CEJOR (2008) 16:425–440 DOI 10.1007/s10100-008-0070-2 ORIGINAL PAPER
Bargaining sets and the core in partitioning games Tamás Solymosi
Published online: 17 June 2008 © Springer-Verlag 2008
Abstract Partitioning games are useful on two counts: first, in modeling situations with restricted cooperative possibilities between the agents; second, as a general framework for many unrestricted cooperative games generated by combinatorial optimization problems. We show that the family of partitioning games defined on a fixed basic collection is closed under the strategic equivalence of games, and also for taking the monotonic cover of games. Based on these properties we establish the coincidence of the Mas-Colell, the classical, the semireactive, and the reactive bargaining sets with the core for interesting balanced subclasses of partitioning games, including assignment games, tree-restricted superadditive games, and simple network games. Keywords Cooperative games · Combinatorial optimization games · Partitioning games · Core · Bargaining set 1 Introduction Partitioning games can be used to model situations when the cooperative possibilities of the agents are restricted by some external factor, so not all of the coalitions can form. It is then natural to ask that only the feasible coalitions be taken into consideration when an outcome is to be found. On the other hand, many unrestricted cooperative games can be viewed as partitioning games, especially those which model situations when the most efficient way of cooperation is determined by the optimal solution of
Prepared during the author’s Bolyai János Research Fellowship. Also supported by OTKA grant T46194. T. Solymosi (B) Department of Operations Research, Corvinus University of Budapest, Pf. 489, 1828 Budapest, Hungary e-mail:
[email protected]
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some combinatorial optimization problem. The coalitional functions of these games are frequently fully determined by the cooperation values of coalitions from a typically small family. Assignment games, for example, depend only on the values of single-player and mixed-pair coalitions, the other coalitional values are generated from these by the well-known assignment optimization problem. Each game obtained this way have a nonempty core, regardless of the values of the generating coalitions. This noteworthy feature inspired Kaneko and Wooders (1982) to introduce partitioning games as a general game model for situations when there is a given collection of basic coalitions and a function that specifies the cooperation values of these coalitions. Kaneko and Wooders (1982) derived necessary and sufficient conditions on the basic collection such that all the generated partitioning games are balanced, no matter how the basic coalitions are valued. Such collections are called strongly balanced by Le Breton et al. (1992), who showed that a collection is strongly balanced, if and only if it defines a normal hypergraph on the player set. Independently, Kuipers (1994) investigated the connection of strongly balanced collections and perfect graphs. Preceding all these works, Vasin and Gurvich (1977) characterized strong balancedness of families, but their paper (written in Russian) remained unnoticed for a long time. In this paper we show that strong balancedness of the basic collection is sufficient (and necessary) also for the coincidence of the core with four bargaining sets in all induced partitioning games. Ordered from largest to smallest, they are the Mas-Colell bargaining set (Mas-Colell 1989), the classical Aumann-Davis-Maschler bargaining set (Aumann and Maschler 1964; Davis and Maschler 1967), the semireactive bargaining set (Sudhölter and Potters 2001), and the reactive bargaining set (Granot 1994). In particular, we get as new results that the core coincides with the Mas-Colell bargaining set in assignment games, and in tree-restricted superadditive games. The equivalence with the classical bargaining set was shown by Solymosi (1999) and Potters and Reijnierse (1995), respectively. The proof we provide is more general than what is just needed to get the above results. We show that the family of partitioning games induced by a fixed basic collection is closed under the strategic equivalence of games, and also for taking the monotonic cover of games. On the one hand, these properties allow us to apply known characterizations, and easily derive the equivalence of the mentioned bargaining sets and the core in the strongly balanced case. On the other hand, the two types of closedness could pave the way to obtain similar results for other well-structured games. Our approach seems to have promising applicability among the many types of combinatorial optimization games known to be balanced. To support this point, we apply the general closedness properties to prove the same equivalence results for simple network games, which are balanced, but not strongly balanced partitioning games. The known equivalence results for this class of games are first by Granot et al. (1997), who showed the coincidence of the core with the reactive bargaining set, and second by Solymosi (1999), who derived the coincidence of the core with the classical bargaining set. The organization of the paper is as follows. We present the necessary background in the next section. In Sect. 3, we introduce the four bargaining sets and recall the general characterizations of their coincidence with the core. Partitioning games are introduced, and the two types of closedness properties are established in Sect. 4. We derive the
