Global Coalitional Games

Università degli Studi di Siena

DIPARTIMENTO DI ECONOMIA POLITICA

GIOVANNI ROSSI

Global Coalitional Games

Abstract - Global coalitional games are TU cooperative games intended to model situations where the worth of coalitions varies across different partitions of the players. Formally, they are real-valued functions whose domain is the direct product of the subset lattice and the lattice of partitions of a finite player set. Therefore, the dimension of the associated vector space grows dramatically fast with the cardinality of the player set, inducing flexibility as well as complexity. Accordingly, some reasonable restrictions that reduce such a dimension are considered. The solution concepts associated with the Shapley value and the core are studied for the general (i.e., unrestricted) case.

Key words: lattice, lattice function, coalition, partition, Shapley value, core.

JEL classification number: C71.

The author thanks Stefano Vannucci for friendly advice and careful guidance. The usual disclaimer applies.

1

Introduction

Traditional TU cooperative game theory deals with situations where each player may decide whether to cooperate or not. Furthermore, attention is usually focused on grand coalitions (i.e., player sets) within which monotonicity holds, so that such grand coalitions must eventually form. Therefore, it is natural to concentrate solely on the solution problem (i.e., how to share the ‘grand cake’). In particular, the core concept is concerned with those additive coalitional games (i.e., sharing rules) that, when considered as payoff vectors, make it rational for each player to join the grand coalition (and for each coalition to join its complement). Conversely, when monotonicity is relaxed, it comes natural to ask what largest coalitions will form (and how will their worth get shared among members). In other words, endogenous coalition formation enters the picture, and nontrivial coalition structures (i.e., different from the grand coalition) do result. In this paper, endogenous coalition formation is not directly approached; yet, most of the concern is on the partition lattice of afinite player set.

Coalition structures are partitions of the players, and constitute the outcome of many situations where cooperation displays synergies and yet the choice of individual players does not merely reduce to whether to cooperate or not, but also concerns the degree of cooperation. For example, in voting situations voters may furnish a finite number of different support levels to the various bills, so that these latter pass whenever a given majority rule is fulfilled by the partition of the voter set obtained by putting any two voters furnishing the same support level into the same block. More generally, Gilboa and Lehrer (1991a) defined global games as real-valued functions whose domain is the partition lattice of the player set, and proposed to use them to model situations where the global (i.e., the world’s) welfare level depends on the (strategic) choice of each player over what block to join. Nevertheless, such a modeling choice does not allow to consider the possibility that for each admissible global welfare level there exist several different distributions of such a welfare over the global population. In order to do so, it is necessary to consider functions taking values on pairs con-sisting of a coalition of players and a partition of the (whole) player set. Such an approach wasfirstly adopted by Thrall and Lucas (1963) and subsequently by Myerson (1977), even though they restricted attention to those pairs consisting of a coalition and a partition embedding the former as one of its blocks. Such a restriction is here abandoned, so that for each coalition a global game gets defined. This implies that each coalition has the strategic possibility not only of forming a corresponding unique block (of some ‘final’ partition), but also of spreading its members over different blocks, possibly containing nonmembers as well. From another viewpoint, this can be seen as an attempt to consider situations where cooperation may occur at both quantitatively as well as qual-itatively different levels. In other words, in global coalitional games players may be seen as cooperating at qualitatively different levels when considered as members of coalitions or else as members of blocks of partitions.

An important remark (applying to the approach here proposed as well as to those of Thrall and Lucas (1963), Myerson (1977) and Gilboa and Lehrer

(1991a)) is the following. For any arbitrarily large (finite integer) numbernof players, the cardinality of the associated subset lattice (that is, the power set of the player set endowed with the set inclusion order relation) is2n_{. Nevertheless,}

there does not exist (as known to the author) an equivalent closed form innfor the cardinality of the associated partition lattice (that is, the set of partitions of the player set endowed with the ‘coarser than’ order relation). In number theory, a partition of any integernis any collectionλ1, . . . , λmsuch thatPmj=1λj=n,

all λj’s being integers as well. The number p(n) of all such partitions was

determined by Hardy and Ramanujan, and can be found in Andrews (1976), ch. 5. It is clearly much greater than2n_{; furthermore, the number of partitions}

of an n-set is much greater than p(n), in that each partition λ1, . . . , λm of

n corresponds to n!₋m!Qm_j=1λj! different partitions of an n-set. In fact,

the number _Bn of partitions of an n-set is determined through recursion by

Bn=Pnk=0−1

¡n−1

k

¢

Bk, withB0:= 1(see Aigner (1979) on the Bell numbers).

Given the above remark, it is easily understood that any kind of n-players game that makes use of the partition lattice becomes rapidly intractable asn

increases. In turn, this implies that, at least in terms of conceivable applications, such games may be useful solely for modeling situations involving a reasonably small number of players. In particular, Gilboa and Lehrer (1991a) propose to model as global games “questions of art and historical treasures preservation, a cure for cancer and AIDS, indeed, the progress of science and art in general, and many other issues [that] -though not unrelated to nations’ political interests-seem to be ‘global’, at least as afirst approximation. [Their] paper models such games and tries to cope with the question of their ‘solution’. [Their idea is that any partition of the player set has an associated global worth (i.e., a worth for the grand coalition), and that, in particular,] the payoffis defined for all players together. (Or, if you will, that the utilities of the players coincide.)” (p. 129). Firstly note that, as previously mentioned, by switching from global to global coalitional games, one may relax this last assumption, i.e., that the utilities of the players coincide. Secondly, consider that for all the above mentioned global situations, one may assume that countries, and not individuals, are the actual players, so that their number allows to model such situations in terms of (games defined on) the partition lattice. Thirdly, in many of such situations it may well be that coalitions of countriesfind it strategically optimal to cooperate, at the partition level, by spreading their members over different blocks containing nonmembers too.

Example 1: environmental clean-up and preservation. For air and water pollutions migrate from polluting to nonpolluting countries, the globe deals with environmental clean-up and preservation by means of international agreements. Furthermore, in practice, at any given time each country signs at most one such agreement and thus a partition of the country set results1_{. Nevertheless, the}

‘worth’ of any agreement (i.e., its efficacy) depends on the residual confi

gura-1_{Alternatively, if some countries had signed more than one agreement, it would still be} possible to define a partition of the country set by ranking agreements from tighter to looser (i.e., in terms of the regulations they encode), and then considering each country only as a member of the tighter agreement it has signed.

tions of agreements (i.e., signed by nonmembers of the former). Furthermore, consider (a coalition of) two important oil-producer countries facing the choice of what agreement to sign. By signing each a different one, they may (try to) achieve some loose regulation for the whole globe, rather than a looser one for a restricted region only. Note that this does not entirely depend on strategic matters. More precisely, such two countries have a common interest they can pursue only at the international level. Furthermore, such an interest may be best pursued by spreading over distinct agreements. In other words, the two countries need not necessarily form a coalition in the usual sense, even though they can definitely agree to do so.

Example 2: currency unions (and areas). When trying to model some

‘international monetary game’, it is rapidly realized that any configuration of currency unions (and areas) within the world economy definitely results in both: (i) a partition of the country set (simply regard any country with a free exchange rate with respect to all other currencies as a one-player union), and (ii) some level of global welfare level. Nevertheless, here again, the worth (however measured) of any currency union does depend on the residual configuration of currency unions. Furthermore, considering the current situation, the United States on the one side, and the European ‘euro-countries’ on the other, typically consti-tute (disjoint) coalitions, as their currencies are both anchors of two different currency areas (and these latter define the blocks of the partition). Neverthe-less, (in the absence of altruism) the utility level attained by their citizens is definitely higher in such a situation than it would be if a unique currency for the world economy was used. Yet,_{BCE, F ED_}cannot be considered a typical two-player coalition. In particular, it can be considered as a coalition that may form at a level of cooperation that differs from that defining the partition2_.

