matemática aplicada ii, escuela superior de ingenierosmbilbao/pdffiles/bicogames.pdf ·...

A SURVEY OF BICOOPERATIVE GAMES

J.M. Bilbao∗, J.R. Fernández, N. Jiménez, J.J. López

Matemática Aplicada II, Escuela Superior de IngenierosCamino de los Descubrimientos s/n, 41092 Sevilla, Spain.

1. IntroductionThe theory of cooperative games studies situations where a group of people/playersare associated to obtain a profit as a result of their cooperation. Thus, a cooperativegame is defined as a pair (N, v) , where N is a finite set of players and v : 2N → R is afunction verifying that v (∅) = 0. For each S ∈ 2N , the worth v (S) can be interpretedas the maximal gain or minimal cost that the players which form coalition S canachieve themselves against the best offensive threat by the complementary coalitionN \ S. Hence, we can say that a cooperative game has orthogonal coalitions (seeMyerson [11]). Classical market games for economies with private goods are examplesof cooperative games.

Games with non-orthogonal coalitions are games in which the worth of coalitionS are not independent of the actions of coalition N \ S. Clearly, social situationsinvolving externalities and public goods are such cases. For instance, we consider agroup of agents with a common good which is causing them expenses or costs. In aexternal or internal way, a modification (sale, buying, etc.) of this good is proposedto them. This action will suppose a greater profit to them in case they all agreewith the change proposed about the actual situation of the good. Moreover, eventhough the patrimonial good can be divisible, we suppose that the greatest value ofthe selling operation is reached if we consider all the common good.

A possibility of modeling these situations may be the following. We consider pairs(S, T ), with S, T ⊆ N and S∩T = ∅. Thus, (S, T ) is a partition of N in three groups.Players in S are defenders of modifying the status quo and they want to accept aproposal; players in T do not agree with modifying the situation and they will takeaction against any change. Finally, the members of N \ (S ∪ T ) are not convinced ofthe profits derived from the proposal and they vote abstention.

Thus, in our model we consider the set of all ordered pairs of disjoint coalitions3N = {(S, T ) : S, T ⊆ N, S ∩ T = ∅} , and define a function b : 3N → R. For each(S, T ) ∈ 3N , the worth b (S, T ) can be interpreted as the maximal gain (wheneverb (S, T ) > 0) or minimal loss (whenever b (S, T ) < 0) that the players of the coalitionS can achieve when they decide to play together against the players of T and theplayers of N \(S ∪ T ) not taking part. This leads us in a natural way into the conceptof bicooperative game introduced by Bilbao [1].

Definition 1. A bicooperative game is a pair (N, b) with N a finite set and b is afunction b : 3N → R with b (∅, ∅) = 0.

∗E-mail: [email protected]

1

A SURVEY OF BICOOPERATIVE GAMES 2

Similarly to the cooperative case in which each coalition S ∈ 2N can be identifiedwith a {0, 1}-vector 1S, each pair (S, T ) ∈ 3N can be identified with the {−1, 0, 1}-vector 1(S,T ) defined, for all i ∈ N, by

1(S,T ) (i) =

⎧⎨⎩1 if i ∈ S,−1 if i ∈ T,0 otherwise.

An especial kind of bicooperative games has been studied by Felsenthal and Ma-chover [5] who consider ternary voting games . This concept is a generalization ofvoting games which recognizes abstention as an option alongside yes and no votes.These games are given by mappings u : 3N → {−1, 1} satisfying the following threeconditions: u (N, ∅) = 1, u (∅, N) = −1, and 1(S,T ) (i) ≤ 1(S0,T 0) (i) for all i ∈ N,implies u (S, T ) ≤ u (S0, T 0) . A negative outcome, −1, is interpreted as defeat and apositive outcome, 1, as passage of a bill.

In Chua and Huang [3] the Shapley-Shubik index for ternary voting games isconsidered. More recently, several works by Freixas [6, 7] and Freixas and Zwicker[8] have been devoted to the study of voting systems with several ordered levelsof approval in the input and in the output. In their model, the abstention is alevel of input approval intermediate between yes and no votes. A new approach tobicooperative games is presented by Grabisch and Lange [10] by using the productof finite distributive lattices. They consider a finite set of players N = {1, . . . , n}and the product of lattices (L1,≤1)× · · · × (Ln,≤n) such that every Li = {−1, 0.1} ,where 1 means voting or playing in favor, −1 means voting or playing against, and 0means abstention.

A one-point solution concept for cooperative games is a function which assignsto every cooperative game a n-dimensional real vector which represents a payoffdistribution over the players. The study of solution concepts is central in cooperativegame theory. The most important solution concept is the Shapley value as proposedby Shapley [13]. The Shapley value assumes that every player is equally likely tojoin to any coalition of the same size and all coalitions with the same size are equallylikely. The Shapley value Φ (v) ∈ Rn of game v is a weighted average of the marginalcontributions of the players and for player i ∈ N, it is given by

Φi (v) =X

S⊆N\{i}

s! (n− 1− s)!

n![v (S ∪ {i})− v (S)] ,

where s = |S| and n = |N | .Another form to introduce the Shapley value is based in the marginal worth

vectors and corresponds to the following interpretation. Suppose the players enter aroom one by one in a randomly chosen order. Each player gets the amount that hecontributes to the coalition S already formed into the room when the player i entersthe room; that is, i gets v (S ∪ {i}) − v (S) . The Shapley value Φ (v) distributes toeach player i ∈ N , the expected amount that he gets by this procedure, that is,

Φi (v) =1

n!

Xπ∈Πn

£v¡πi ∪ {i}

¢− v

¡πi¢¤.


where Πn is the set of all permutations of N and πi is the set of the predecessors ofplayer i in the order π.

A solution concept for cooperative games is a function which assigns to everycooperative game (N, v) with |N | = n, a subset of n-dimensional real vectors whichrepresent the payoff distribution over the players. The core is one of the most studiedsolution concepts. The core of a cooperative game (N, v) consists of all payoff vectorswhich distribute the total savings v (N) among players and secure to every coalitionS ∈ 2N at least the amount it can obtain by operating on its own, that is,

C (N, v) =©x ∈ Rn : x (N) = v (N) and x (S) ≥ v (S) for all S ∈ 2N

ª,

where x (S) =P

i∈S xi and x (∅) = 0.Although it is considered a very natural solution concept, has the trouble that,

in many cases, it is empty. For the class of convex games [14], it can be affirmedthat it is a nonempty set; nevertheless, even though the core is not empty, it couldbe small for obtaining reasonable solutions to certain games. This leads to considerother solution concepts. In 1978, Weber [16] proposed as a solution concept for acooperative game, a set that contains the core, is always nonempty and easier tocompute. Its definition is based in the marginal worth vectors. Each permutation ofthe elements of N, π = (i1, i2, . . . , in), can be interpreted as a sequential process offormation of the grand coalition N. Beginning from the empty set, first the player i1is incorporated, next the player i2 and so until the incorporation of the player in giverise to the coalition N . In each one of these processes, the corresponding marginalworth vector, aπ (v) ∈ Rn, evaluates the marginal contribution of every player to thecoalition formed by his predecessors, that is,

aπi (v) = v¡πi ∪ {i}

¢− v

¡πi¢for all i ∈ N,

where πi is the set of the predecessors of player i in the order π. The Weber set ofgame v is the convex hull of all marginal worth vectors, that is,

W (N, v) = conv {aπ (v) : π ∈ Πn} .