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coincidence of the bargaining sets and the core for strongly balanced partitioning games, and apply it to assignment games and to tree-restricted superadditive games in Sect. 5. In Sect. 6, we demonstrate that although simple network games are not of the strongly balanced type of partitioning games, the results from Sect. 4 can be used to establish the same coincidences. A brief summary concludes the paper. 2 Preliminaries A transferable utility cooperative game (N , v) consists of a nonempty finite set N of players and a coalitional function v : 2 N → R satisfying v(∅) = 0. We use the standard notation x(S) := i∈N aiS xi for the total payoff to coalition S ⊆ N at payoff allocation x ∈ R N , where a S ∈ R N denotes the membership vector of coalition S (i.e., aiS = 1 if i ∈ S, aiS = 0 if i ∈ S). We denote by N the collection of nonempty coalitions, i.e., N = 2 N \{∅}. Given a game (N , v), the excess e(S, x) := v(S) − x(S) of a coalition S at a payoff allocation x is the usual measure of gain (or loss if negative) to S, if its members depart from the proposed x in order to form their own coalition. Note that e(∅, x) = 0 for all x ∈ R N . A payoff allocation x ∈ R N is called efficient, if e(N , x) = 0; individually rational, if e({i}, x) ≤ 0 for all i ∈ N ; and coalitionally rational, if e(S, x) ≤ 0 for all S ⊆ N . The set I∗ of efficient payoffs is called the preimputation set, the set I of individually rational preimputations the imputation set, and the set C of coalitionally rational preimputations the core of the game. On player set N , games v and w are called strategically equivalent, if there exist α > 0 and b ∈ R N such that w(S) = αv(S) + b(S) for all S ⊆ N . In particular, the transformation of v with α = 1 and b = (−v({i}) : i ∈ N ) gives its 0-normalization v 0 . Note that the core, the imputation set, and the preimputation set are all covariant with transformations which yield strategically equivalent games. More precisely, for any α > 0 and b ∈ R N , the transformed sets αC + b, αI + b, and αI∗ + b equal, respectively, the core, imputation set, and preimputation set of the transformed game αv + b, defined by (αv + b)(S) = αv(S) + b(S) for all S ⊆ N . On player set N , the superadditive cover v˜ of game v is defined by v(∅) ˜ := 0, and for S ∈ N by v(S) ˜ := max
v(T ) : T ⊆ N ,
T ∈T
a =a T
S
.
(1)
T ∈T
The totally balanced cover vˆ of v is given by v(∅) ˆ := 0, and for S ∈ N by v(S) ˆ := max
T ∈N
λT v(T ) :
λT a = a , λT ≥ 0 ∀ T ∈ N T
S
.
(2)
T ∈N
Of course, v(S) ≤ v(S) ˜ ≤ v(S) ˆ for all S ⊆ N . The game v is called superadditive, if v = v; ˜ totally balanced, if v = v; ˆ and balanced, if v(N ) = v(N ˆ ). It is well known that for transferable utility games balancedness and nonemptiness of the core
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are equivalent properties. Note that the classes of superadditive, of balanced, and of totally balanced games are closed under strategic equivalence. A third cover will also be used. On player set N , the monotonic cover of a game v is the game v defined by v(S) := max v(T ) T ⊆S
for all S ⊆ N .
(3)
The definition immediately implies that (i) v(∅) = 0; (ii) S ⊆ T ⇒ v(S) ≤ v(T ), i.e., v is monotonic; (iii) v(S) ≥ v(S) ∀ S ⊆ N , i.e., v covers v. Notice that v is the minimum game which satisfies the latter two properties. Also note that in general v and v are strategically not equivalent. We shall need the following straightforward generalizations of (total) balancedness. We consider coalitions only from a fixed nonempty collection B ⊆ N , assumed to contain all single-player coalitions, i.e., B ⊇ B1 := {{i} : i ∈ N }. Let us define ∆+ (B) = {(λT )T ∈N : λT ≥ 0 ∀ T ∈ B, λT = 0 ∀ T ∈ N \B} and
+
Λ(B, S) = (λT )T ∈N ∈ ∆ (B) :
λT a = a T
S
.
(4)
T ∈N
We call an element (λT )T ∈N of the set Λ(B, S) a B-balanced profile of weights for S ⊆ N . Notice that Λ(B, S) = ∅ for all S ⊆ N , and that it consists of a single profile for each S ∈ B1 ∪ {∅}. A collection G ⊆ N is said to be B-balanced on S, if it is the carrier of a B-balanced profile of weights for S, i.e., if there is a (λT )T ∈N ∈ Λ(B, S) for which G = {T ∈ N : λT > 0}. A B-balanced collection is minimal, if it contains no proper subcollection that is B-balanced. The minimal B-balanced collections on S are precisely the carriers of extreme elements of Λ(B, S), among them are the carriers of integral (hence 0–1) component B-balanced profiles of weights for S, called B-partitions of S. They form the collection (B, S) = T ⊆ B :
a =a T
S
.