The paper is organized as follows. Section 2 contains a formalization of the setting, identifies the vector space of global coalitional games and recalls some general results of Gilboa and Lehrer (1991a) on lattice functions. Section 3 considers both the global games of Gilboa and Lehrer (1991a) and the games in partition function form of Thrall and Lucas (1963) and Myerson (1977); in particular, the Möbius transform of the latter is derived and expressed in terms of the former’s one, so to show what vector (proper) subspace (i.e., of the vector space of global coalitional games) the latter games belong to. Sections 4 and 5 deal, respectively, with the Shapley value and the core concepts. The Shapley (1953) axioms are adapted to global coalitional games and shown (following Weber (1988)) to characterize a unique value function. As previously mentioned, any global coalitional game associates to each coalition a global game (i.e., a number of real quantities that equals the number of partitions of the players), so that defining the core may be (and has been so far) approached as a typical aggregation problem, here solved by means of the Choquet integral. Section 6 contains some concluding remarks and possible future research topics.

2_{The application of global coalitional games (graph-restricted and in partition function} form; see section 3) to currency unions is done in a forthcoming paper. Furthermore, most likely currency areas could be dealt with by means of the MLE (multilinear extension) of global coalitional games, to be defined.

2

Preliminary notations and results

A complete lattice is any set X endowed with an order (binary) relation < satisfying reflexivity (x<xfor allx_∈X), antisymmetry (y<x, x<y_⇒y=x

for allx, y _∈ X) and transitivity (z < y, y < x_⇒ z < xfor all x, y, z _∈ X), and such that there exist the least upper bound _∨x∈S ∈ X and the greatest

lower bound _∧x∈S ∈ X (with respect to <) for all S ⊆ X, in which case

x_⊥, x> _{∈ X such that x}

⊥ 4 x, x> < x for all x ∈ X are bottom and top

elements respectively.

Any finite player set N = _{1, . . . , n_} defines two main complete lattices. One is the subset lattice, that is2N =_{A_|N_⊇A_}, which is in fact endowed with the set inclusion relation_⊇ by definition. The other is the partition lat-tice, i.e.,_P = ½ {A1, . . . , Am}⊂2N\∅| ∪ 1≤j≤mAj =N,j∈J⊆{∩1,... ,m}Aj=∅ ¾ en-dowed with the coarser than relation _≥. Recall that for any P, P0 _{∈ P, with}

P=_{A1, . . . , Am}andP0 ={A01, . . . , A0m0}, the order relation≥is defined by:

P _≥P0 _{if for eachj}0_{∈{1, . . . , m}0_{}it holds A}0

j0 ⊆Aj for somej∈{1, . . . , m}.

Also recall that A _⊃ B _⇔ A _⊇ B, A ₆= B for all A, B _⊆ N, as well as

P > Q _⇔ P _≥ Q, P ₆= Q for all P, Q _{∈ P}. The subset lattice may be de-noted¡2N,_∩,_∪¢. Similarly, the partition lattice is usually denoted (_P,_∧,_∨), where P _∧P0 _{(P ∨P}0_{) denotes the coarsest (finest) partition finer (coarser)} than both P, P0 _{∈ P. Formally, P ∧P}0 ₌ ©_A

j∩A0j0 |Aj∩A0j0 6=∅ ª and P_∨P0₌ ½ ∪ 1≤j≤m,1≤j0_≤m0 ¡ Aj∪A0j0 ¢ |Aj∩A0j0 6=∅ ¾

. The bottom elements are

∅ ∈2N _andP

0={{1}, . . . ,{n}}∈P, while the top one is3 N ∈2N,P.

Coali-tional and global games onN are0-normalized and real-valued lattice functions with domains 2N _{and P respectively, so that C}N ₌©_{v: 2}N_{→R|v(∅) = 0}ª

andGN _{={f :P→R|f(P}

0) = 0}denote the sets of such games.

Now consider the Cartesian product2N_{× P, and endow it with the order}

relation4 < such that (A, P) < (B, Q) if A _⊇ B, P _≥ Q for any two pairs

(A, P),(B, Q)_∈2N_{× P}. This assures that<satisfies reflexivity, antisymmetry and transitivity, and thus that¡2N_{× P},<¢is a well defined complete lattice, its bottom and top elements being(_∅, P0)and(N, N)respectively. The additional

binary relation5 _Â∗_{, defined by(A, P)Â}∗_{(B, Q)if(A, P)Â(B, Q)but for no}

(B0_{, Q}0_{)it holds(A, P)Â(B}0_{, Q}0_{)Â(B, Q), will also be useful.}

A global coalitional game (grounded) onNis a0-normalized lattice function

h: 2N_{× P →R; thus h(∅, P}

0) = 0. It is easily seen thath is an application

naturally embedding both: a number_{| P |}of coalitional games (i.e., on2N_{), each}

consisting of the collection©h(A, P)_|A_∈2Nª_{forP ∈P, and a number2}n _of

global games (i.e., on_P), each consisting of the collection _{h(A, P)_|P _∈P}

forA _∈2N_{. In the sequel, the restriction h(∅, P) = 0 for all P ∈P is shown}

3_{Parentheses for the coarsest partition {N}=N∈Pare omitted whenever no confusion} is possible.

4_{The order relation<is denotedÀin Myerson (1977).}

5_{The binary relations⊃}∗_on2N _and>∗_{onPused in the sequel are defined analogously.} In fact,Â∗_{results to be the ‘sum’ of⊃}∗_and>∗_.

to result in no loss of generality. LetGCN denote the set of global coalitional games onN, with the embedding implyingCN_⊂GCN_⊃GN_.

CN_{, G}N _{and GC}N _{can be treated as vector spaces. Concerning their}

di-mensions, Shapley (1953) showed that the set ©uA|A∈2N\∅

ª

of unanimity (coalitional) games, defined by uA(B) = 1 ifB ⊇ A and 0otherwise,

consti-tutes a basis of CN _{⊆ R}2n

−1_{. Similarly, Gilboa and Lehrer (1991a) showed}

that _{gP |P ∈P\P0}, wheregP(Q) = 1ifQ≥P and 0otherwise, is a basis

ofGN _⊆R|P|−1_{. Here the dimension ofGC}N _{clearly is| 2}N_{× P | −1, and a}

similar result obtains6_.

Proposition 1 The set of games ©gA,P |(∅, P0)6= (A, P)∈2N× Pª, defined

by gA,P(B, Q) = 1 if (B, Q) < (A, P) and 0 otherwise for any two pairs

(A, P),(B, Q)_∈2N_{× P, is a linear basis of GC}N_.

Proof. Following Gilboa and Lehrer (1991a), linear independence is shown by considering thatP_{(∅,P0)6=(A,P)∈2}N_×PαA,PgA,P = 0iffαA,P = 0for all pairs

(_∅, P0)6= (A, P)∈2N× P, in thatP(∅,P0)6=(A,P)∈2N×PαA,PgA,P(B, Q) = 0iff

αB,Q= 0for all(B, Q)Â∗(∅, P0), and the argument proceeds by induction. For

its cardinality is in fact_|2N_{×P |−1, the set}©_g

A,P |(∅, P0)6= (A, P)∈2N× P

ª

is now easily seen to be a basis ofGCN_.

Treating global coalitional games as generic lattice functions leads to define anyh_∈GCN _{to be}

nonnegative: ifh(A, P)_≥0for all(A, P)_∈2N_{× P;}

monotone: if(A, P)<(B, Q)impliesh(A, B)_≥h(B, Q),

convex: ifh(A, P) +h(B, Q)_≤h(A_∪B, P _∨Q) +h(A_∩B, P_∧Q), additive: if its convexity holds with equality,

for all(A, P),(B, Q)_∈2N_{× P};

m-positive: if for every(A1, P1), . . . ,(Am, Pm)∈2N× P

h µ ∪ 1≤i≤mAi,1≤∨i≤mPi ¶ ≥ X ∅6=I⊆{1,... ,m} (₋1)|I|+1h µ ∩ i∈IAi,i∧∈IPi ¶ ,

totally positive: ifm-positivity holds for allm_∈{2, . . . ,2n_{× | P |}}.