Let us outline the contents of our work. In the next section, we study someproperties and characteristics of the distributive lattice 3N . The aim of the thirdsection is to introduce the Shapley value for a bicooperative game. We obtain anaxiomatization of the Shapley value in this context as well as a nice formula tocompute it. This value is the only one that satisfies our five axioms. Four of themare extensions of the classical axioms for the Shapley value: linearity, symmetry,dummy and efficiency. The fifth axiom is refereed to the structure of the family ofsigned coalitions. In the fourth section we define the above solutions concepts forbicooperative games and prove that the core is always contained in the Weber set. Inthe relation between the Weber set and the core, the bisupermodular games, which aredefined in the fifth section, play an important role. We see that the bisupermodulargames are the only ones for which their Weber set and the core coincide, establishinga characterization of these games. Throughout this chapter, we will write S ∪ i andS \ i instead of S ∪ {i} and S \ {i} respectively.


2. The lattice 3N

Let N = {1, . . . , n} be a finite set and let 3N = {(A,B) : A,B ⊆ N, A ∩B = ∅} .Grabisch and Labreuche [9] proposed a relation in 3N given by

(A,B) v (C,D)⇐⇒ A ⊆ C, B ⊇ D.

The set¡3N ,v

¢is a partially ordered set (or poset) with the following properties:

1. (∅, N) is the first element: (∅, N) v (A,B) for all (A,B) ∈ 3N .2. (N, ∅) is the last element: (A,B) v (N, ∅) for all (A,B) ∈ 3N .3. Every pair of elements of 3N has a join

(A,B) ∨ (C,D) = (A ∪ C,B ∩D) ,

and a meet(A,B) ∧ (C,D) = (A ∩ C,B ∪D) .

Moreover,¡3N ,v

¢is a finite distributive lattice. Two pairs (A,B) and (C,D) are

comparable if (A,B) v (C,D) or (C,D) v (A,B) ; otherwise, (A,B) and (C,D) areincomparable. A chain of 3N is an induced subposet of 3N in which any two elementsare comparable. In

¡3N ,v

¢, all maximal chains have the same number of elements

and this number is 2n+ 1. Thus, we can consider the rank function

ρ : 3N → {0, 1, . . . , 2n}

such that ρ [(∅, N)] = 0 and ρ [(S, T )] = ρ [(A,B)]+1 if (S, T ) covers (A,B) , that is, if(A,B) @ (S, T ) and there no exists (H,J) ∈ 3N such that (A,B) @ (H,J) @ (S, T ) .

For the distributive lattice 3N , let P denote the set of all nonzero ∨-irreducibleelements. Then P is the disjoint union C1 +C2 + · · ·+ Cn of the chains

Ci = {(∅, N \ i), (i,N \ i)}, 1 ≤ i ≤ n = |N |.

An order ideal of P is a subset I of P such that if x ∈ I and y ≤ x, then y ∈ I.The set of all order ideals of P , ordered by inclusion, is the distributive lattice J(P ),where the lattice operations ∨ and ∧ are just ordinary union and intersection. Thefundamental theorem for finite distributive lattices (see [15, Theorem 3.4.1]) statesthat the map ϕ : 3N → J(P ) given by (A,B) 7→ {(X,Y ) ∈ P : (X,Y ) v (A,B)} isan isomorphism (see Figure 1).

Example. Let N = {1, 2}. Then P = {(∅, {1}), (∅, {2}), ({2}, {1}), ({1}, {2})} is thedisjoint union of the chains (∅, {1}) @ ({2}, {1}) and (∅, {2}) @ ({1}, {2}). We willdenote a = (∅, {1}), b = ({2}, {1}), c = (∅, {2}), d = ({1}, {2})}, and hence

J(P ) = {∅, {a}, {c}, {a, c}, {a, b}, {c, d}, {a, b, c}, {a, c, d}, {a, b, c, d}}.


•

• •

• • •

• •

•

{a, b, c, d}

{a, b, c} {a, c, d}

{a, b}{a, c}

{c, d}

{a} {c}

∅

¡¡¡¡¡¡

¡¡¡¡¡¡

¡¡¡¡¡¡

@@

@@

@@

@@

@@

@@

@@

@@

@@•

• •

• • •

• •

•

({1, 2}, ∅)

({2}, ∅) ({1}, ∅)

({2}, {1})(∅, ∅)

({1}, {2})

(∅, {1}) (∅, {2})

(∅, {1, 2})

¡¡¡¡¡¡

¡¡¡¡¡¡

¡¡¡¡¡¡

@@

@@

@@

@@

@@

@@

@@

@@

@@

Fig. 1

In the following, we will denote by c¡3N¢the number of maximal chains in 3N and

by c ([(A,B) , (C,D)]) the number of maximal chains in the sublattice [(A,B) , (C,D)] .

Proposition 1. The number of maximal chains of 3N is (2n)!/2n, where n = |N |.

Proof. The number of maximal chains of 3N is equal to the number of maximalchains of J(P ) and this number is also equal to the number of extensions e(P ) of Pto a total order (see Stanley [15, Section 3.5]).

Since P = C1 + · · · + Cn, where the chain Ci satisfies |Ci| = 2 for 1 ≤ i ≤ n, wecan apply the enumeration of lattice paths method from Stanley [15, Example 3.5.4],and obtain

c¡3N¢= e(P ) =

µ2n

2, . . . , 2

¶=(2n)!

2n. ¤

Proposition 2. For all (A,B) ∈ 3N , the number of maximal chains of the sublattice[(∅, N) , (A,B)] is (n+ a− b)!/2a, where a = |A| and b = |B| .

Proof. Given the sublattice [(∅, N) , (A,B)] , we consider N \ B = {i1, . . . , in−b}and hence there are n− b elements (∅, N \ i) with i /∈ B (see Figure 2).

•

• • •

•HHHH

HHHH

HHHH

HHH

@@

@@

@@

@@

©©©©

©©©©

©©©©

©©©

(i1, N \ i1), i1 ∈ A

(∅, N \ i1) (∅, N \ i2) (∅, N \ in−b)

(∅, N)

Fig. 2


Since A ⊆ N \B, then a ≤ n− b and thus, the set of the irreducible elements ofthe sublattice can be written as

P[(∅,N),(A,B)] = C1 + · · ·+Ca + Ca+1 + · · ·+ Ca+(n−b−a)

where for all ij ∈ A, 1 ≤ j ≤ a and ia+k /∈ A ∪B, 1 ≤ k ≤ n− b− a, we obtain

Cj = {(∅, N \ ij) , (ij ,N \ ij)} ,Ca+k = {(∅, N \ ia+k)} .

That is, there are a chains such that |Cj | = 2 and there are n− b− a chains suchthat |Ca+k| = 1. Since

|C1|+ · · ·+ |Ca|+ |Ca+1|+ · · ·+¯̄Ca+(n−b−a)

¯̄= 2a+ (n− b− a),

we can apply the enumeration of lattice paths method from Stanley [15, Section 3.5]and we obtain

c ([(∅, N) , (A,B)]) =µ2a+ (n− b− a)

2, . . . , 2, 1, . . . , 1

¶=(n+ a− b)!