(5)
T ∈T
Notice that (B, S) = {∅} if and only if S = ∅, and also that {S} ∈ (B, S) if and only if S ∈ B. Given N and B, a game v is called B-balanced on S ∈ N , if v(S) ≥ max
λT v(T ) : (λT )T ∈N ∈ Λ(B, S) ;
(6)
T ∈N
simply B-balanced, if B-balanced on N ; and totally B-balanced, if B-balanced on every S ∈ N . Of course, (total) N -balancedness is precisely (total) balancedness in the standard sense.
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Analogously to (4) and (5), we define for B ⊆ N and S ⊆ N the set Λ≤ (B, S) = (λT )T ∈N ∈ ∆+ (B) :
λT a T ≤ a S
T ∈N
of B-subbalanced profiles of weights for S, the carriers of which are the B-subbalanced collections on S; and the collection ≤
(B, S) = T ⊆ B :
a ≤a T
S
T ∈T
of B-subpartitions of S. Again, notice that ≤ (B, S) = {∅} if and only if S = ∅, and also that {S} ∈ ≤ (B, S) if and only if S ∈ B. Given N and B, a game v is called B-subbalanced on S ∈ N , if v(S) ≥ max
≤
λT v(T ) : (λT )T ∈N ∈ Λ (B, S) ;
(7)
T ∈N
simply B-subbalanced, if B-subbalanced on N ; and totally B-subbalanced, if B-subbalanced on every S ∈ N . Clearly, (7) implies (6). In the games we shall deal with in this paper, the converse also holds. It is an easy exercise to show the following observation. Lemma 1 Given N and B ⊇ B1 , let v({i}) ≥ 0 for all i ∈ N . Then for each S ∈ N , v is B-subbalanced on S if and only if v is B-balanced on S. 3 Bargaining sets Weaker and more dynamic versions of stability than the one leading to the core were introduced by Aumann and Maschler (1964). They argue that an allocation outside the core might also be realized if all objections raised at that allocation can be neutralized by counter-objections, so none of the objections can be justified. Aumann and Maschler (1964) suggested several plausible notions of objections and counterobjections, and thus introduced several bargaining sets. The variant M1i turned out to be fundamental, for Davis and Maschler (1967) proved that M1i is not empty whenever the imputation set I is not empty. Given x ∈ I(N , v) and players i = j, we say that a pair (S, y), S ⊆ N and y ∈ R N , is an (i, j)-objection at x if i ∈ S j, y(S) = v(S), and yk > xk ∀ k ∈ S. Notice that although formally included in the definition, the payoffs to players outside the objecting coalition are actually irrelevant. We further say that a pair (T, z), T ⊆ N and z ∈ R N , is a counter-objection to the (i, j)-objection (S, y) at x if j ∈ T i, z(T ) = v(T ), and z k ≥ yk ∀ k ∈ T ∩ S, z k ≥ xk ∀ k ∈ T \S. Also notice the irrelevance of payoffs to the players not in the counter-objecting coalition.
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The classical bargaining set M1i (N , v) of the game (N , v) is defined as the set of imputations at which every objection between any two players can be countered. Formally, M1i = {x ∈ I : for every objection at x there is a counter-objection}. The core is always a subset of the classical bargaining set, for at core allocations there are no objections, hence there are no justified objections. Below we cite a characterization of those superadditive games in which the two solutions coincide, but we need some preparations. Given a game (N , v), for each fixed allocation x ∈ R N the excess function induces another game, called the excess game at x, on the same player set N by the coalitional function ex (S) := e(S, x) = v(S) − x(S) for all S ⊆ N . Recall that e(∅, x) = 0 for all x ∈ R N , so ex is indeed a coalitional function. Note that ex is always strategically equivalent to v. The monotonic cover of ex will be called the maximal excess game at x, and denoted by wx , i.e., wx (S) := max e(T, x) T ⊆S
for all S ⊆ N .