Monotonicity is a standard assumption in cooperative game theory, for it translates the idea that the larger the number of players that cooperate, the greater the worth that gets produced through cooperation. Thus, such an as-sumption is innocuous when referred to coalitional games. On the other hand,

6_{In fact, the results reported in this section have been shown to hold for lattice functions} in general by Gilboa and Lehrer (1991a), and thus they are here merely reproduced for the sake of completeness.

global games associate a worth to each partition, so that coarser partitions may be seen as corresponding to situations where cooperation occurs within (collections of) larger coalitions. Yet, global coalitional games allow to com-pare pairs such as¡A,_{A_}∪PAc¢

and ¡A,_{A_}∪QAc¢

, where Ac _{=N\A and}

P(Ac_{) 3 P}Ac

, QAc

denotes the set of partitions of any Ac _{⊂ N. Consider}

the case PAc 0 ≤ PA c < QAc ≤ {Ac_{}, where P}Ac 0 ,{Ac} ∈ P(Ac) constitute,

respectively, thefinest and coarsest partition of Ac_{. For such pairs}

monotonic-ity requires h¡A,_{A_}∪PAc¢ _{≤ h}¡_A,{A}∪QAc¢_{, even though the worth of}

A, every time it constitutes a block on its own, might (intuitively) be greater when the remaining playersi_∈Ac _{are more dispersed (i.e., over more blocks).}

In fact, most likelyh¡A,_{A_}∪PAc¢_{< h}¡_N,{A}∪PAc¢_{< h}¡_N,{A}∪QAc¢_.

Yet, playersi_∈Acappear better organized for bargaining (i.e., displaying more cohesion) underQAc rather than underPAc.

GCN can clearly be endowed with addition and (positive) scalar multipli-cation in the usual manner, and any subset of global coalitional games that is closed under such two operations constitutes a cone in_R2n

×|P|−1 _⊇GCN_{. In}

particular, following Shapley (1971), it can be noticed that the set of convex games constitutes a cone that contains the subspace of additive games. Sim-ilarly, following Gilboa and Lehrer (1991a), it can be noticed that the set of totally positive and monotone games constitutes a cone that includes that of convex games. The next two sections are devoted to define reasonable subspaces of (and cones in)GCN_.

Theorem 2 h_∈GCN _{is totally positive and monotone iffα}

A,P(h)≥0for all

(A, P)_∈2N_{× P.}

Proof. Consider the coneGCN

T P M ⊂GCNof totally positive and monotone

global coalitional games. As in Gilboa and Lehrer (1991a), if global coalitional unanimity games are such that gA,P ∈ GCT P MN , then the ‘if’ part is proved.

For gamesgA,P are monotone by definition, they must be shown to be totally

positive, that is, any collection(A, P),(B1, Q1), . . . ,(Bm, Qm)∈2N× P must

satisfy gA,P µ ∪ 1≤i≤mBi,1≤∨i≤mQi ¶ ≥ X ∅6=I⊆{1,... ,m} (₋1)|I|+1gA,P µ ∩ i∈IBi,i∧∈IQi ¶ .

Let J _⊆{1_≤j _≤m_|(A, P)4(Bj, Qj)}, noting that if J = ∅ the right side

vanishes and the inequality holds. Otherwise,

gA,P µ ∪ 1≤i≤mBi,1≤∨i≤mQi ¶ − X ∅6=I⊆J (₋1)|I|+1gA,P µ ∩ i∈IBi,i∧∈IQi ¶ = = 1₋P_∅6=I⊆J(₋1)|I|+1 =P_I⊆J(₋1)|I|= (1₋1)|J|= 0. Let_{(B1, Q1), . . . ,(Bm, Qm)}= © (B, Q)_∈2N_{× P |(A, P)Â}∗_{(B, Q)}ª_for

requires the following result: h(A, P)₋ X ∅6=I⊆{1,... ,m} (₋1)|I|+1h µ ∩ i∈IBi,i∧∈IQi ¶ =αA,P(h) (∗).

To see this, recall the representation

h(A, P) = X (B,Q)∈2N_×P αB,Q(h)gB,Q(A, P) = X (B,Q)4(A,P) αB,Q(h),

so that(_∗)is immediately seen to hold true as

X (B,Q)≺(A,P) αB,Q(h)− X ∅6=I⊆{1,... ,m} (₋1)|I|+1 X (B0,Q0)4 _i∩ ∈IBi,i∧∈IQi αB0_,Q0(h) = = X (B,Q)≺(A,P) αB,Q(h) ⎛ ⎜ ⎜ ⎜ ⎝1− X ∅6=I⊆{1,... ,m}|(B,Q)4 _i∩ ∈IBi,i∧∈IQi (₋1)|I|+1 ⎞ ⎟ ⎟ ⎟ ⎠= =P_{(B,Q)≺(A,P)}αB,Q(h) ³ 1₋P_{∅6=I⊆JB,Q}(₋1)|I|+1´= 0, in that JB,Q 6=∅for all(B, Q)₆= (_∅, P0), whereJB,Q ={1≤j≤m|(B, Q)4(Bj, Qj)}.

The proof ends by considering a totally positive and monotone game hand

(A, P) _Â∗ _(B

1, Q1), . . . ,(Bm, Qm). If m = 1, then nonnegativity is implied

by monotonicity, as (_∗) yields αA,P(h) = h(A, P)−h(B1, Q1). Otherwise,

nonnegativity is implied by(_∗)and total positivity.

The product lattice¡2N_{× P,<}¢_{is rich, that is, whenever the levell(A, P)}

of a pair is strictly greater than unity its degree d(A, P) also is so, where

l(A, P) =ksuch that

(A, P) _Â ∗(B, Q)_1Â∗(B, Q)_2Â∗. . ._Â∗(B, Q)_k= (_∅, P0),

whiled(A, P) = _|©(B, Q)_∈2N_{× P |}(A, P)_Â∗(B, Q)ª_|.

For d(_∅, P0) = l(∅, P0) = 0, each (A, P) has l(A, P) > 1 ⇒ d(A, P) > 1

holding. In particular, observation 3.4 of Gilboa and Lehrer (1991a) applies.

Proposition 3 For anyh_∈GCN_{such thath(∅, P)6= 0for at least oneP ∈P,}

there exists a_∅-normalized gameh_∅∈GCN_{such thath}

∅(∅, P) = 0for allP∈P

andh_∅(A, P)₋h_∅(B, P) =h(A, P)₋h(B, P)for allP _∈P andA, B_⊆N.

Proof. For h_∈GCN_{, defineh}

∅(A, P) = h(A, P)−PQ≤Pα∅,Q(h)for all

(A, P)_∈2N_{× P; then h}

∅(∅, P) =h(∅, P)−h(∅, P) = 0for allP ∈P, as well

ash_∅(A, P)₋h_∅(B, P) =

for allP _∈P andA, B_⊆N.

Let _R(2+n−1)×|P| ⊇ GC_∅N ⊂ GCN denote the set of ∅-normalized and

non-negative global coalitional games; its dimension is (2n₋1)_{× | P |} for a basis clearly is©gA,P |(A6=∅, P)∈2N× P

ª

⊂©gA,P |(∅, P0)6= (A, P)∈2N× P

ª

.