2a. ¤

Proposition 3. Let (A,B) , (C,D) ∈ 3N with (A,B) v (C,D) . The number ofmaximal chains of the sublattice [(A,B) , (C,D)] is equal to the number of maximalchains of the sublattice [(D,C) , (B,A)] .

Proof. First of all, note that if (A,B) v (C,D) , then A ⊆ C, B ⊇ D and hence(D,C) v (B,A) . Therefore, [(D,C) , (B,A)] is a sublattice of 3N .

Let ϕ :¡3N ,v

¢→¡3N ,v

¢be the map defined by ϕ (A,B) = (B,A) . This map

is one to one since

ϕ (A,B) = ϕ (C,D)⇐⇒ (B,A) = (D,C)⇐⇒ B = D, A = C ⇐⇒ (A,B) = (C,D) .

Clearly, if (A,B) @ (A1, B1) @ · · · @ (Ak, Bk) @ (C,D) is a maximal chain in thesublattice [(A,B) , (C,D)] then

(D,C) @ (Bk, Ak) @ · · · @ (B1, A1) @ (B,A)

is a maximal chain in the sublattice [(D,C) , (B,A)] . Finally, it follows that

(X,Y ) ∈ [(A,B) , (C,D)]⇐⇒ (Y,X) ∈ [(D,C) , (B,A)] . ¤

3. The Shapley value for bicooperative gamesWe denote by BGN the real vector space of all bicooperative games on N, that is

BGN =©b : 3N → R, b (∅, ∅) = 0

ª.

We consider the identity games©δ(S,T ) : (S, T ) ∈ 3N , (S, T ) 6= (∅, ∅)

ª, the superior

unanimity games©u(S,T ) : (S, T ) ∈ 3N , (S, T ) 6= (∅, ∅)

ªand the inferior unanimity

gamesnu(S,T ) : (S, T ) ∈ 3N , (S, T ) 6= (∅, ∅)

o, which are defined, for any (S, T ) ∈ 3N

such that (S, T ) 6= (∅, ∅) as follows.


The identity game δ(S,T ) : 3N → R is defined by

δ(S,T ) (A,B) =

½1 if (A,B) = (S, T ) ,0 otherwise.

The superior unanimity game u(S,T ) : 3N → R is given by

u(S,T ) (A,B) =

½1 if (S, T ) v (A,B) , (A,B) 6= (∅, ∅) ,0 otherwise.

The inferior unanimity game u(S,T ) : 3N → R is defined by

u(S,T ) (A,B) =

½−1 if (A,B) v (S, T ) , (A,B) 6= (∅, ∅) ,0 otherwise.

It is easy to prove (see [2]) that all the above collections are bases of BGN .A value on BGN is a function Φ : BGN → Rn, which associates to each bicoopera-

tive game b a vector (Φ1 (b) , . . . ,Φn (b)) which represents the value that every playerhas in the game b. In order to define a reasonable value for a bicooperative game andfollowing the same issue and interpretation of the Shapley value in the cooperativecase, we consider that a player i estimates his participation in game b, evaluatinghis marginal contributions b(S ∪ i, T ) − b(S, T ) in those signed coalitions (S ∪ i, T )that are formed from others (S, T ) when i is incorporated to S and his marginalcontributions b(S, T ) − b(S, T ∪ i) in those (S, T ) that are formed when i leaves thecoalition T ∪ i.

Thus, a value for player i can be written as

Φi(b) =X

(S,T )∈3N\i

hpi(S,T ) (b(S ∪ i, T )− b(S, T )) + pi

(S,T )(b(S, T )− b (S, T ∪ i))

i,

where for every (S, T ), the coefficient pi(S,T ) can be interpreted as the subjective

probability that the player i has of joining the coalition S and pi(S,T )

as the subjective

probability that the player i has of leaving the coalition T ∪i. Thus, Φi (b) is the valuethat the player i can expect in the game b.

Figure 3 shows the different sequential orders corresponding to the different chainsfrom (∅, N) to (N, ∅) which contain (S, T ) and (S ∪ i, T ) and all chains that containthe signed coalitions (S, T ∪ i) and (S, T ) .


•

• • ... • •

•

•

• •

...

• •

•

(N, ∅)

(S ∪ i, T )pi(S,T )

(S, T )

(∅,N)

¡¡

¡¡

¡¡

¢¢¢¢¢¢

AAAAAA

@@@@@@

HHHH

HH

@@

@

¡¡¡

©©©©

©©

©©©©©©

¡¡

¡

@@@

HHHHHH

@@

@@

@@

AAAAAA

¢¢¢¢¢¢

¡¡¡¡¡¡

•

• • ... • •

•

•

• •

...

• •

•

(N, ∅)

(S, T )pi(S,T )

(S, T ∪ i)

(∅, N)

¡¡

¡¡

¡¡

¢¢¢¢¢¢

AAAAAA

@@@@@@

HHHH

HH

@@

@

¡¡¡

©©©©

©©

©©©©©©

¡¡

¡

@@@

HHHHHH

@@

@@

@@

AAAAAA

¢¢¢¢¢¢

¡¡¡¡¡¡

Fig. 3

If we assume that all sequential orders or chains have the same probability, wecan deduce formulas for these probabilities pi(S,T ) and pi

(S,T )in terms of the number

of chains which contain to these coalitions. Applying Propositions 2 and 3, we obtain

pi(S,T ) =c ([(∅,N) , (S, T )]) c ([(S ∪ i, T ) , (N, ∅)])

c (3N )

=

(n+ s− t)!

2s· (n+ t− s− 1)!

2t

(2n)!

2n

=(n+ s− t)! (n+ t− s− 1)!

(2n)!2n−s−t,

pi(S,T )

=c ([(∅, N) , (S, T ∪ i)]) c ([(S, T ) (N, ∅)])

c (3N )

=

(n+ t− s)!

2t· (n+ s− t− 1)!

2s

(2n)!

2n

=(n+ t− s)! (n+ s− t− 1)!

(2n)!2n−s−t.

Taking into account that pi(S,T ) and pi(S,T )

are independent of player i, and only

depend of s = |S| and t = |T | , we can establish the following definition.Definition 2. The Shapley value for the bicooperative game b ∈ BGN is given, foreach i ∈ N, by

Φi(b) =X

(S,T )∈3N\i

hps,t (b(S ∪ i, T )− b(S, T )) + p

s,t(b(S, T )− b (S, T ∪ i))

i,


where, for all (S, T ) ∈ 3N\i,

ps,t =(n+ s− t)! (n+ t− s− 1)!

(2n)!2n−s−t

and

ps,t=(n+ t− s)! (n+ s− t− 1)!

(2n)!2n−s−t.

With the aim to characterize the Shapley value for bicooperative games, we con-sider a set of reasonable axioms and we prove that the Shapley value is the uniquevalue on BGN which satisfies these axioms.

Linearity axiom. For all α, β ∈ R, and b, w ∈ BGN ,

Φi(αb+ βw) = αΦi(b) + βΦi(w).