In addition to the general properties of a monotonic cover, for every allocation x ∈ R N we also have (i) wx (S) ≥ e+ x (S) := max{ex (S), 0} for all S ⊆ N ; (ii) x is individually rational iff wx ({i}) = 0 for all i ∈ N ; (iii) x is coalitionally rational iff wx (N ) = 0. The following result is due to Solymosi (Theorem 1 in Solymosi 1999). Theorem 1 In a superadditive game (N , v), let x be an imputation. If x ∈ M1i (N , v)\ C(N , v) then (N , wx ) is not balanced. The classical bargaining set formalizes a process when the objecting player must reveal all details (coalition and payoff distribution) of his/her objection at the outset. In contrast, in the definition of the reactive bargaining set Mr (Granot 1994), the objecting player is allowed to retain the specifics of his/her objection until the objected player has announced his/her defending coalition. This way the objector is able to react to the move of his/her opponent, thus it becomes easier to raise justified objections. The result is a subset of the classical bargaining set, that still contains the core. Moreover, Mr is also not empty whenever the imputation set I is not empty (Granot 1994). An intermediate variant is the semireactive bargaining set Msr (Sudhölter and Potters 2001), the set of imputations which survive a process where the objecting player must immediately announce the coalition he/she objects with, but may postpone revealing the payoff distribution until the objected player has announced his/her defending coalition. It follows that the semireactive bargaining set is also a subset of the classical bargaining set, but a superset of the reactive bargaining set, implying the nonemptiness of Msr in all games with a nonempty imputation set. Applied to preimputations, the above bargaining protocols define the classical M1i∗ , ∗ prebargaining sets. Of course, they are in the reactive Mr∗ , and the semireactive Msr the same hierarchy as their counterparts. We consider yet another solution based on objections and counter-objections. It ∗ , that was introduced by Mas-Colell is the (pre)bargaining set, denoted here by MMC
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(1989) mainly for atomless exchange economies, but the definition applies to cooperative games as well. Because of its original context (where agents have negligible effects individually), here coalitions rather than pairs of players bargain, furthermore, the individual rationality requirement for payoff allocations is abandoned. The meaning of objections and counter-objections are also altered. Given a preimputation x ∈ I∗ (N , v), we say that a pair (S, y), ∅ = S ⊆ N and y ∈ R N , is a weak objection at x if y(S) = v(S) and yk ≥ xk ∀ k ∈ S, with at least one strict inequality for some k ∈ S. We further say that a pair (T, z), T ⊆ N and z ∈ R N , is a strong counter-objection to objection (S, y) at x if z(T ) = v(T ) and z k ≥ yk ∀ k ∈ T ∩ S, z k ≥ xk ∀ k ∈ T \S, with at least one strict inequality for some k ∈ T . Here too, notice the irrelevance of payoffs to the players not in the objecting / counter-objecting coalitions. The Mas-Colell bargaining set is defined as ∗ = {x ∈ I∗ : every weak objection at x can be strongly countered}. MMC ∗ is not empty in all games (the preimputation set Mas-Colell (1989) proved that MMC is never empty), and it is a superset of the core (core allocations also prevent weak objections). The following general equivalence result, analogous to Theorem 1, is due to Holzman (Corollary 2.3 in Holzman 2000).
Theorem 2 In an arbitrary game (N , v), let x be an imputation. Then x ∈ M1i (N , v)\ C(N , v) if and only if (N , wx ) is not balanced. As a corollary of Theorems 1 and 2, we get that for superadditive games, M1i is a ∗ . A straightforward extension1 of Theorem 1 gives Mi∗ ⊆ M∗ . The subset of MMC 1 MC relations among the various solutions are summarized in Corollary 1 In every superadditive game, ∗ ∗ ∗ and C ⊆ Mr∗ ⊆ Msr ⊆ M1i∗ ⊆ MMC . C ⊆ Mr ⊆ Msr ⊆ M1i ⊆ MMC
An interesting feature of the above solutions is that they might not depend on all coalitional values: nothing happens if certain coalitions are neglected in their determination. Coalition S ∈ N is called essential in game v, if v(S) = v(S) ˜ and only the trivial partition {S} of S is optimal in (1), and inessential otherwise. Let Ev denote the collection of essential coalitions in v. Observe that in any game v, B1 ⊆ Ev , and that for each inessential S there is an Ev -partition T of S such that T ∈T v(T ) ≥ v(S). The core is well known to be independent of inessential coalitions. Closing this section, we remark that, like the core, all (pre)bargaining sets in Corollary 1 are covariant with transformations which yield strategically equivalent games. More precisely, if the set M equals any of the above (pre)bargaining sets of the game v, then for arbitrary α > 0 and b ∈ R N , the transformed set αM + b equals the (pre)bargaining set of the transformed game αv + b. 1 Indeed, in their excellent book, Peleg and Sudhölter state Theorem 1 explicitly for preimputations and for the classical prebargaining set (Theorem 4.3.2 in Peleg and Sudhölter 2003) and use the same proof as in Solymosi (1999).
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4 Partitioning games A cooperative situation (N , B, f ) consists of the set of agents N , the collection B ⊆ 2 N of basic coalitions, and the basic function f : B → R. We again assume that B contains the collection B1 of single-player coalitions, but ∅ ∈ B. Following Kaneko and Wooders (1982), we associate with situation (N , B, f ) the game (N , v(B, f ) ) whose coalitional function is defined by v(B, f ) (S) := max
f (T ) : T ∈ (B, S)
(8)
T ∈T
for each S ∈ N , and by v(B, f ) (∅) := 0. The game (N , v(B, f ) ) will be called the B-partitioning game induced by basic function f . Given N and B, we also say that v is a B-partitioning game, if v = v(B, f ) for some basic function f . Clearly, for every cooperative situation (N , B, f ), v(B, f ) is the minimum superadditive game that satisfies v(B, f ) (B) ≥ f (B) for all B ∈ B. Notice that v(B, f ) (B) = f (B) for all B ∈ B1 , since for single-player coalitions only the trivial partition is optimal in (8). Larger basic coalitions with the same property are also crucial for the induced game, so for cooperative situation (B, f ) we introduce the collection E(B, f ) =
B ∈ B : {{B}} = arg max
f (T ) : T ∈ (B, B)
T ∈T
of essential coalitions in (B, f ). It is easy to check that ours is in accordance with the standard terminology, because E(B, f ) = Ev(B, f ) . The straightforward proof of the following observation is also left to the reader. Proposition 1 For any cooperative situation (B, f ), v(B, f ) v(E(B, f ) ,v(B, f ) ) = v(B,v(B, f ) ) .