3

Global games and the partition function form

Different reasonable approaches may be adopted for reducing the dimension of

GCN

∅ , one of whose basis is in fact

© e gA,P |(A6=∅, P)∈2N× P ª defined by e

gA,P(B, Q) = 1 if(B, Q) = (A, P) and 0otherwise. An important reduction

occurs through the partition function form approach adopted by Thrall and Lucas (1963) and Myerson (1977). Let_E =©(A, P)_∈2N_{× P |A∈P}ª_denote

the subset of pairs where the partition embeds the coalition. Then, an ‘n -person game in partition function form’ is any functionbh:_{E →R}, and it can be extended as bhext _{: 2}N_{× P → R to the whole domain of global coalitional}

games by letting _{(B1, Q1), . . . ,(Bm, Qm)} = {(B, Q)∈E |(A, P)<(B, Q)}

for each(A, P)_∈2N_{× P. In fact, this allows to define both the set of indices}

JA,P =

½

1_≤j_≤m_|(Bi, Qi) _or4

¨ (Bj, Qj), i∈{1, . . . , m} \ {j}

¾

and the extension bhext_{(A, P) =} P

j∈JA,Pbh(Bj, Qj) for all (A, P) ∈ 2

N _{× P.}

Thus, _{(Bj, Qj)|j∈JA,P} denotes the set of pairs such that: (i) Bj ∈ Qj,

(ii)(A, P)<(Bj, Qj), and (iii) there is no(Bi, Qi)Â(Bj, Qj), i∈{1, . . . , m}

satisfying (i) and (ii). In other terms, the pairs (Bj, Qj), j ∈ JA,P are <

-maximals of_{(B1, Q1), . . . ,(Bm, Qm)}. First note that, apart from the trivial

caseJ_∅,P =∅for allP ∈P, it may be either|JA,P |= 1, or else|JA,P |>1. In

particular, for any(A₆=_∅, P)_∈2N_{× P}, letPA_∈P(A)denote the partition of

Ainduced by ©B1, . . . , B_|P|

ª

=P _∈P, that is ©A_∩B1, . . . , A∩B_|P|

ª

. Note that _| PA _|=_| JA,P |= 1 either when (A, P) ∈ E, or else when A ⊂ Bj ∈ P

for some 1 _≤ j _≤| P _|. In the former case hbext_{(A, P) =} b_{h(A, P), while in}

the latter bhext_{(A, P) =} b_h¡_{A, P}A_∪PAc¢_{, in that P}A _{= {A}∪{B}

j\A} and

thus ¡A, PA_∪PAc¢ _{∈ E. On the other hand, | P}A _{|=| J}

A,P |> 1 implies

bhext_{(A, P) =}P A∈PAbh

³ b

A, PA_∪PAc´_{. Also note that, from another}

perspec-tive, one may regard the extensionbhext_{of games}b_{hin partition function as the}

restriction of global coalitional gamesh_∈GCN _{that coincide with}b_honE. Definition 4 The _E-restriction (in partition function form) ofh_∈GCN is

h/E(A, P) =

X

j∈JA,P

h(Bj, Qj) for all (A, P)∈2N× P.

Clearly, h/E ∈ GC∅N for any h ∈ GCN. Also, the vector space GCEN of E-restricted global coalitional games constitutes a (proper) subspace ofGC_∅N; in fact, its dimension is shown to be_{| E |}<(2n_{−1)× | P |in the sequel.}

Global games have been already introduced as0-normalized partition func-tionsf :_{P →R|}f(P0) = 0. Note that aE-restricted global coalitional game

h/E can be treated as a mapping that associates to each nonvoid coalition A a global subgamefA :P(Ac)→Rdefined by fA

¡ PAc¢ =h/E ¡ A,_{A_}∪PAc¢ (andh/E ¡ A,_{A_}∪PAc¢ =h¡A,_{A_}∪PAc¢ ) for allPAc ∈P(Ac_).

Neverthe-less, this clearly requires to abandon the assumption of 0-normalization, i.e.,

fA

¡

PAc

0

¢

R0. The remainder of this section focuses on the Möbius transforms of these latter global subgames (i.e., the sets of reals©αPAc(f_A)|PA

c

∈P(Ac)ª

for each nonvoidA; see Gilboa and Lehrer (1991a)), and on the Möbius trans-form ofh/E (i.e., the set of reals

© αA,P ¡ h/E ¢ |(A, P)_∈2N_{× P}ª_{as defined by}

(_∗)in the proof of theorem 2). For any ©B1, . . . , B|P|

ª

=P _{∈ P}, let _{Q1, . . . , QkP}={Q∈P |P >∗Q} denote the set of partitions covered byP. Then, any Qj,1≤j ≤kP has form

Qj =PB c i∪QBi j , withQ Bi j ∈P(Bi)(2)andP(Bi)(2)={Q0∈P(Bi)|2 =|Q0 |}

denoting the set of 2-block partitions of Bi ∈P (and Bic =N\Bi). In words,

anyQj covered byP must equal this latter for all blocksBi0_6=i, while dividing

some blockBi,1≤i ≤|P | in two (new) blocks, i.e.,{B0i, Bi\Bi0}=Q Bi

j with

∅6=B0

i ⊂ Bi. In particular, eachP ∈ P coverskP =−|P |+PB∈P2|B|−1

partitions Qj,1 ≤j ≤ kP (see Aigner (1979), ex. I.4 #6, p. 29), and clearly

the same applies to anyPA_∈P(A)for nonvoidA. Thus, any(A, P)_∈2N_{× P}

covers pairs either of the form(B, Q) = (A_\i, P) withi_∈A (i.e.,A_⊃∗B), or else of the form(B, Q) = (A, Qj)withP >∗Qj and1≤j≤kP as above. For

there are_|A_|+kP such covered pairs, let{(Bj, Qj)|1≤j≤|A|+kP}=

{(B, Q)_|(A, P)_Â∗(B, Q)_}=©(B1, P), . . . , ¡ B_|A|, P ¢ ,(A, Q1), . . . ,(A, QkP) ª

for each(A, P) _∈ 2N_{× P}. Also recall that any coalitional gamev _∈ CN has Möbius transform©αA(v)|A∈2Nªgiven by

αA(v) = X B⊆A (₋1)|A\B|v(B) =v(A)₋ X ∅6=I⊆{1,... ,|A|} (₋1)|I|+1v µ ∩ i∈IBi ¶ ,

whereA_⊃∗Bi=A\ifor eachi∈AwhenA={1, . . . ,|A|}⊆N. Analogously,

any global gamef _∈GN has Möbius transform_{αP(f)|P ∈P}given by

αP(f) =f(P)− X ∅6=I⊆{1,... ,kP} (₋1)|I|+1f µ ∧ i∈IQi ¶ , where_{Q1, . . . , QkP}={Q∈P |P >∗Q}as above.

Theorem 5 Any h_∈GCN is_E-restricted (that is, h_∈GC_EN or, equivalently,

h=h/E) iff αA,P(h) = αPAc(f_A)−P_B

⊂AαPBc(f_B) for all (A, P) ∈E and

Proof. Firstly consider the ‘if’ part for(A, P)_∈E, in which case7 αA,P¡h/E ¢ =h(A, P)₋ |A_X|+kP k=1 (₋1)k+1 X I∈{1,... ,|A|+kP}(k) h/E µ ∩ i∈IBi,i∧∈IQi ¶ = =h(A, P)₋ k_{P A}c X k=1 (₋1)k+1 X I∈_{1,... ,k_{P A}c}(k) h µ A, _∧ i∈IQi ¶ + − |A_X|+kP k=1 (₋1)k+1 X I∈{1,... ,|A|+kP}(k) A /∈ _i∧ ∈IQi ∩ i∈IBi h/E µ ∩ i∈IBi,i∧∈IQi ¶ ,

where each Qi,1 ≤ i ≤ kPAc has form Q_i = {A}∪QA c

i with PA

c

>∗ _QAc

i .

