S be the total order on  S w consistent with , which is defined by the following requirement: If T  T ′ ∈  satisfy T ≺ T ′ and T ∩ S =  = T ′ ∩ S, then T ∩ S ≺S T ′ ∩ S By RGP and RCP of the positive core (see Lemma 3.2), +S S w = y ∈
S
y xN \S ∈ + N  v  and thus, XS S w = y ∈
S
y xN \S ∈ X N  v  Therefore, v Sx = wS . Hence S S w = xS by RGP of the prenucleolus of games with coalition structures (see Peleg (1988, Theorem 5.2.7)). RECONFIRMING THE PRENUCLEOLUS 289
the following “cyclic” four-person game (3) To show that  is not a subsolution of
M u is defined by M = 1     4 and  if S ∈ 1 2 2 3 3 4 4 1  1 if S ∈  M uS = 0  −2 otherwise Note that M u is transitive. (A game is transitive if its symmetry group—i.e., the group of permutations of N that do not change the game—is transitive.) Indeed, the cyclic permutation, which maps 1 to 2, 2 to 3, 3 to 4, and 4 to 1, is a symmetry. Hence, by AN and PO, we obtain M u = 0 ∈
M . Therefore, x ∈  + M u , iff xS = 0 for every S = 1 2     4 1, and xT ≥ −2 for every T ⊆ M. These inequalities show that  + M u = convex hull −1 1 −1 1  1 −1 1 −1  Also,  M u = k
k ∈ M.
M u = −1 1 −1 1  1 −1 1 −1 .  Thus
Corollary 3.8. There is a minimal solution that satisfies NE, PO, COV, ETP, RGP, AN, and that does not coincide with the prenucleolus.
that satisfy NE, PO, Proof. Let $ denote the partially ordered set of subsolutions of

∈ $. To show that a chain $0 (a subset COV, ETP, RGP, and AN. By Theorem 3.7,
of comparable elements) has a lower bound we verify that
0 , defined by
0 N  v =  0
∈$0
N  v , belongs to $. The solution
satisfies PO, COV, ETP, RGP, and AN, because all members of the chain satisfy these axioms. Moreover,
0 satisfies NE, because any
∈ $ is finite-valued. Hence, by Zorn’s lemma, $ has a minimal element. By Theorem 3.7,   $.  In Gurvich et al. (1994, §4.4) the following question is raised: Let N  v be a game. Is  ṽ such that (a) N ⊆ N , (b) ℘N  ṽ is a singleton consisting of the there any game N N y prenucleolus y only, and (c) ṽ = v? A positive answer to this question would yield a new proof Sobolev’s or Orshan’s axiomatization of the prenucleolus. Theorem 3.7 shows that the answer to this question is negative. Moreover, the game M u defined in the last part of the proof of this theorem is an explicit “counter” example. (Note that a one-parameter set of games that contains M u is discussed in Orshan 1994 and a variant of M u is used to prove the main result of Sughölter and Peleg 2002.) 4. An axiomatization of the prenucleolus. This section is devoted to show that the prenucleolus is axiomatized by NE, COV, ETP, and RCP, provided
U
= . Let
be a solution. Let % 2 be the set of games with at most two persons. The following lemmata are useful. Lemma 4.1. Assume that
U
≥ 2 and that
satisfies NE, COV, ETP, and RCP. Then
N  v = N  v for every N  v ∈ % 2 . Proof. Let M u be a two-person game. By NE there exists x ∈
M u . By ETP and COV there exists a ∈
such that xi = u i + a for i ∈ M. Let i ∈ M and ui = u ix . By NE there exists y ∈
 i ui . By COV, y ! = !y − ui  i + ui  i ∈
 i ui for every ! > 0. By RCP, y !  xM\ i ∈