= v(E(B, f ) , f ) =
The key properties of partitioning games in terms of the coalitional function are summarized in Corollary 2 If v is a B-partitioning game then (i) v(S) = max for all S ⊆ N , T ∈T v(T ) : T ∈ (B, S) (ii) v is balanced ⇔ v is B-balanced (on N ), (iii) v is balanced ⇔ the LP (6) for N has an integral optimal solution. Statement (i) is simply a reformulation of v(B, f ) = v(B,v(B, f ) ) . A B-partition of S which maximizes the right hand side in (i) is called a maximal B-partition of S in the game v. Claim (ii) is implied by the inessentiality of non-basic coalitions in partitioning games. Quint (1991) proves (iii), and that if v is balanced, C(v) coincides with the set of optimal solutions to the dual of the LP for N in (6). One quickly observes that the additive structure of a B-partitioning game is preserved under positive affine transformations, thus for a given collection B, the class of B-partitioning games is closed under strategic equivalence. In particular, the
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0-normalization and all excess games related to a B-partitioning game are also 0 B-partitioning games. Let us define the 0-normalization f Nof basic function f by 0 f (B) := f (B) − i∈B f ({i}) for all B ∈ B. Also, for x ∈ R , let the basic function f − x be defined by ( f − x)(B) := f (B) − x(B) for all B ∈ B. Proposition 2 Let v be a B-partitioning game induced by basic function f . Then for any α > 0 and b ∈ R N , αv + b is a B-partitioning game induced by basic function α f + b. Moreover, for each S ∈ N , the maximal B-partitions of S in v and in αv + b are the same. Schematically, (B, f ) ⏐ ⏐ α(·)+b
(8 ) −−−−−−−−→
v(B, f ) ⏐ ⏐α(·)+b
(B, α f + b) −−−−−−−−→ v(B,α f +b) = αv(B, f ) + b (8 ) Proof For any α > 0, b ∈ R N , and ∅ = S ⊆ N , we have max (α f + b)(T ) = max f (T ) + b(T ) α T ∈(B,S)
T ∈T
T ∈(B,S)
=α
max
T ∈(B,S)
T ∈T T ∈T f (T ) + b(S) T ∈T
= αv(S) + b(S) = (αv + b)(S). For S = ∅ there is nothing to show.
Corollary 3 Let v be a B-partitioning game induced by basic function f . Then the 0-normalization v 0 of v is a B-partitioning game induced by the 0-normalization f 0 of f . Also, for any x ∈ R N , the excess game ex is a B-partitioning game induced by basic function f − x. Next we show that for a given collection B, the class of B-partitioning games is closed for taking the monotonic cover. Applied to excess games, we get that all maximal excess games related to a B-partitioning game are also B-partitioning games. We associate with basic function f its positive part f + defined for B ∈ B by f + (B) := max{ f (B), 0}, and another basic function f 1+ defined for B ∈ B1 by f 1+ (B) := f + (B), and for B ∈ B\B1 by f 1+ (B) := f (B). Proposition 3 Let v be a B-partitioning game induced by basic function f . Then its monotonic cover v is a B-partitioning game induced by basic functions f + or f 1+ . Schematically, (8) (B, f ) −−−−−−−−→ ⏐ ⏐ (·)+ (·)+ 1
v(B, f ) ⏐ ⏐ (3)
+ (B, f (1) ) −−−−−−−−→ v(B, f + ) = v (B, f ) (1) (8)
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Proof First we prove the proposition for f + . Let u denote the B-partitioning game induced by basic function f + , i.e., u(S) = maxT ∈(B,S) T ∈T f + (T ) for all S ∈ N . From the definitions we immediately get that v(S) = 0 ⇔ v(T ) ≤ 0 ∀ T ∈ 2 S ⇔ f + (B) = 0 ∀ B ∈ B ∩ 2 S ⇔ u(S) = 0. v(S) > 0. Then there is an S ∗ ⊆ S such that v(S) = v(S ∗ ). By assumpAssume now tion, v(S ∗ ) = T ∈T ∗ f (T ) for some B-partition T ∗ of S ∗ . In this decomposition f (T ) ≥ 0 ∀ T ∈ T ∗ , for otherwise, by discarding the negative components we would get a subset of S ∗ ⊆ S with v value strictly greater than v(S ∗ ) = v(S), a contradiction to the definition of cover. It follows that f (T ) = f + (T ) for all T ∈ T ∗ , the monotonic ∗ + and so v(S ) = T ∈T ∗ f (T ). Since T ∗ ∪ { { j} : j ∈ S \ S ∗ } is a B-partition of S, we get that v(S) = v(S ∗ ) ≤
f + (T ) +
f + ({ j}) ≤ u(S).