We now introduce the dummy axiom, understanding that a player is a dummyplayer when his contributions to signed coalitions (S ∪ i, T ) formed with his incorpo-ration to S and his contributions to signed coalitions (S, T ) formed with his desertionof T ∪ i coincide exactly with his individual contributions, that is, a player i ∈ N isa dummy in b ∈ BGN if, for every (S, T ) ∈ 3N\i, it holds

b(S ∪ i, T )− b(S, T )) = b ({i} , ∅) ,b(S, T )− b (S, T ∪ i) = −b (∅, {i}) .

Note that if i ∈ N is a dummy in b ∈ BGN then, for all (S, T ) ∈ 3N\i,

b(S ∪ i, T )− b (S, T ∪ i) = b({i} , ∅)− b (∅, {i}) .

Since a dummy player i in a game b has no meaningful strategic role in the game,the value that this player should expect in the game b must exactly be the sum upof his contributions.

Dummy axiom. If player i ∈ N is dummy in b ∈ BGN , then

Φi(b) = b ({i} , ∅)− b (∅, {i}) .

In the similar way to the cooperative case, for the comparison of roles in a game tobe meaningful, the evaluation of a particular position should depend on the structureof the game but not on the labels of the players.

Symmetry axiom. For all b ∈ BGN and for any permutation π over N, it holds thatΦπi(πb) = Φi(b) for all i ∈ N , where πb (πS, πT ) = b (S, T ) and πS = {πi : i ∈ S} .

In a cooperative game, it is assumed that all players decide to cooperate amongthem and form the grand coalition N. This leads to the problem of distributingthe amount v (N) among them. Taking into account different situations that canbe modelled by a bicooperative game b, we suppose that the amount b(N, ∅) is themaximal gain and b (∅, N) is the minimal loss obtained by the players when theydecide full cooperation. Then the maximal global gain is given by b(N, ∅)− b (∅, N) .From this perspective, the value Φ must satisfy the following axiom.


Efficiency axiom. For every b ∈ BGN , it holdsXi∈NΦi(b) = b(N, ∅)− b (∅, N) .

It is easy to check that our Shapley value for bicooperative games verifies the aboveaxioms. But this value is not the unique value which satisfies these four axioms. Forinstance, the value Φ(b) defined, for b ∈ BGN and i ∈ N, by

Φi(b) =X

S⊆N\i

s! (n− s− 1)!n!

[b (S ∪ i,N \ (S ∪ i))− b (S,N \ S)] ,

also verifies these axioms. However, note that, for any bicooperative game b ∈ BGN ,this value is the Shapley value corresponding to the cooperative game (N, v) , wherev : 2N → R is defined by v (A) = b (A,N \A) if A 6= ∅, and v (∅) = 0. This valueis not satisfactory for any bicooperative game in the sense that only consider thecontributions to signed coalitions in which all players take part. Moreover, there isan infinity of different bicooperative games which give rise to the same cooperativegame.

For these reasons, if we want to obtain an axiomatic characterization of our Shap-ley value for bicooperative games, we need to introduce an additional axiom. Previ-ously, we show that a value on BGN that satisfy the above four axioms is given bythe expression

Φi(b) =X

(S,T )∈3N\i

hps,t (b(S ∪ i, T )− b(S, T )) + p

s,t(b(S, T )− b (S, T ∪ i))

i,

where ps,t and ps,t satisfy some conditions. We prove this result in several steps. Firstof all, we show that a value for player i satisfying the linearity and dummy axiomscan be expressed as a linear combination of his contributions.

Theorem 4. Let Φi be a value for player i ∈ N which satisfies linearity and dummyaxioms. Then, for every b ∈ BGN ,

Φi(b) =X

(S,T )∈3N\i

hpi(S,T ) (b(S ∪ i, T )− b(S, T )) + pi

(S,T )(b(S, T )− b (S, T ∪ i))

i,

whereX

(S,T )∈3N\ipi(S,T ) = 1, and

X(S,T )∈3N\i

pi(S,T )

= 1.

Proof. The set of identity games is a basis of BGN , and each game b ∈ BGN canbe written as

b =X

{(S,T )∈3N :(S,T )6=(∅,∅)}b(S, T )δ(S,T ).

By the linearity axiom,

Φi(b) =X

{(S,T )∈3N :(S,T )6=(∅,∅)}Φi(δ(S,T ))b(S, T ).


We denote by ai(S,T ) = Φi¡δ(S,T )

¢for all (S, T ) 6= (∅, ∅) and thus, the value Φi(b) is

given byX(S,T )∈3N

ai(S,T )b(S, T )

=X

(S,T )∈3N\iai(S,T )b(S, T ) +

X{(S,T )∈3N :i∈S}

ai(S,T )b(S, T ) +X

{(S,T )∈3N :i∈T}ai(S,T )b(S, T )

=X

{(S,T )∈3N\i:(S,T )6=(∅,∅)}ai(S,T )b(S, T ) +

X(S,T )∈3N\i

ai(S∪i,T )b(S ∪ i, T )

+X

(S,T )∈3N\iai(S,T∪i)b(S, T ∪ i)

=X

{(S,T )∈3N\i:(S,T )6=(∅,∅)}

³ai(S,T )b(S, T ) + ai(S∪i,T )b(S ∪ i, T ) + ai(S,T∪i)b(S, T ∪ i)

´+ai({i},∅)b({i} , ∅) + ai(∅,{i})b(∅, {i}).

Let us consider the games wi(A,B) : 3

N → R where, for each (A,B) ∈ 3N\i, thegame wi

(A,B) is defined by

wi(A,B) (S, T ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩wi(A,B) (S \ i, T ) if i ∈ S,

wi(A,B) (S, T \ i) if i ∈ T,

1 if i /∈ S ∪ T, (∅, ∅) 6= (S, T ) v (A,B) ,0 otherwise.

Clearly, player i is a dummy in wi(A,B) for each (A,B) ∈ 3N\i and hence Φi(wi

(A,B)) = 0

by the dummy axiom. If we apply the above equality to the game wi(A,B) we getX

{(S,T )∈3N\i:(∅,∅)6=(S,T )v(A,B)}

³ai(S,T ) + ai(S∪i,T ) + ai(S,T∪i)

´= 0.

We show, by induction on ρ [(S, T )] , the rank of the signed coalitions, that for all(S, T ) ∈ 3N\i, (S, T ) 6= (∅, ∅) , it holds that ai(S,T ) + ai(S∪i,T ) + ai(S,T∪i) = 0. Note that

the first element in¡3N\i,v

¢is (∅, N \ i) , and so ρ [(∅, N \ i)] = 0. HenceX

{(S,T )∈3N\i:(S,T )v(∅,N\i)}

³ai(S,T ) + ai(S∪i,T ) + ai(S,T∪i)

´= ai(∅,N\i)+a

i({i},N\i)+a

i(∅,N) = 0.