M u , thus y ! = xi for all ! > 0. We conclude that y ! = y for all ! > 0, hence the proof is complete.  For any game N  v and any x ∈
N let 'x v = maxS⊆N eS x v . Lemma 4.2. Under the assumptions of Lemma 4.1 and the additional assumption that
U
= , the following assertions are valid: (1)
satisfies PO. (2) Let N  v be a game, let x ∈
N  v , and let i ∈ N . Then there exist S i  S −i ⊆ N such that i ∈ S i , i  S −i , and eS i  x v = eS −i  x v = 'x v . 290 G. ORSHAN AND P. SUDHÖLTER Proof. Let N  v be a game. As
U
=  we may assume that N ⊆ U \ 1 2 and  = 1 2 ∪ N ⊆ U . By NE there exists x ∈
N  v . Define ṽS  S ⊆ N , by ṽS = that N  ṽ arises from N  v by adding the two null-players 1 and 2.) vS \ 1 2 . (That is, N  ṽ . By ETP, x̃1 = x̃2 . By NE there exists x̃ ∈
N  = ṽN  : Let ṽ1 = ṽ 1x̃ . By Lemma 4.1,
 1 ṽ1 = ṽ1  1 . By Claim 1. x̃N  ṽ . By ETP, ṽ1  1 = x̃2 = x̃1 , thus x̃N  = ṽN . RCP, ṽ1  1  x̃N \ 1 ∈
N Claim 2. x̃1 = x̃2 = 0: For every i ∈ N let ṽi1 = ṽ 1i . Assume the contrary. Then two cases may occur:  attain 'x̃ ṽ . (1) x̃1 = x̃2 < 0: Then e 1 x̃ ṽ = −x̃1 > 0, thus 'x̃ ṽ > 0. Let S ⊆ N  By Claim 1,  = S = N . By our assumption, 1 2 ⊆ S. Let i ∈ N \ S. By Lemma 4.1,  ṽ . ỹ =  1 i ṽi1 ∈
 1 i ṽi1 . Then ỹ1 > x̃1 . By RCP we have ỹ x̃N\ 1i ∈
N A contradiction to ETP is obtained, because ỹ1 > x̃1 = x̃2 .  \ 1 x̃ ṽ = x̃1 > 0 by Claim 1, thus 'x̃ ṽ > 0. Let S ⊆ N  (2) x̃1 = x̃2 > 0: Then eN  attain 'x̃ ṽ . By Claim 1,  = S = N . By our assumption, 1 2 ∩ S = . Let i ∈ S. By Lemma 4.1, ỹ =  1 i ṽi1 ∈
 1 i ṽi1 . Then ỹ1 < x̃1 . By RCP, ỹ x̃N\ 1i ∈  ṽ . A contradiction to ETP is obtained, because ỹ1 < x̃1 = x̃2 .
N Now assertion (1) of our lemma can be deduced. By Claim 2, N  ṽN x̃ = N  v , thus  ṽ by RCP. By Claim 1, xN = vN . x̃ = 0 0 x ∈
N To prove assertion (2), we have to show that the following conditions are satisfied: (4.1) (4.2) N = =   S ⊆ N
eS x v = 'x v  S ⊆ N
eS x v = 'x v  Assume the contrary. Then two cases may occur:  (1) There exists i ∈ N \ S ⊆ N
eS x v = 'x v : By Lemma 4.1, ỹ =  1 i ṽ 1ix̃ ∈
 1 i ṽ 1ix̃  The fact that ỹ1 > 0 = x̃2 , is in contradiction to ETP.  (2) There exists i ∈ S ⊆ N
eS x v = 'x v : By Lemma 4.1, ỹ =  1 i ṽ 1ix̃ ∈
 1 i ṽ 1ix̃  The fact that ỹ1 < 0 = x̃2 , is in contradiction to ETP.  Now the main theorem of this section can be proved. Theorem 4.3. The prenucleolus is the unique solution that satisfies NE, COV, ETP, and RCP, provided
U
= . Proof. The prenucleolus satisfies the desired properties. Indeed, it satisfies RCP, because RCP and RGP are equivalent for single-valued solutions. To show the opposite direction let
be a solution that satisfies the desired axioms. By Lemma 4.1,
satisfies PO. In view of Theorem 2.2 it suffices to show that
satisfies SIVA. Let N  v be a game. Take a disjoint copy N ∗ ⊆ U of N , i.e., N ∩ N ∗ =  and N → N ∗  i → i∗  is a bijection Choose any real number ! satisfying ! > n2 + n maxP Q⊆N vP − vQ “replicated” game N ∪ N ∗  v̂ by v̂S ∪ T ∗ = vS  if T = S −! otherwise and define a 291 RECONFIRMING THE PRENUCLEOLUS where S T ⊆ N . Let z ∈