j∈S\S ∗
∗
T ∈T
Now assume u(S) > 0. Let T ∈ (B, S) such that u(S) = T ∈T f + (T ). Then the set S + := ∪ T : T ∈ T , f + (T ) > 0 is a nonempty subset of S, and the collection T + := {T ∈ T : f + (T ) > 0} is a B-partition of S + . Thus, u(S) =
T ∈T
f + (T ) =
T ∈T
f + (T ) =
+
T ∈T
f (T ) ≤ v(S + ) ≤ v(S).
+
Therefore, v(S) = u(S) for all S ⊆ N , so indeed v = v(B, f + ) . To see that v is also induced by f 1+ , it suffices to note that if f + (B) = f 1+ (B) then f (B) < 0. For such a basic coalition B, we have f 1+ (B) = f (B) < B ∈ B\B1 and + + + 0 = f + (B) ≤ j∈B f ({ j}) = j∈B f 1 ({ j}), thus B is inessential both for f + and f 1 . By Proposition 1, the essential coalitions for the basic function completely determine the induced partitioning game, thus v = v(B, f + ) = v(B, f + ) , and the proof 1 is complete. Combined with Corollary 3, Proposition 3 immediately gives Corollary 4 Let v be a B-partitioning game induced by basic function f . Then for each x ∈ R N , the maximal excess game wx is a B-partitioning game induced by basic functions ( f − x)+ or ( f − x)+ 1. 5 Strongly balanced partitioning games Kaneko and Wooders (1982) proved (among other necessary and sufficient conditions) that for a collection B, all induced B-partitioning games are balanced regardless of the basic function, if and only if every minimal B-balanced collection on N is a B-partition of N . Following Le Breton et al. (1992) we call such a collection strongly balanced, and the induced partitioning games strongly balanced partitioning games.
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Various characterizations of strong balancedness were independently obtained by Vasin and Gurvich (1977), Kaneko and Wooders (1982), Le Breton et al. (1992) and Kuipers (1994), among others. We summarize the polyhedral ones in the first part of Theorem 3 On player set N , let B ⊆ N contain all single-player coalitions. Then the following properties are equivalent: (i) (ii) (iii) (iv) (v) (vi)
v(B, f ) is balanced for every basic function f , every minimal B-balanced collection on N is a B-partition of N , every extreme point of Λ(B, N ) is integral, every extreme point of Λ≤ (B, N ) is integral, for all S ∈ N , every extreme point of Λ≤ (B, S) is integral, v(B, f ) is totally balanced for every basic function f .
We extend the list with some equivalent game-theoretic properties: (vii) (viii) (ix) (x)
v(B, f ) v(B, f ) v(B, f ) v(B, f )
is balanced for every 0-normalized basic function f , is subbalanced for every 0-normalized basic function f , is totally subbalanced for every 0-normalized basic function f , is totally balanced for every 0-normalized basic function f .