Now assume the property for (H,J) ∈ 3N\i with ρ [(H,J)] ≤ k− 1 and suppose that(S, T ) ∈ 3N\i has ρ [(S, T )] = k. Then

Φi(wi(S,T )) =

X{(H,J)∈3N\i:(∅,∅)6=(H,J)v(S,T )}

³ai(H,J) + ai(H∪i,J) + ai(H,J∪i)

´= ai(S,T ) + ai(S∪i,T ) + ai(S,T∪i)

+X

{(H,J)∈3N\i:(∅,∅)6=(H,J)@(S,T )}

³ai(H,J) + ai(H∪i,J) + ai(H,J∪i)

´= ai(S,T ) + ai(S∪i,T ) + ai(S,T∪i) = 0,


where the last but one equality follows from the induction hypothesis, and the lastone follows from the dummy axiom. Now for each (S, T ) ∈ 3N\i, define

pi(∅,∅) = ai({i},∅), pi(∅,∅) = −a

i(∅,{i}), pi(S,T ) = ai(S∪i,T ), pi

(S,T )= −ai(S,T∪i),

and we compute

Φi(b) =X

(S,T )∈3N\i

h³pi(S,T )

− pi(S,T )

´b(S, T ) + pi(S,T )b(S ∪ i, T )− pi

(S,T )b (S, T ∪ i)

i=

X(S,T )∈3N\i

hpi(S,T ) (b(S ∪ i, T )− b(S, T )) + pi

(S,T )(b(S, T )− b (S, T ∪ i))

i.

Finally, it is easy to check that player i is a dummy in the games u({i},N\i) andu(N\i,{i}), and henceX

(S,T )∈3N\ipi(S,T ) =

X(S,T )∈3N\i

ai(S∪i,T ) =X

{(S,T )∈3N :i∈S}ai(S,T )

=X

{(S,T )∈3N :i∈S}Φi¡δ(S,T )

¢= Φi

⎛⎝ X{(S,T )∈3N :i∈S}

δ(S,T )

⎞⎠= Φi

¡u({i},N\i)

¢= u({i},N\i) ({i} , ∅)− u({i},N\i) (∅, {i}) = 1.

X(S,T )∈3N\i

pi(S,T )

=X

(S,T )∈3N\i−ai(S,T∪i) =

X{(S,T )∈3N :i∈T}

−ai(S,T )

=X

{(S,T )∈3N :i∈T}−Φi

¡δ(S,T )

¢= Φi

⎛⎝ X{(S,T )∈3N :i∈T}

−δ(S,T )

⎞⎠= Φi

³u(N\i,{i})

´= u(N\i,{i}) ({i} , ∅)− u(N\i,{i}) (∅, {i}) = 1. ¤

Now, we show that if add the symmetry axiom to the linearity and dummy axioms,the coefficients pi(S,T ) and pi

(S,T )only depend of the cardinality of S and T.

Theorem 5. Let Φi be a value for player i ∈ N defined, for every game b ∈ BGN ,by

Φi(b) =X

(S,T )∈3N\i

hpi(S,T ) (b(S ∪ i, T )− b(S, T )) + pi

(S,T )(b(S, T )− b (S, T ∪ i))

i.

If Φi satisfies the symmetry axiom, then pi(S,T ) = ps,t and pi(S,T )

= ps,tfor all (S, T ) ∈

3N\i with s = |S| and t = |T | .

Proof. Let Φi be a value for player i given by

Φi(b) =X

(S,T )∈3N\i

hpi(S,T ) (b(S ∪ i, T )− b(S, T )) + pi

(S,T )(b(S, T )− b (S, T ∪ i))

i.


Let (S1, T1) and (S2, T2) be signed coalitions in 3N\i such that (S1, T1) 6= (∅, ∅) 6=(S2, T2) satisfying that |S1| = |S2| < n − 1 and |T1| = |T2| < n − 1. Consider apermutation π of N that takes πS1 = S2 and πT1 = T2 while leaving i fixed. Thenπδ(S1,T1) = δ(S2,T2) and

pi(S1,T1) = Φi(δ(S1∪i,T1)) = Φi(δ(S2∪i,T2)) = pi(S2,T2),

pi(S1,T1)

= −Φi(δ(S1,T1∪i)) = −Φi(δ(S2,T2∪i)) = pi(S2,T2)

,

where the second equality follows from the symmetry axiom.Now, let i, j ∈ N, i 6= j and let (S, T ) ∈ 3N\{i,j}. Let us consider the permutation

π of N that interchanges i and j while leaving the remaining players fixed. Thenπδ(S,T ) = δ(S,T ) and

pi(S,T ) = Φi(δ(S∪i,T )) = Φj(δ(S∪j,T )) = pj(S,T ),

pi(S,T )

= −Φi(δ(S,T∪i)) = −Φj(δ(S,T∪j)) = pj(S,T )

.

Moreover,

pi(N\i,∅) = Φi(δ(N,∅)) = Φj(δ(N,∅)) = pj(N\j,∅),

pi(∅,N\i) = −Φi(δ(∅,N)) = −Φj(δ(∅,N)) = pj

(∅,N\j).

Hence, for every (S, T ) ∈ 3N\i there exist ps,t and ps,tsuch that pi(S,T ) = ps,t and

pi(S,T )

= ps,tfor all i ∈ N. ¤

The following theorem characterizes the values Φ = (Φ1, . . . ,Φn) which satisfythe above axioms and are efficient.

Theorem 6. Let Φ = (Φ1, . . . ,Φn) be a value on BGN defined, for every game b andfor all i ∈ N , by

Φi(b) =X

(S,T )∈3N\i

hps,t (b(S ∪ i, T )− b(S, T )) + p

s,t(b(S, T )− b(S, T ∪ i))

i.

Then, the value Φ satisfies the efficiency axiom if and only if it is satisfied

pn−1,0 =1

n, p

0,n−1 =1

n,

and(n− s− t) ps,t + tp

s,t−1 = (n− s− t) ps,t+ sps−1,t

for all 0 ≤ s, t ≤ n− 1 and 0 < s+ t ≤ n− 1.


Proof. For every b ∈ BGN we have thatP

i∈N Φi(b) is equal toXi∈N

X(S,T )∈3N\i

hps,t (b(S ∪ i, T )− b(S, T )) + p

s,t(b(S, T )− b(S, T ∪ i))

i=

Xi∈N

X(S,T )∈3N\i

hps,tb(S ∪ i, T )− p

s,tb(S, T ∪ i) +

³−ps,t + p

s,t

´b (S, T )

i=

X(S,T )∈3N

b (S, T )hsps−1,t − tp

s,t−1 + (n− s− t)³−ps,t + p

s,t

í= b (N, ∅)npn−1,0 − b (∅, N)np

0,n−1

+X

(S,T )∈3N(S,T )/∈{(∅,∅),(∅,N),(N,∅)}

b (S, T )hsps−1,t − tp

s,t−1 + (n− s− t)³−ps,t + p

s,t

í.

If the coefficients satisfy the relations for the coefficients, then Φ satisfies theefficiency axiom.

Conversely, fix (S, T ) ∈ 3N , (S, T ) 6= (∅, ∅) , and applying the preceding equalityto the identity game δ(S,T ), we have

Xi∈NΦi(δ(S,T )) =

⎧⎪⎨⎪⎩npn−1,0 if (S, T ) = (N, ∅) ,−np

0,n−1 if (S, T ) = (∅, N) ,sps−1,t − tp

s,t−1 + (n− s− t)³ps,t− ps,t

ótherwise.

Thus, if Φ satisfies the efficiency axiom, the relations for the coefficients are true.¤

As we have already indicated, these four axioms are not sufficient to characterizethe Shapley value for bicooperative games. Now, we introduce an additional axiomand prove that our Shapley value is the unique value on BGN that verifies the fiveaxioms. This new axiom will take into account the structure of the set of the signedcoalitions.