N ∪ N ∗  v̂ . It is the aim to show that the reduced game N  u with respect to N and z (defined by u = v̂N z ) is given by (4.3) uS = v̂S ∪ S ∗ − zS = vS − zS ∀S ⊆ N  To prove (4.3), first note that for any i ∈ N the players i and i∗ are substitutes. Hence, by ETP, zi = zi∗ for all i ∈ N . Claim 1. For all i ∈ N , zi ≥ minP Q⊆N vP − vQ : Assume, on the contrary, that there exists i0 ∈ N such that zi0 < minP Q⊆N vP − vQ . Choose any coalition S ∪ T ∗ attaining 'z v̂ . In view of the fact that e i0  i0∗  z v̂ = v i0  − 2zi0 > −zi0 > 0 the maximal excess cannot be attained by  or by N ∪ N ∗ . By Lemma 4.2, Claim 1 is shown as soon as i0 ∈ S is verified. Assume, on the contrary, i0 ∈ S. If i0 ∈ T , then the observation eS ∪ i0  ∪ T ∗ ∪ i0∗  z v̂ − eS ∪ T ∗  z v̂ = −2zi0 > 0 if S = T  vS ∪ i0  − vS − 2zi0 > 0 if S = T yields the desired contradiction in this case. If i0 ∈ T , then the observation eS ∪ i0  ∪ T ∗  z v̂ − eS ∪ T ∗  z v̂ = yields the desired contradiction. Claim 2. zi ≤ n maxP Q⊆N vP − vQ Let i0 ∈ N be a player. Observe that −zi0 > 0 if S = T \ i0  vS ∪ i0  + ! − zi0 > 0 if S = T \ i0  ∀i ∈ N vN = 2zN = 2zi0 + 2zN \ i0  ≥ 2zi0 + 2n − 1 min vP − vQ P Q⊆N by PO, ETP, and Claim 1. Thus our claim follows immediately.  = i ∈ N
zi < 0 and observe that Now the proof can be finished. Put S (4.4)   ∀ = S
N  uS = max vS − zS  −! − zS Let S be a nontrivial ( = S
N ) coalition. Then vS − zS ≥ vS − n − 1 max vP − vQ P Q⊆N and  ≤ −! + n2 max vP − vQ −! − zS P Q⊆N ≥ −n max vP − vQ P Q⊆N < −n max vP − vQ  P Q⊆N where the last inequality is implied by the definition of !. Hence u is given by uS = vS − zS ∀S ⊆ N  By NE there exists x ∈
N  v . COV implies x − zN ∈
N  u , thus x − zN  zN ∗ ∈
N ∪ N ∗  v̂ by RCP. ETP implies x − zN = zN , thus x = 2zN is the unique member of
N  v .  Four examples are presented that show that each of the axioms (1) NE, (2) COV, (3) ETP, and (4) RCP is logically independent of the remaining axioms in Theorem 4.3. Let N  v be a game. Let
i  i = 1 2 be defined by
1 N  v = 
2 N  v = x ∈ XN  v
xi = xj ∀i j ∈ N  292 G. ORSHAN AND P. SUDHÖLTER Let  be a total order of U . For every finite set N ⊆ U let N be the restriction of  to N and let ≤Nlex be the induced lexicographical order on
N . Then
3 is defined by
3 N  v = x ∈  + N  v
x ≤Nlex y ∀y ∈  + N  v  Also, let
4 be the Shapley value. It is straightforward to check that
i  i = 1     4 satisfies all properties except the ith one. If
U
≥ 4, then none of the solutions coincides with the prenucleolus. The examples also show that each axiom employed in Sobolev’s or Orshan’s characterization of the prenucleolus is logically independent of the remaining axioms. Remark 4.4. The prekernel satisfies NE, COV, AN, ETP, and RGP. Thus RCP cannot be replaced by RGP in Theorem 4.3. The positive core (see Definition 3.1) satisfies NE, COV, AN, RGP, and RCP, hence ETP cannot be replaced by AN (as in the “classical”) axiomatization. Another well-known solution is the least core. The least core of a game N  v is defined by     ℒN  v = x ∈ XN  v  max eS x v = max eS N  v  v  =S
N =S
N It is well known and easy to verify that ℒ satisfies NE, COV, and AN. It also satisfies RCP because the maximal excess of nontrivial coalitions in a reduced game is not larger than the maximal excess of nontrivial coalitions in the game. The least core does not satisfy RGP. Remark 4.5. The infinity assumption on