Thus, thefirst line equalsαPAc(f_A). On the other hand, the second line groups all terms such that h/E

µ ∩

i∈IBi,i∧∈IQi

¶

6

= h³A,Qe´ for some Qe _{∈ P}, in that

µ ∧

i∈IQi

¶_i∩

∈IBi

denotes the partition of _∩

i∈IBi induced by i∧∈IQi. Now, for any

B_⊂A, consider allI_⊆{1, . . . ,_|A_|+kP}such that±h

¡

B, PB_∪PBc¢

appears as a summand in the second line. Firstly, this occurs for_|I_|= 1, i.e.,I=_{i0

1}, when (a) Bi0 1 = A and Qi01 = P B _∪PBc = _{B_}∪{A_\B_}∪PAc , and thus with sign ₋. Secondly, it occurs for _| I _|=_| A_\B _|, i.e., I = ©i1, . . . , i|A\B|

ª

(and thus with sign (₋1)|A\B|+1), when (b) Bij = A\j, with j ∈ A\B and

1 _≤ j _≤| A_\B _| (and thus A_\B = _{1, . . . ,_|A_\B_{|} ⊂} N), while Qij = P for all ij ∈ I. Thirdly (and lastly), it occurs for (c) | I |=| A\B | +1, i.e.,

I=_{i0

1}∪

©

i1, . . . , i|A\B|

ª

(and thus with sign(₋1)|A\B|+2). Clearly, cases (b) and (c) cancel out, in that they display opposite sign for any_|B_|. Therefore,

αA,P ¡ h/E ¢ =α_PAc(f_A)−P_B ⊂Ah ¡ B, PB_∪PBc¢ + − |A_X|+kP k=1 (₋1)k+1 X I∈{1,... ,|A|+kP}(k) A /∈ _i∧ ∈IQi ∩ i∈IBi (B,PB∪PBc)6= _i∩ ∈IBi,i∈∧IQi for allB⊂A h/E µ ∩ i∈IBi,i∧∈IQi ¶ =

7_{For anyfinite setS, in combinatoricsS}(k)_{denotes thek-th level set of2}S_{, i.e., the set of}

allk-cardinal subsets ofS, thus|S(k)_|=|S|

k

=αPAc(f_A) + − X B⊂A ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ h³B, PB_∪PBc´₋ k_{P B}c X k=1 (₋1)k+1 X I∈{1,... ,|A|+kP}(k) ∧ i∈IQi B =PB PBc_>∗_QBc i for alli∈I h µ B, _∧ i∈IQi ¶ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = =α_PAc(f_A)−P_B ⊂AαPBc(f_B)as desired.

Secondly, the ‘if’ part for (A, P) _∈/ _E requires to distinguish between case (i)_|PA _|=|J

A,P |= 1, and case (ii)|PA |=|JA,P |>1. For the former case,

letJA,P ={jA,P}andJbA,P =©1≤j ≤|A|+kP |(Bj, Qj)<¡BjA,P, QjA,P

¢ª

, noting thatJbA,P 6=∅for all(A, P)∈/E such that|PA|>1. Therefore,

αA,P ¡ h/E ¢ = (1₋1)|JA,P|_α BjA,P,QjA,P ¡ h/E ¢ = 0,

whereαBjA,P,QjA,P

¡ h/E ¢ = =h¡BjA,P, QjA,P ¢ − X ∅6=I⊆ 1,... ,|B_jA,P|+k_Qj A,P (₋1)|I|+1h/E µ ∩ i∈IBi,i∧∈IQi ¶ , while(1₋1)|JA,P| = 1 +P_∅6=I⊆J A,P6=∅(−1)

|I|_{= 0. Similarly, for case (ii), let}

JA,P = n j1 A,P, . . . , j| PA | A,P o as well as b J_A,Pi =n1_≤j_≤|A_|+kP |(Bj, Qj)< ³ B_ji A,P, QjA,Pi ´o

for1_≤i_≤|PA_{|, noting thatJ}bi

A,P 6=∅, alli. Therefore,

αA,P¡h/E ¢ = |_XPA| i=1 (1₋1)|JA,Pi |_α B_ji A,P,QjiA,P ¡ h/E ¢ = 0

Eventually, concerning the ‘only if’ part, note that ifh_∈GCN has Möbius transform as defined by the theorem, thenh(A, P) =

= X (B,Q)4(A,P) αB,Q(h) = X j∈JA,P X E3(B0_,Q0_)4(Bj,Qj) αB0_,Q0(h) = = X j∈JA,P X E3(B0_,Q0_)4(Bj,Qj) Ã α_Q0B0c(f_B0)− X B00_⊂B0 α_Q0B00c(f_B00) ! = = X j∈JA,P ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ h(Bj, Qj)−P ∅6=I⊆ ⎧ ⎨ ⎩1,... ,kQ Bc_j j ⎫ ⎬ ⎭ (₋1)|I|+1h µ Bj, ∧ i∈IQi ¶ + −PB0_⊂Bjα_QB0c j (fB 0) + + P E3(B,Q)≺(Bj,Qj) ⎛ ⎝α_QBc ¡ f_B¢₋ P B⊂B α QBc ³ f B ´⎞ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = =P_j∈J A,Ph(Bj, Qj)as wanted.

The idea that the larger cooperation the greater the worth that gets pro-duced, which is roughly captured by monotonicity, is made more precise and enforced by convexity, and even more by total positivity. In particular, in a con-vex global coalitional game any two disjoint coalitionsA, B, considered under any two partitionsP, Q, alwaysfind it convenient to merge underP_∨Q. Thus, merging must be (weakly) convenient even when(P_∨Q)A∪B=PA∪B_=QA∪B_.

Nevertheless, in such a case solely(A_∪B)c, and notA_∪B, increases its cohesion by switching fromP, QtoP_∨Q. More generally, the representation coefficients

αA,P(h)quantify the net surplus of worth produced by(A, P)with respect to

all(B, Q)_≺(A, P). Therefore, a totally positive and monotone game describes a situation where cooperation most widely displays synergies, making it inter-esting to consider the intersection ofGC_{T P M}N with subspaces ofGC_∅N. In fact,

GCN

T P M ∩GCEN is the cone consisting of those hthat for all(A, P)∈2N× P

satisfy αA,P(h) ≥ 0 if A ∈ P and αA,P(h) = 0 if A /∈ P. In particular, if

a_∈R+andh, h0 ∈GCT P MN ∩GCEN, thenah(A, P) =

P

j∈JA,Pah(Bj, Qj)and

(h+h0_{) (A, P) =} P

j∈JA,P(h+h

0_{) (Bj, Qj), as well as αA,P(ah) =aαA,P(h)} and αA,P(h+h0) =αA,P(h) +αA,P(h0) for all(A6=∅, P) ∈2N× P,

imply-ingah,(h+h0_)∈GCN

T P M ∩GCEN. It is also evident thatGCEN corresponds to

the subspace spanned by©gA,P (oregA,P)|(A6=∅, P)∈E⊂2N× P

ª , and thus with dimensionP_{P∈P |}P _|=_{| P |}+P_P∈P(_|P_|−1) =_{| E |}<(2n_{−1)× | P |.} Interestingly,h_∈GCN T P M∩GCENimpliesαPAc(f_A)≥P_B ⊂AαPBc(f_B)for all

(A, P)_∈E, as well as monotonicity and total positivity (i.e., total monotonicity) of coalitional gamesv_∈CN _implyv(A)≥P

B⊂AαB(v)for allA⊆N.