First of all, note that the signed coalitions (S \ j, T ) and (S, T ∪ i) where j ∈ Sand i /∈ S ∪ T have the same rank

ρ [(S \ j, T )] = ρ [(S, T ∪ i)] = n+ s− t− 1.

However, the number of maximal chains in the sublattice [(∅, N) , (S \ j, T )] is not thesame that the number of maximal chains in [(∅, N) , (S, T ∪ i)] since, by Proposition2,

c ([(∅,N) , (S \ j, T )]) =(n+ s− 1− t)!

2s−1,

c ([(∅, N) , (S, T ∪ i)]) =(n+ s− t− 1)!

2s.

Hence, beginning from the signed coalition (∅, N) , the probability of formationof the signed coalition (S, T ) with the incorporation of one player j to (S \ j, T ) must


be distinct to the probability of formation (S, T ) with the desertion of one player iin (S, T ∪ i) .

In analogous form, if we consider (S, T \ k) with k ∈ T and (S ∪ i, T ) which havethe same rank, the number of maximal chains in [(S, T \ k) , (N, ∅)] is not equal tonumber of maximal chains in [(S ∪ i, T ) , (N, ∅)] . Therefore the probability of forma-tion of (N, ∅) beginning from (S, T \ k) when one player k leaves the coalition T mustbe distinct to the probability of formation of (N, ∅) when one player i form the signedcoalition (S ∪ i, T ) .

•

• •

•

• •

•

(N, ∅)

(S, T \ k), k ∈ T (S ∪ i, T )

(S, T )

(S \ j, T ), j ∈ S (S, T ∪ i), i /∈ S ∪ T

(∅, N)¡¡¡

¢¢¢

AAA

@@

@

©©©©

©©©©

©©©©

HHHH

HHHH

HHHH

@@

@

¡¡¡

¡¡¡

¢¢¢

@@

@

AAA

¡¡

¡

@@@

¢¢¢

AAA

AAA

¢¢¢

··········

··········

··········

··········

·····

·····

Fig. 4

Taking into account these considerations, the values that one player must obtainin the identity games must be proportional to the number of maximal chains in thecorresponding sublattices. It must be also considered that one value verifying theabove four axioms assigns a non-negative real number to one player i in the identitygame δ(S,T ) if this player belongs S and a non-positive real number if the player ibelongs T. From this point of view, our value must be satisfied the following axiom(see Figure 4).

Structural axiom. For every (S, T ) ∈ 3N\i, j ∈ S and k ∈ T, it holds

c ([(∅, N) , (S \ j, T )])c ([(∅, N) , (S, T ∪ i)]) = −

Φj(δ(S,T ))

Φi¡δ(S,T∪i)

¢ , c ([(S, T \ k) , (N, ∅)])c ([(S ∪ i, T ) , (N, ∅)]) = −

Φk(δ(S,T ))

Φi¡δ(S∪i,T )

¢ .


Theorem 7. Let Φ be a value on BGN . The value Φ is the Shapley value if andonly if Φ satisfies the efficiency axiom and each component satisfies linearity, dummy,symmetry and structural axioms.

Proof. If Φ is a value that satisfies linearity, dummy, symmetry and efficiency,then

Φi(b) =X

(S,T )∈3N\i

hps,t (b(S ∪ i, T )− b(S, T )) + p

s,t(b(S, T )− b (S, T ∪ i))

iand the coefficients ps,t and p

s,tsatisfy

pn−1,0 =1

n, p

0,n−1 =1

n,

and(n− s− t) ps,t + tp

s,t−1 = (n− s− t) ps,t+ sps−1,t. (1)

Taking into account that the value Φ verifies the structural axiom then

ps−1,t = 2ps,t (2)

ps,t−1 = 2ps,t (3)

We prove that these coefficients, verifying all above conditions, are determined inunique form. Indeed, consider a coalition (S, T ) with |S| = n− 1 and |T | = 0. If weapply the equation 1 to this coalition, we obtain

pn−1,0 = pn−1,0 + (n− 1) pn−2,0

and by 2, pn−2,0 = 2pn−1,0. Taking into account that pn−1,0 =1n and combining the

above equalities, we have that

1

n= (1 + 2 (n− 1)) p

n−1,0

and hence

pn−1,0 =

1

n (2n− 1) =1! (2n− 2)!2n−1 (2n)!

2n, pn−2,0 =2

n (2n− 1) =1! (2n− 2)!2n−2 (2n)!

2n.

In similar way, if we apply 1 and 2 to a signed coalition (S, T ) with |S| = n − 2and |T | = 0, we get

2pn−2,0 = 2pn−2,0 + (n− 2) pn−3,0,

pn−3,0 = 2pn−2,0,

and hence

pn−2,0 =

2! (2n− 3)!2n−2 (2n)!

2n, pn−3,0 =2! (2n− 3)!2n−3 (2n)!

2n.


If we assume that

ps+1,0

=(n− s− 1)! (n+ s)!

2s+1 (2n)!2n, ps,0 =

(n− s− 1)! (n+ s)!

2s (2n)!2n

then, for |S| = s and |T | = 0, applying 1 and 2,

(n− s) ps,0 = (n− s) ps,0+ sps−1,0,

ps−1,0 = 2ps,0,

and combining both expressions, we obtain, for 1 ≤ s ≤ n− 1,

ps,0=(n− s)! (n+ s− 1)!

2s (2n)!2n, ps−1,0 =

(n− s)! (n+ s− 1)!2s−1 (2n)!

2n.

If we apply the same reasoning with the equalities 1 and 3 beginning with acoalition (S, T ) with |S| = 0 and |T | = n− 1, we obtain, for 1 ≤ t ≤ n− 1,

p0,t =(n− t)! (n+ t− 1)!

2t (2n)!2n, p

0,t−1 =(n− t)! (n+ t− 1)!

2t−1 (2n)!2n.

If we now consider (S, T ) with |S| = s and |T | = 1, we apply 1 and 3,

(n− s− 1) ps,1 + ps,0= (n− s− 1) p

s,1+ sps−1,1,

ps,1 =1

2ps,0, ps−1,1 =

1

2ps−1,0,

and substitute the values already obtained, then

ps−1,1 =(n− s+ 1)! (n+ s− 2)!

2s (2n)!2n, p

s,1=(n− s+ 1)! (n+ s− 2)!

2s+1 (2n)!2n.

If we assume that

ps−1,t−1 =(n− s+ t− 1)! (n+ s− t)!

2s+t−2 (2n)!2n, p

s,t−1 =(n− s+ t− 1)! (n+ s− t)!

2s+t−1 (2n)!2n,

then applying ps,t−1 = 2ps,t (3) we obtain, for all 0 ≤ s, t ≤ n− 1 and s+ t ≤ n− 1,

ps,t =(n+ s− t)! (n+ t− s− 1)!

2s+t (2n)!2n.

Finally, applying 1 and 2,

(n− s− t) ps,t + tps,t−1 = (n− s− t) p

s,t+ sps−1,t,

ps−1,t = 2ps,t,

it holds that

ps,t=(n+ t− s)! (n+ s− t− 1)!