Proof Since v(B, f ) is balanced if and only if its 0-normalization v(0B, f ) is balanced, furthermore, by Corollary 3, v(0B, f ) = v(B, f 0 ) , we get that property (i) is equivalent to (vii), that in turn, by Lemma 1, is equivalent to (viii). Now we show that (viii) implies, hence is equivalent to (ix). Suppose not, and there Then there is an S ∈ N is a 0-normalized f such that v(B, f ) is not totally subbalanced. and a (λT )T ∈N ∈ Λ≤ (B, S) for which v(B, f ) (S) < T ∈N λT v(B, f ) (T ). The carrier of (λT ) contains only subsets of S. We define another 0-normalized basic function g : B → R by g(B) := f (B) if B ⊆ S, and Then we clearly have g(B) := 0 if B ⊆ S. v(B,g) (N ) = v(B,g) (S) = v(B, f ) (S) < T ∈N λT v(B, f ) (T ) = T ∈N λT v(B,g) (T ), a contradiction to the assumed subbalancedness of every 0-normalized B-partitioning game. Again by Lemma 1, property (ix) is equivalent to property (x), that in turn is equivalent to property (vi), since total balancedness is preserved under positive affine transformations. Therefore, we established a chain of equivalent game-theoretic properties between properties (i) and (vi). We are now in position to put together the pieces and prove Theorem 4 Let the fixed family B ⊆ 2 N contain all single-player coalitions. Then (i) if B is strongly balanced then all (pre)bargaining sets listed in Corollary 1 coincide with the core for all B-partitioning games, and conversely, (ii) if any of the (pre)bargaining sets listed in Corollary 1 coincides with the core for all B-partitioning games, then B is strongly balanced. Proof Every partitioning game is superadditive, so claim (i) follows from Theorem 2 and Corollary 1, since, by Corollary 4, the maximal excess game wx at any allocation
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x ∈ R N of a strongly balanced partitioning game is also a strongly balanced partitioning game, hence is balanced. The superadditivity of partitioning games and the existence results cited in Sect. 3 imply that in partitioning games none of the (pre)bargaining sets listed in Corollary 1 is empty. Claim (ii) is then immediate from property (i) in Theorem 3. We now apply Theorem 4 to two well-known classes of games. Both of them are strongly balanced partitioning games. Assignment games, introduced by Shapley and Shubik (1972), are the inspiring examples for partitioning games. The player set is partitioned into two nonempty subsets, N = J ∪ K , the basic collection B consists of the single-player coalitions, {i} for all i ∈ N , and of the “mixed-pair” coalitions, { j, k} for all j ∈ J and k ∈ K . It is well known that the membership matrix of such a collection B is totally unimodular, hence the collection B is strongly balanced [for a direct proof of the latter see Le Breton et al. (1992)]. So, any game that is built up by (8) from such a B has a nonempty core regardless of the basic function f . This seems to be somewhat stronger than the balancedness of all assignment games established by Shapley and Shubik (1972), since they required f ({i}) = 0 for all i ∈ N , and f ({ j, k}) ≥ 0 for all j ∈ J and k ∈ K . In light of Theorem 3 (vii) and Proposition 3, however, we see that taking only 0-normalized and nonnegative basic functions means no loss in generality. Granot and Granot (1992) proved directly that the class of assignment games is closed for taking the maximal excess games for all imputations. Based on this, Granot (1994) showed the equivalence of the core with the reactive bargaining set Mr . Combined with Theorem 1, Solymosi (1999) obtained the coincide with the classical bargaining set M1i . Now, using Theorem 4, we easily get the coincidence of the core even with the Mas-Colell bargaining set and all of its subsolutions. Corollary 5 For assignment games, all the (pre)bargaining sets listed in Corollary 1 coincide with the core. Naturally, the same holds for the larger class of assignment games with arbitrary individual reservation values and arbitrary mixed-pair cooperation values, the model considered, for example, by Owen (1992). The class of permutation games—which is not of a strongly balanced partitioning type—contains this larger class of assignment games, so Corollary 5 also follows from the main result in Solymosi et al. (2003) which asserts the same coincidence for permutation games. Graph-restricted games, pioneered by Myerson (1977) and Owen (1986), are derived from cooperative situations where the communication structure between the players is given by an undirected graph Γ = (N , E), with the natural interpretation that players i, j ∈ N can communicate if and only if {i, j} ∈ E. Cooperation requires communication, so the coalitions which can actually form are the node sets of the connected subgraphs of Γ . If coalitional function v specifies the potential proceeds of cooperation, the Γ -restricted game vΓ derived from v is defined for S ∈ N by vΓ (S) := T ∈S/Γ v(T ), where S/Γ denotes the set of connected components of S in Γ . Owen (1986) proves that if v is superadditive, so does vΓ . Let BΓ denote the set of connected coalitions in Γ . Le Breton et al. (1992) show that BΓ is strongly balanced, if and only if Γ is a forest. Potters and Reijnierse (1995) prove that if Γ is a
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tree and v is superadditive and 0-normalized, the classical bargaining set M1i coincides with the core in the derived vΓ , called in this case a Γ -component additive game. Before we cast graph-restricted superadditive games as partitioning games, let us remark that from a modeling perspective the superadditivity of v, hence of vΓ , is a fairly natural assumption: why deny the breaking up of a connected coalition into connected subcoalitions if it is beneficial, and why force the break-up of a non-connected coalition into its inclusion-maximal connected subcoalitions? Through (8), the above BΓ and v induce the partitioning game v(BΓ ,v) . Clearly, v(BΓ ,v) ≥ vΓ , and equality holds if v is superadditive. If Γ is a tree, BΓ is strongly balanced, thus from Theorem 4 we readily get Corollary 6 For tree-restricted superadditive games, all the (pre)bargaining sets listed in Corollary 1 coincide with the core. 6 Simple network games It is easily seen that an assignment game can be considered as a game induced by a network flow problem. Indeed, in the context of assignment games, the generating optimization problem (8) can be formulated as a maximum revenue problem on a bipartite capacitated directed network, where the player set augmented by a source and a sink is the set of nodes, the generating collection B is the set of arcs, all with unit capacity and with an objective coefficient specified by the coalitional function. In this section we consider another type of games induced by maximum-revenue unitcapacitated network flow problems with the difference that the player set is identified with the set of arcs, and the underlying network is not bipartite. Consider a single-source, single-sink directed network G with all arcs having unit capacity and a (possibly negative) weight coefficient. Suppose the set of arcs is identified with the set N of players. Then the (simple) network G with weight vector c defines the simple network game v c as follows: for each S ⊆ N let G S denote the subnetwork with arcs belonging to S, and let v c (S) be the weighted value of a sourceto-sink flow in G S which is maximal with respect to the weight vector c. Kalai and Zemel (1982) proved that all simple network games are totally balanced, irrespective of the arc-weights. Since for any S ⊆ N there is an optimum flow in G S which is composed of unit flows on arc-disjoint simple paths, it is easily seen that any simple network game defined on the simple network G is a partitioning game generated by the collection of simple source-to-sink paths and individual arcs in G. It follows that simple network games are superadditive. Consider, for example, the network G depicted in Fig. 1. The related simple network game v c is the partitioning game v(BG , f ) generated by the basic family BG = {163, 15273, 1528, 4273, 428, 1, 2, 3, 4, 5, 6, 7, 8} and the (0-normalized) basic function: f (S) = 0, if |S| = 1; and f (S) = i∈S ci , if S is one of the five coalitions which are related to the source-to-sink paths in G.