U
in Theorem 4.3 is crucial. Indeed, if
U
< , then define for any game N  v ,  N  v  if N
U 
N  v = Sx x ∈ ℘KN  v
xS = S v ∀ = S
N  if N = U  Then
satisfies all axioms of Theorem 4.3. Also, there are examples that show that the prenucleolus is a proper subsolution of
when
U
≥ 4. Acknowledgments. The authors are grateful to the referees of this paper for their comments. The second author was supported by the Center for Rationality and Interactive Decision Theory at The Hebrew University of Jerusalem and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). References Balinski, M., H. P. Young. 1982. Fair Representation, Yale University Press, New Haven, CT. Gurvich, V. A., O. R. Menshikova, I. S. Menshikov. 1994. On the nucleolus, balanced sets of coalitions and Sobolev’s theorem. Mimeograph. Hwang, Y-A., P. Sudhölter. 2001. Axiomatizations of the core on the universal domain and other natural domains. Internat. J. Game Theory. 29 597–623. Kohlberg, E. 1971. On the nucleolus of a characteristic function game. SIAM J. Appl. Math. 20 62–66. Maschler, M., B. Peleg, L. S. Shapley. 1972. The kernel and bargaining set for convex games. Internat. J. Game Theory 1 73–93. Orshan, G. 1993. The prenucleolus and the reduced game property: Equal treatment replaces anonymity. Internat. J. Game Theory 22 241–248. . 1994. Non-symmetric prekernels. Discussion Paper 60, Center for Rationality, The Hebrew University of Jerusalem, Jerusalem, Israel. Peleg, B. 1986. On the reduced game property and its converse. Internat. J. Game Theory 15 187–200. . 1988. Introduction to the theory of cooperative games, Chapter 5. Research Memorandum 84, Center for Research in Mathematical Economics and Game Theory, The Hebrew University of Jerusalem, Jerusalem, Israel. RECONFIRMING THE PRENUCLEOLUS 293 Schmeidler, D. 1969. The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17 1163–1170. Shimomura, K.-I. 1992. Individual rationality and collective optimality. Ph.D. thesis, University of Rochester, Rochester, New York. Sobolev, A. I. 1975. The characterization of optimality principles in cooperative games by functional equations (Russian). N. N. Vorobiev, ed. Mathematical Methods in the Social Sciences, Vol. 6, Academy of Sciences of the Lithuanian SSR, Vilnius, Lithuania, 95–151. Sudhölter, P. 1993. Independence for characterizing axioms of the pre-nucleolus. Working Paper 220, Institute of Mathematical Economics, University of Bielefeld, Bielefeld, Germany. , B. Peleg. 2002. A note on an axiomatization of the core of market games. Math. Oper. Res. 27 441–444. G. Orshan: Dept. of Agriculture Economics and Management, The Hebrew University of Jerusalem, P.O. Box 12, Rehovot, 76100, Israel, and Dept. of Mathematics, The Open University of Israel, 16 Klaussner Street, P.O.B. 39382, Ramat Aviv, Tel Aviv, 61392, Israel; e-mail: orshan@agri.huji.ac.il P. Sudhölter: Department of Economics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark; e-mail: psu@sam.sdu.dk