2s+t (2n)!2n

for all 0 ≤ s, t ≤ n− 1 and s+ t ≤ n− 1. ¤


4. The core and the Weber setNow, some solution concepts for bicooperative games are introduced, understandingas a solution concept any subset of vectors in Rn that provide an equitable distributionof the total saving among the players. Taking into account different situations thatcan be modelled by a bicooperative game (N, b), the amount b (N, ∅) is the maximalgain and b (∅, N) is the minimal loss obtained by the players when they decide fullcooperation and so, the maximal global gain is given by b (N, ∅)− b (∅,N) . A vectorx ∈ Rn which satisfies

Pi∈N xi = b (N, ∅)− b (∅, N) is called efficient vector and the

set of all efficient vectors is called preimputation set which is defined by

I∗(N, b) =

(x ∈ Rn :

Xi∈N

xi = b (N, ∅)− b (∅, N)).

The imputations for game b are the preimputations that satisfy the individualrationality principle for all players, that is, each player gets at least the differencebetween the amount that he can attain for himself taking the rest of players againstand the value of the signed coalition (∅,N) ,

I(N, b) = {x ∈ I∗(N, b) : xi ≥ b(i,N \ i)− b (∅, N) for all i ∈ N} .

A satisfactory distribution criterion could be that every signed coalition (S, T ) ∈3N receives at least the amount it can contribute to the coalition (∅, N) , that is, theamount b(S, T )− b (∅, N) . It leads us to define the notion of the core of the game bas the set

C(N, b) =

½x ∈ I∗(N, b) : x = y + z with

y (S) + z (N \ T ) ≥ b(S, T )− b (∅, N) ∀ (S, T ) ∈ 3N¾.

This definition can be interpreted in the following manner. For each (S, T ) ∈ 3N , theplayers who are not in the coalition T have contributed to the formation of (S, T )since they will not act against the player of the coalition S and for this, they must bereceived a payoff given by the vector z. Moreover, those players of N \ T who are inthe coalition S must get a different payoff to the rest of players, given by the vectory since these players have contributed to the formation of (S, T ) in a different way.

In order to extend the idea of the Weber set to a bicooperative game (N, b) , itis assumed that all players estimate that (N, ∅) is formed as a sequential processwhere in each step a different player is incorporated to the defender coalition or adifferent player leaves the detractor coalition. These sequential processes are obtainedconsidering the different chains from (∅,N) to (N, ∅) . In each one of these processes,a player can evaluate his contribution when is incorporated to the defenders or hiscontribution when leaves the detractors. This can be reflected in the vectors of Rn

denominated superior marginal worth vectors and inferior marginal worth vectors.With the aim to formalize this idea, we introduce the following notation.

Given N = {1, . . . , n} , let N = {−n, . . . ,−1, 1, . . . , n} .We can define an isomor-phism Λ : 3N → 2N as follows: For each (S, T ) ∈ 3N , Λ (S, T ) = S∪{−i : i ∈ N \ T} ∈2N . For instance, Λ (∅,N) = ∅ and Λ (N, ∅) = N. Since S ∩ T = ∅ ⇔ S ⊆ N \ T wesee that i ∈ Λ (S, T ) and i > 0 imply −i ∈ Λ (S, T ) .


In the lattice¡3N ,v

¢, we consider the set of all maximal chains which going from

(∅, N) to (N, ∅) and denote this set by Θ¡3N¢. If θ ∈ Θ

¡3N¢is the maximal chain

(∅, N) @ (S1, T1) @ · · · @ (Sj , Tj) @ · · · @ (S2n−1, T2n−1) @ (N, ∅) ,

then we can write the following associated chain of sets in 2N ,

∅ ⊂ {i1} ⊂ · · · ⊂ {i1, . . . , ij} ⊂ · · · ⊂ {i1, . . . , i2n−1} ⊂ N,

where {i1, . . . , ij} = Λ (Sj , Tj) for j = 1, . . . , 2n. We define the vector θ (ij) =(i1, . . . , ij) , where the last component ij ∈ N satisfies the following property: ifij > 0 then the player ij ∈ Sj and ij /∈ Sj−1, that is, ij is the last player who joinsSj and if ij < 0, then the player −ij /∈ Tj and −ij ∈ Tj−1, that is, −ij is the lastplayer who leaves Tj−1. Equivalently, the elements in θ (ij) = (i1, . . . , ij) are writtenfollowing the order of incorporation in the defenders coalitions or desertion of thedetractors coalition (depending on the sign of each ik) in the signed coalitions inchain θ . Moreover, we write

θ (ij) \ ij = (i1, i2, . . . , ij−1) = θ (ij−1)

and ik ∈ θ (ij) when ik is one component of the vector θ (ij) , that is 1 ≤ k ≤ j.Note that an equivalence between maximal chains and vectors θ = (i1, . . . , i2n) isobtained. Fix an order θ = (i1, . . . , i2n) , we also define α [θ (ij)] = (Sj , Tj) suchthat Λ (Sj , Tj) = {i1, . . . , ij}. Moreover, α [θ (ij) \ ij ] = α [θ (ij−1)] = (Sj−1, Tj−1) .In particular, α [θ (i2n)] = (N, ∅) and α [θ (i1) \ i1] = (∅, N) .

For example, let N = {1, 2, 3} and θ ∈ Θ¡3N¢given by

(∅, N) @ (∅, {1, 3}) @ ({2} , {1, 3}) @ ({2} , {1}) @ ({2} , ∅) @ ({2, 3} , ∅) @ (N, ∅) .

Its associated chain of sets in 2N is given by

∅ ⊂ {−2} ⊂ {−2, 2} ⊂ {−2, 2,−3} ⊂ {−2, 2,−3,−1} ⊂ {−2, 2,−3,−1, 3} ⊂ N.

and the maximal chain can be also represented by the order θ = (−2, 2,−3,−1, 3, 1) .One signed coalition, for instance ({2} , ∅) , can be also represented by α [θ (−1)] andby Λ−1 ({−2, 2,−3,−1}) .

Definition 3. Let θ ∈ Θ¡3N¢and b ∈ BGN . We call inferior and superior marginal

worth vectors with respect to θ to the vectorsmθ (b) , Mθ (b) ∈ Rn respectively where

mθi (b) = b (α [θ (−i)])− b (α [θ (−i) \−i]) ,

Mθi (b) = b (α [θ (i)])− b (α [θ (i) \ i]) ,

for all i ∈ N.We call marginal worth vector with respect to θ to the vector aθ (b) ∈ Rn

obtained as the sum of inferior and superior marginal worth vectors, that is,

aθi (b) = mθi (b) +Mθ

i (b) , for all i ∈ N.

The following result show that the marginal worth vectors are preimputations.


Proposition 8. For any b ∈ BGN and θ ∈ Θ¡3N¢we haveX

i∈Naθi (b) = b (N, ∅)− b (∅, N) .