4

The Shapley value

A solution is any functionφ:GC_∅N _→ACN. In words, a solutionφassociates ann-dimensional payoffvector φ(h) = (φ_i(h))_{i∈N ∈ R}n _{(that is, an additive}

coalitional gameφ(h)_∈ACN) to eachh_∈GC_∅N. Attention is usually placed on solutions satisfying certain requisites; in particular, consider the following axioms:

linearity: φ(h+h0_{) =φ(h) +φ(h}0_{)andφ(ah) =aφ(h)for allh, h}0_∈GCN

∅

anda_∈R;

nonnegativity: φ_i(h)_≥φ_i(h0_{)≥0for all i∈N and for allh, h}0 _∈GCN W M

such thath_≥h0 _{(i.e.,h(A, P)≥h}0_{(A, P)for all(A, P)∈2}N_{× P);}

symmetry: φ(h) = πφ(πh) for all π _∈ ΠN_{, where Π}N _{denotes the set of}

permutations ofN and game πh is defined by πh(πA, πP) = h(A, P) for all

(A, P)_∈2N_{× P;}

dummy: if h(A, P) = h¡A_\i,_{i_}∪PN\i¢_{+h({i}, P}

0) for i ∈ N and all

(A, P) _∈ 2N_{× P such that i ∈ A, then φ}

i(h) = h({i}, P0), and i ∈ N is a

dummy player inh;

efficiency: P_i∈Nφ_i(h) =h(N, N).

Linearity, symmetry, efficiency and dummy are as usual. In particular, this latter axiom simply formalizes the idea of a dummy player in a global coalitional game. Also note thatπP =_{πB1, . . . , πBm}when P ={B1, . . . , Bm}as well

asπφ(πh) = (φ_πi(πh))_i∈N. On the other hand, nonnegativity is somehow new. It is clearly intended to substitute the traditional monotonicity axiom, in that restricting attention to monotone global coalitional games has the above men-tioned undesired implications. In particular, in the presence of nonnegativity ofh_∈GCN

∅ , weak monotonicity seems sufficient for requiring nonnegativity of

each player’s payoff, while the implicationh_≥h0 _{⇒φ(h)≥φ(h}0_)is theoreti-cally acceptable as well as technitheoreti-cally useful in a proof. Following Weber (1988), solutions satisfying linearity, dummy and nonnegativity may be defined to be probabilistic. Furthermore, probabilistic solutions that also satisfy symmetry may be defined to be semivalues, while adding efficiency leads to the class of values. In the sequel, letC+_⊆GCN

∅ be any cone.

Theorem 6 If φ:C+ _→Rn satisfies linearity, then there exist real constants

© b

pi

A,P |(A6=∅, P)∈2N× P

ª

for each i_∈N such that

φ_i(h) = X

(A6=∅,P)∈2N_×P

b

pi_A,Ph(A, P).

Proof. See Weber (1988), theorem 1, p. 104.

This enables to immediately concentrate on the dummy axiom.

Theorem 7 If φ : C+ _{→ R}n _{satisfies linearity and dummy, then there exist}

real constants©pi

A,P |(A, P)∈2N× P, A3i

ª

for each i_∈N such that

φ_i(h) = X

(A,P)∈2N_×P|A3i

pi_A,Phh(A, P)₋h³A_\i,_{i_}∪PN\i´i

Proof. Firstly note that linearity alone implies φi(gN,N) = pbiN,N, while

dummy and linearity together imply

φ_i¡gN\i,{i}∪{N\i}

¢

= 0 =p_bi_N,N+p_bi_N\i,N +_bpi_{N,{i}∪{N\i}}+p_bi_{N\i,{i}∪{N\i}}, that isp_bi_N,N =₋³p_bi_N\i,N+p_bi_{N,{i}∪{N\i}}+p_bi_{N\i,{i}∪{N\i}}´, for alli_∈N (where {i_}∪{N_\i_}=_{i, N_\i_}∈P(2) _=P(2)_{(N)). Secondly, consider any A ⊆N\i}

andP _∈P with_{i_}∈P, so that

0 = φ_i(gA,P) = X (B,Q)<(A,P) b pi_B,Q= = X (B,Q)<(A,P)|B3i ³ b pi_B,Q+p_bi_B\i,Q+p_bi_B,{i}∪QN\i+pbi_B\i,{i}∪QN\i ´ ,

and thus, by induction,p_bi

B,Q=− ³ b pi B\i,Q+pbiB,{i}∪QN\i+bpi_B\i,{i}∪QN\i ´ for all

(B, Q)_∈2N_{× P such thatB 3i. Also,{i}∈Qimpliespb}i

B,Q =−pbiB\i,Q, and in particularp_bi {i},P0 =−pb i ∅,P0, where pb i

∅,P0 may be arbitrarily set. Therefore,

simply setpi

{i},P0 =pb

i

{i},P0 for alli∈N as well as

pi_A,P =p_bi_A,P =₋³p_bi_A,{i}∪PN\i+pb i A\i,P +pb i A\i,{i}∪PN\i ´

for all(A, P)_∈2N_{× P such thatA3i. Eventually, noting that}

φ_i¡g_{i},P0 ¢

= 1 = X

(A,P)∈2N_×P|A3i

pi_A,P

for alli_∈N completes the proof.

Theorem 8 Ifφ:C+_→Rn satisfies linearity, dummy and nonnegativity, then

there exist real constants ©pi_{A,P |}(A, P)_∈2N_{× P}, A₃iªfor each i_∈N such

that

φ_i(h) = X

(A,P)∈2N_×P|A3i

pi_A,Phh(A, P)₋h³A_\i,_{i_}∪PN\i´i,

P

(A,P)∈2N_×P|A3ip_A,Pi = 1 andpi_A,P ≥0for all P∈P.

Proof. For any playeri_∈N, consider any (possibly void) coalitionA_⊆N_\i

and any partitionP _∈P, and let gamesg_bA,P andbbgA,P be defined by

b gA,P(B, Q) = ½ 1ifB_⊃A, Q_≥P 0otherwise ,bbgA,P(B, Q) = ½ 1ifB _⊃A, Q > P 0otherwise

for all(B₆=_∅, Q)_∈2N_{×P. Note that bothbg}

A,P andbbgA,P are weakly monotone

and that _bgA,P(B, Q)≥bbgA,P(B, Q) for all (B6=∅, Q)∈2N× P. Thus, when

{i_}∈P, nonnegativity requires φ_i(_bgA,P) = X Q≥P pi_A∪i,Q≥φ_i³bbg_A,P´= X Q>P pi_A∪i,Q≥0,

implyingpi_{A∪i,P ≥}0for allA_⊆N_\i, P _∈P,_{i_}∈P. Similarly,_{i_}∈/P yields

φ_i(g_bA,P) = X B⊃A,B3i Q≥P pi_B,Q≥φ_i³bbg_A,P´= X B⊃A,B3i Q>P pi_B,Q≥0, so thatpi

B,P ≥0for allB⊆N, B3i, P ∈P,{i}∈/P as wanted.

As already mentioned, these last three results enable to characterize the class of probabilistic solutions. In fact, such solutions may actually be defined as those satisfying linearity, dummy and nonnegativity (see Weber (1988), theorems 4 and 5), and clearly identify situations where i’s payoff is the expectation of random variable©h(A, P)₋h¡A_\i,_{i_}∪PN\i¢_{|(A, P)∈2}N_{× P, A3i}ª_with

respect toi’s subjective probability©pi_{A,P |}(A, P)_∈2N_{× P}, A₃iª, alli_∈N. Attention is now focused on the symmetry and efficiency axioms. In par-ticular, the former characterizes semivalues by requiring thepi

A,P’s to depend

solely on A’s, P’s, and P’s blocks’ cardinalities (i.e., _| A _|,_| P _| and _| Bj |

for 1_≤ j _≤| P _|), while the latter characterizes values by means of an addi-tional normalization condition. In particular, this latter results to be very much similar to the normalization condition found by Gilboa and Lehrer (1991a) as characterizing their Shapley value of global games.