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Fig. 1 Network G with unit arc-capacities and arc-weights c
Since all simple network games are totally balanced (Kalai and Zemel 1982), any BG -partitioning game which is induced by a similar “additive” 0-normalized basic function is balanced. However, the family BG is not strongly balanced. Indeed, the collection {1528, 163, 4273, 4, 5, 6, 7, 8} is a minimal balanced BG -collection (with balancing weights all equal to 1/2) but not a BG -partition. Thus, simple network games are (totally balanced) partitioning games but not strongly balanced partitioning games. Nevertheless, we can show that the maximal excess game of a simple network game taken at any preimputation is balanced. To this end, let G be a fixed unit-capacitated simple network. We can assume without loss of generality that there is no source-sink arc in G, so for arbitrary arcweight vector c, the induced simple network game v c is 0-normalized. Let A denote the set of single-player coalitions, and P the set of “source-to-sink path” coalitions. On basic family BG = A ∪ P, the 0-normalized basic function f (S) = 0, if S ∈ A;
f (S) =
ci , if S ∈ P
i∈S
generates v c . By Corollary 4, for each x ∈ R N , the maximal excess game wxc is the BG -partitioning game induced by basic function g = ( f − x)+ 1 , i.e. g(S) = (−xi )+ , if S = {i}; g(S) =
(ci − xi ), if S ∈ P. i∈S
If x is an imputation then g is 0-normalized, so wxc is the simple network game v c−x , hence it is balanced. However, if x is not an imputation then wxc is not 0-normalized, so it cannot be a simple network game on G. But it is strategically equivalent to a simple network game, hence it is also balanced. Indeed, by Corollary 3, the 0-normalization (wxc )0 of the maximal excess game wxc is the BG -partitioning game induced by the 0-normalization g 0 of the basic function g, i.e. g 0 (S) = 0, if S ∈ A; g 0 (S) =
(ci − xi − (−xi )+ ), if S ∈ P.
i∈S
From xi + (−xi )+ = xi+ we get the following
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Corollary 7 Let x be a preimputation in the simple network game v c . Then the 0-normalization (wxc )0 of the maximal excess game wxc is the simple network game + v c−x . A direct proof of this observation in case x is an imputation was given by Granot et al. (1997). They used it to show the coincidence of the reactive bargaining set Mr and the core for simple network games. Corollary 7, the well-known balancedness of simple network games (Kalai and Zemel 1982), and the closedness of balanced games under 0-normalization makes Theorem 2 applicable. Thus we easily obtain the following stronger result. Corollary 8 For simple network games, all the (pre)bargaining sets listed in Corollary 1 coincide with the core. 7 Summary We established the coincidence of the Mas-Colell bargaining set (and consequently, of the classical, the semireactive, and the reactive bargaining sets) with the core for wellknown balanced subclasses of partitioning games, namely for assignment games, for tree-restricted superadditive games, and for simple network games. In order to apply the general characterization by Holzman (Theorem 2) in a more general setting— and derive the above-mentioned coincidences as easy corollaries, we showed that the family of partitioning games defined on a fixed basic collection is closed under the strategic equivalence of games, and also for taking the monotonic cover of games. We believe that our approach can be successfully applied not only to various other types of balanced combinatorial optimization games, but also for games which model situations with restricted cooperative possibilities between the agents. Acknowledgments I thank two anonymous referees for their comments and suggestions on how to improve the presentation.
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