Theorem 9 Any probabilistic solution satisfying symmetry has constantspi_A,P

that depend solely on_|A_|=a,_|P _|=m,_{|B1|, . . . ,|Bm|}={bj}_1≤j≤m, i.e.,

piA,P =

⎧ ⎨ ⎩

p{i}∈

(a,m,{bj}_1≤j≤m) for allA⊆N, A3i andP ∈P such that {i}∈P,

p{i}∈/

(a,m,{bj}1≤j≤m)

for allA_⊆N, A₃i andP _∈P such that _{i_}∈/P.

Furthermore, p{i}∈ (a,m,{bj}_1≤j≤m)=p {j}∈ (a,m,{bj}_1≤j≤m)=p ∈ (a,m,{bj}_1≤j≤m) p{i}∈/ (a,m,{bj}1≤j≤m) =p{j}∈/ (a,m,{bj}1≤j≤m) =p∈/ (a,m,{bj}1≤j≤m) for all i, j_∈N andA_⊆N, P _∈P.

Proof. LetA_⊆N_\i, P _∈P and consider a permutationπ_∈ΠN _{such that}

πi=i. Also letB=πA, Q=πP and games_bg,bbgbe defined as before. Then

with the last equality following from symmetry. Firstly consider the case where

iconstitutes a block on her own inP (and thus inQas well), i.e.,P _{3 {}i_}∈Q, yielding φi ³ bbgB,Q ´ = X Q0_>Q piB∪i,Q0 =φi ³ bbgA,P ´ = X P0_>P piA∪i,P0.

Nevertheless, for game_bgsymmetry imposes

φi(gbB,Q) = X Q0_≥Q piB∪i,Q0 =φi(gbA,P) = X P0_≥P piA∪i,P0, so that φ_i(_bgB,Q)−φi ³ bb g_B,Q´=φ_i(_bgA,P)−φi ³ bb g_A,P´ impliespi

B∪i,Q = piA∪i,P for allA, B ⊆ N\i such that | A |=| B | and for all

P, Q_∈P such thatQ=πP,_{i_}∈P, Q. On the other hand, if_{i_}∈/P, Q, then

φ_i³bbg_B,Q´ = X B0⊃B,B03i Q0>Q pi_B0_,Q0 =φ_i ³ bb g_A,P´= X A0⊃A,A03i P0>P pi_A0_,P0 φ_i(_bgB,Q) = X B0_⊃B,B0_3i Q0≥Q pi_B0_,Q0 =φ_i(bgA,P) = X A0_⊃A,A0_3i P0≥P pi_A0_,P0, so that φ_i(_bgB,Q)−φi ³ bb g_B,Q´ = φ_i(g_bA,P)−φi ³ bb g_A,P´ implies pi B0_,Q = pi_A0_,P

for allA0, B0 _⊆N, A0, B0 ₃i such that_| A0 _|=_|B0 _|and for all P, Q_∈P such that _{i_}∈/ P, Q. These results show that both when _{i_}∈ P as well as when {i_}∈/ P thepi_A,P’s depend solely on A’s, P’s and P’s blocks’ cardinalities for all i _∈ N. Therefore, the next step consist in showing that pj_A∪j,P = pi

A∪i,P

for all i, j _∈ N whenever (A, P) _∈ 2N _{× P is such that i, j /∈ A and either}

{i_},_{j_}∈P, or else_{i_},_{j_}∈/P. In words,iand j must be considered when each of them (separately) joins some (possibly void) coalition A _⊆ N_{\ {}i, j_}, and for any partition P where either they both constitute a 1-element block, or else each of them belongs to some larger block (that is, _{i, j_{} ⊆} B _∈ P

or i _∈ B _∈ P ₃ B0 ₃ j, with _| B _|,_| B0 _|≥ 2). Let π _∈ ΠN be such that

πi = j, πj = i, πk = k for k _∈ N_{\ {}i, j_}. Also let A _⊆ N_{\ {}i, j_}, implying

πA=A, andfirstly consider the case where_{i_},_{j_}∈P _∈P. Then symmetry requires

φ_πi³πbbg_A,P´=φ_πi³bbg_πA,πP´=φ_j³bbg_A,P´=φ_i³bbg_A,P´=φ_πj³πbbg_A,P´, where the third equality follows from symmetry. Therefore,

φ_j³bbg_A,P´ = X P0_>P pj_A∪j,P0 =φi ³ bbg_A,P´= X P0_>P pi_A∪i,P0 φj(bgA,P) = X P0_≥P pj_A∪j,P0 =φi(bgA,P) = X P0_≥P pi_A∪i,P0.

Thus,φ_j(_bgA,P)−φj ³ bb g_A,P´=φ_i(_bgA,P)−φi ³ bbg_A,P´impliespj_A∪j,P =pi A∪i,P for

allA_⊆N_{\ {}i, j_}, P _∈P,_{i_},_{j_}∈P. Now consider that when_{i_},_{j_}∈/ P

symmetry imposes φ_j³bbg_A,P´ = X A0⊃A,A03j P0_>P pj_A0_,P0 =φi ³ bb g_A,P´= X A0⊃A,A03i P0_>P pi_A0_,P0 φ_j(_bgA,P) = X A0_⊃A,A0_3j P0≥P pj_A0,P0 =φi(bgA,P) = X A0_⊃A,A0_3i P0≥P pi_A0_,P0, so thatpj_A∪j,P =pi

A∪i,P for allA⊆N\ {i, j}, P ∈P,{i},{j}∈/P. Clearly, the

coarsest partition such that_{i_},_{j_}∈/P isP =N _∈P, thus let

piN,N =pjN,N =φi(gN,N) =φj(gN,N) =p∈₍/_n,1,{1})

for alli, j_∈N. On the other hand, thefinest partition such that_{i_},_{j_}∈P is

P=P0. Thus, for eachi∈N letpi_{i},P0 =φi ¡ g_{i},P0 ¢ −P(A,P)∈2N ×P (A,P)Â({i},P0) pi A,P = = 1₋ ⎛ ⎜ ⎜ ⎝ X (A,P)Â({i},P0) {i}∈P p∈ (a,m,{bj}1≤j≤n) + X (A,P)Â({i},P0) {i}∈/P p∈/ (a,m,{bj}1≤j≤n) ⎞ ⎟ ⎟ ⎠. By symmetry,φi ¡ g_{i},P0 ¢ =φj ¡ g_{j},P0 ¢

for alli, j_∈N, so that the proof gets complete simply by settingpi

{i},P0=p ∈ ⎛ ⎜ ⎝1,n, ⎧ ⎪ ⎨ ⎪ ⎩1| {z }, . . . ,1 n ⎫ ⎪ ⎬ ⎪ ⎭ ⎞ ⎟ ⎠ for alli_∈N.

Theorem 10 Any probabilistic solution satisfying efficiency has constantspi A,P satisfying X i∈A pi_A,P = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1if A=N, P =N 0if ½ A=N, P ₆=N or A₆=N,_{j_∈Ac_{| {j}∈P}=∅} P j∈Ac {j}∈P P Q≥P {j}∪QN\j_=P pj_A∪j,Q ifA₆=N,_{j _∈Ac_{| {j}∈P} 6=∅} for all(A₆=_∅, P)_∈2N_{× P.}

Proof. Efficiency implies1 =P_i∈Nφ_i(gN,N) =Pi∈NpiN,N. Furthermore,

φi(gN,P) =piN,N+

P

N6=Q≥PpN,Qi for alli∈N andP 3P < N, so that

1 =X i∈N φ_i(gN,P) = X i∈N pi_N,N+X i∈N X N6=Q≥P pi_N,Q= 1 +X i∈N X N6=Q≥P pi_N,Q,

and thus P_i∈NP_N6=Q≥Pp_N,Qi = 0 by efficiency, and P_i∈Npi_N,P = 0 for all

P₆=N by induction. Next consider that

X i∈N φ_i(h) = X i∈N X (A,P)∈2N×P A3i pi_A,Phh(A, P)₋h³A_\i,_{i_}∪PN\i´i= = X (A,P)∈2N_×P h(A, P) ⎡ ⎢ ⎢ ⎣ X i∈A pi A,P − X j∈Ac {j}∈P X Q≥P {j}∪QN\j_=P pj_A∪j,Q ⎤ ⎥ ⎥ ⎦

for allh, so that any probabilistic solution satisfying the conditions of the the-orem is also efficient. Now consider any(A, P)_∈2N_{× P such thatA}c_6=∅and

{j_∈Ac_{| {j}∈P}=∅, so that(g} A,P−gbA,P) (N, N) = 0 = X i∈A X (B,Q)<(A,P) pi_B,Q+ X j∈Ac X (B,Q)Â(A,P) B3j pj_B,Q+ −X i∈A X (B,Q)Â(A,P) B⊃A pi_B,Q+ X j∈Ac X (B,Q)Â(A,P) B3j pj_B,Q, and thereforeP_i∈AP_Q≥Ppi A,Q= 0, implying P

i∈ApiA,P = 0for all such pairs

(A, P)by induction. On the other hand, if Ac _{6=∅6={j∈A}c_{| {j}∈P}, then}

defineeeg_A,P byeeg_A,P(B, Q) = 1 ifB_⊃A, Q_≥P orB _⊇A, Q > P or both, and

0otherwise, so thatP_i∈Nφ_i³eeg_A,P´= X i∈A ⎛ ⎜ ⎜ ⎝ X (B,Q)Â(A,P) B⊇A,Q>P pi_B,Q+ X (B,Q)Â(A,P) B⊃A,Q≥P pj_B,Q− X (B,Q)Â(A,P) B⊃A,Q>P pj_B,Q ⎞ ⎟ ⎟ ⎠+ + X j∈Ac {j}∈/P X (B,Q)Â(A,P) B3j pj_B,Q+ X j∈Ac {j}∈P X Q≥P {j}∪QN\j=P pj_A∪j,Q+, and therefore X i∈N φ_i³gA,P −eegA,P ´ = 0 =X i∈A pi_{A,P −} X j∈Ac {j}∈P X Q≥P {j}∪QN\j_=P pj_A∪j,Q

for all such pairs(A, P)as wanted.

constantspi_A,P satisfying pi_A,P = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 n =p / ∈ (n,1,{1}) if (A, P) = (N, N) 0if ½ A=N, P ₆=N or A₆=N, P ₆=_{A_}∪P₀Ac 1 n(n−1 n−a) =p∈⎛/ ⎜ ⎜ ⎝a,n−a+1, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1, . . . ,1 | {z } n−a ,a ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎠ ifA₆=N, P =_{A_}∪PAc 0 for all(A₆=_∅, P)_∈2N_{× P, i∈A.}

Proof. It is clear that efficiency and symmetry together imply

1 n =p / ∈ (n,1,{1})=piN,N =φi(gN,N) and 0 =p∈/or∈ (n,m,{bj}1≤j≤m) =pi N,P ifP 6=N for alli_∈N.

Thus, letj _∈N, noting thatP_i∈N\jpi_{N\j,{j}∪{N\j}} =pj_{N,{j}∪{N\j}}+pj_N,N by efficiency. Adding symmetry leads top∈₍/_{n−1,2,{1,n−1})}= _n−1₁¡0 + 1_n¢= 1

n(n−1 1 )

. On the other hand, efficiency alone clearly implies pi_N\j,P = 0 for all P _{∈ P}

such that_{j_}∈/P, for alli_∈N_\j. Eventually, note that

X i∈N\j pi_N\j,{j}∪PN\j =p j N,{j}∪PN\j + X B∈PN\j pj_N, {B∪j}∪PN\(B∪j) = 0

for all_{N_\j_{} 6}=PN\j _{∈P(N\j). Therefore,p}i

N\j,P = 0for allP 6={j}∪{N\j},

for alli_∈N_\j. In order to use induction, assume the theorem holds true for all

A0_{6=Nsuch that|A}0 _|≥a0_{, and consider anyA6=N such that|A|=a=a}0_−1, with P _{∈ P}. Then P_i∈Api A,P = P j∈Ac_|{j}∈P P Q≥P|{j}∪QN\j_=Pp j A∪j,Q by efficiency, butpj_A∪j,Q = 1 n( n−1 n−a−1) ifQ =_{A_∪j_}∪P₀Ac∪j and 0otherwise by assumption, thereforeP_i∈Api A,P = P j∈Acp j A∪j,{A∪j}∪P0Ac∪j ifP =_{A_}∪PAc 0

and0otherwise. Adding symmetry leads to

ap∈⎛/ ⎜ ⎜ ⎝a,n−a+1, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1, . . . ,1 | {z } n−a ,a ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎠ = (n₋a) 1 n¡_n−n−_a−1₁¢, and thusp∈⎛/ ⎜ ⎜ ⎝a,n−a+1, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1, . . . ,1 | {z } n−a ,a ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎠ =n−_aa 1 n( n−1 n−a−1) = 1 n(n−1 n−a) as wanted. LetφSh :GCN

as obtained above. It is clear thatφShi (h) = = X A⊆N A3i (a₋1)! (n₋a)! n! h h³A,_{A_}∪P₀Ac´₋h³A_\i,_{A_\i_}∪P₀Ac∪i´i = X A⊆N A3i (a₋1)! (n₋a)! n! ⎡ ⎢ ⎣ X (A\i,{A\i}∪PAc∪i 0 )≺(B,Q)4(A,{A}∪P0Ac) αB,Q(h) ⎤ ⎥ ⎦

for alli_∈N and allh_∈GC_∅N. It is also evident that

φSh_i (gA,P) =φShi µ g Dc A,P,{DcA,P}∪P DA,P 0 ¶ = ( ₁ |Dc A,P| ifi∈D c A,P 0otherwise

for all (A₆=_∅, P) _∈ 2N _{× P, where D}

A,P = {j∈Ac| {j}∈P} is the set of

dummy players ingA,P, whileDcA,P =N\DA,P.

Thus, as in Gilboa and Lehrer (1991a)8_{, the Shapley value of any global}

coalitional gameh_∈ GCN

∅ coincides with the Shapley value of the coalitional

game _evh ∈ CN defined by evh(A) = h

¡

A,_{A_}∪PAc

0

¢

for all _∅6=A _⊂ N and

e

vh(N) =h(N, N). As shown above, this means that φSh is determined by a

proper subset of©αA,P(h)|(A, P)∈2N× Pª. Therefore, such a solution seems

inappropriate when applied to global coalitional games, in that it (arbitrarily) reduces these latter to coalitional games, making it somehow useless to formalize the cooperative situation in terms of the partition lattice.

5

The core

The core_CCN(v)of any coalitional gamev∈CNis the set of additive coalitional gamesφ= (φi)i∈N∈ACN ⊆Rnsuch thatφ(A) =

P

i∈Aφi≥v(A)forA⊆N,

with equality forA=N, that is _CCN(v) =©φ∈ACN |v≤φ, v(N) =φ(N)ª (thus _CCN : CN ³ ACN, with ³ denoting a correspondence). In terms of bargaining, this means that each coalition A will accept to cooperate so to form the grand coalition N =A_∪Ac _{only if its (coalitional) payoff is greater}

than ‘the best payoff it can achieve without help from other players’ (Shap-ley (1971), p. 13; see also footnote 3 p. 11), that is v(A). Nevertheless, such a definition does not straightforwardly apply here. In fact, as observed by Gilboa and Lehrer (1991a), any partition of N involves all the n players, and thus defining what coalitions can achieve without help from other players becomes arbitrary. In general, defining the core of any global coalitional game

h amounts to define some lower-bound coalitional game vh such that vh(A)

represents the minimum coalitional payoff required by any coalition A _⊆ N

for joining any coalition B _⊆ Ac_{. This general problem was obviously dealt}