Proof. Let b ∈ BGN and θ ∈ Θ¡3N¢. It holds thatX

i∈Naθi (b) =

Xi∈N

hmθ

i (b) +Mθi (b)

i=

Xi∈N

[b (α [θ (−i)])− b (α [θ (−i) \−i]) + b (α [θ (i)])− b (α [θ (i) \ i])]

=2nXj=1

[b (α [θ (ij)])− b (α [θ (ij) \ ij ])]

= b (α [θ (i1)])− b (α [θ (i1) \ i1]) +2nXj=2

[b (α [θ (ij)])− b (α [θ (ij−1)])]

= b (N, ∅)− b (∅, N) . ¤

Proposition 9. Let b ∈ BGN and θ ∈ Θ¡3N¢. Then,X

j∈SMθ

j (b) +X

j∈N\Tmθ

j (b) = b (S, T )− b (∅, N) ,

for every (S, T ) in the chain θ.

Proof. Let θ ∈ Θ¡3N¢and (S, T ) in the chain θ with |S| = s, |T | = t, s + t ≤ n

and such that Λ (S, T ) = {i1, i2, . . . , in+s−t} where the ij are written following theorder of incorporation in θ, that is, θ (ij) = (i1, i2, . . . , ij) for all 1 ≤ j ≤ n + s − t.Then,X

j∈SMθ

j (b) +X

j∈N\Tmθ

j (b) =X

{ij∈Λ(S,T ):ij>0}Mθ

ij (b) +X

{ij∈Λ(S,T ):ij<0}mθ−ij (b)

=X

ij∈Λ(S,T )[b (α [θ (ij)])− b (α [θ (ij) \ ij ])]

=n+s−tXj=1

[b (α [θ (ij)])− b (α [θ (ij) \ ij ])]

= b (S, T )− b (∅, N) .

Note that for (S, T ) = (N, ∅), we havePj∈N

hmθ

j (b) +Mθj (b)

i= b (N, ∅) − b (∅, N) .

¤

Definition 4. Let b ∈ BGN be a bicooperative game. The Weber set of b is theconvex hull of the marginal worth vectors, that is

W (N, b) = convnaθ (b) : θ ∈ Θ

¡3N¢o

.


As the preimputation set is a convex set, it is evident that W (N, b) ⊆ I∗ (N, b) .However, in general, the vectors of the Weber set are not imputations. For example,let (N, b) with N = {1, 2} and b : 3N → R defined as

b (∅, N) = −5, b (∅, i) = −4, b (i, j) = −1, b (i, ∅) = 1, b (N, ∅) = 2,

for all i, j ∈ N. If we consider θ = (−2, 2,−1, 1) , then aθ1 (b) = mθ1 (b) +Mθ

1 (b) = 3.As b (1, 2)− b (∅, N) = 4, then aθ1 (b) < b (1, N \ 1)− b (∅, N) and aθ (b) /∈ I (N, b) .

It is easy to see, taking into account that I (N, b) is a convex set, that W (N, b) ⊆I (N, b) if all marginal worth vectors are imputations. For this, a sufficient conditionis that the game b is zero-monotonic, a concept that is defined as follows.

Definition 5. A bicooperative game b ∈ BGN is monotonic when for all signed coali-tions (S1, T1) , (S2, T2) with (S1, T1) v (S2, T2) it holds that b (S1, T1) ≤ b (S2, T2) .

Definition 6. The zero-normalization of a bicooperative game b ∈ BGN is the gameb0 ∈ BGN defined by

b0 (S, T ) = b (S, T )−Xj∈S

[b (j,N \ j)− b (∅, N)] , for all (S, T ) ∈ 3N .

Definition 7. A bicooperative game b ∈ BGN is called zero-monotonic if its zero-normalization is monotonic.

Proposition 10. Let b ∈ BGN be a zero-monotonic bicooperative game. Then, forevery θ ∈ Θ

¡3N¢, the marginal worth vector associated to θ is an imputation for the

game b.

Proof. Let θ ∈ Θ¡3N¢. Since the vector aθ (b) is efficient, we prove that aθi (b) ≥

b (i,N \ i)− b (∅, N), for all i ∈ N. Indeed,

aθi (b) = b (α [θ (i)])− b (α [θ (i) \ i]) + b (α [θ (−i)])− b (α [θ (−i) \−i])= b0 (α [θ (i)]) +

X{ij∈θ(i):ij>0}

[b (ij ,N \ ij)− b (∅,N)]

−b0 (α [θ (i) \ i])−X

{ij∈θ(i)\i:ij>0}[b (ij , N \ ij)− b (∅, N)]

+b0 (α [θ (−i)]) +X

{ij∈θ(−i):ij>0}[b (ij , N \ ij)− b (∅, N)]

−b0 (α [θ (−i) \−i])−X

{ij∈θ(−i)\−i:ij>0}[b (ij , N \ ij)− b (∅, N)]

= b0 (α [θ (i)])− b0 (α [θ (i) \ i]) + b0 (α [θ (−i)])−b0 (α [θ (−i) \−i]) + b (i,N \ i)− b (∅, N)

≥ b (i,N \ i)− b (∅, N) ,

where the inequality follows the zero-monotonicity of the bicooperative game b. ¤Now we prove that the core of a bicooperative game is always included in its

Weber set. It should be noted that the proof of this result is closely related to theproof in [4] of the inclusion of the core in the Weber set for cooperative games.


Theorem 11. If b ∈ BGN , then C (N, b) ⊆W (N, b) .

Proof. Assume that there exists x ∈ C (N, b) such that x /∈ W (N, b). As x ∈C (N, b) , then

Pi∈N xi = b (N, ∅)− b (∅, N) and x = y + z with y (S) + z (N \ T ) ≥

b(S, T ) − b (∅,N) for all (S, T ) ∈ 3N . Since the Weber set W (N, b) is convex andclosed we apply the Separation Theorem (see Rockafellar [12]), and so there existsu ∈ Rn such that

w · u > x · u for all w ∈W (N, b) . (4)

In particular for all marginal worth vectors w = aθ (b) with θ ∈ Θ¡3N¢. If the

components of vector u are ordered in non increasing order

ui1 ≥ ui2 ≥ · · · ≥ uin−1 ≥ uin ,

let θ ∈ Θ¡3N¢be the maximal chain given by θ = (−i1, i1,−i2, i2, . . . ,−in, in) . Note

that θ (ij) \ ij = θ (−ij) for all 1 ≤ j ≤ n, θ (−ij) \ −ij = θ (ij−1) for all 2 ≤ j ≤ nand α [θ (−i1) \−i1] = (∅,N) . Then

aθ (b) · u =nX

j=1

aθij (b)uij =nX

j=1

hMθ

ij (b) +mθij (b)

iuij

=nX

j=1

uij [b (α [θ (ij)])− b (α [θ (ij) \ ij ]) + b (α [θ (−ij)])− b (α [θ (−ij) \−ij ])]

=nX

j=1

uij [b (α [θ (ij)])− b (α [θ (ij−1)])]

= uinb (N, ∅) +n−1Xj=1

uijb (α [θ (ij)])− ui1b (∅, N)−nX

j=2

uijb (α [θ (ij−1)])

= uinb (N, ∅)− ui1b (∅, N) +n−1Xj=1

¡uij − uij+1

¢b (α [θ (ij)])

≤ uinb (N, ∅)− ui1b (∅, N) +n−1Xj=1

¡uij − uij+1

¢ " jXk=1

yik +

jXk=1

zik + b (∅, N)#

= uin

"nX

k=1

yik +nX

k=1

zik + b (∅, N)#− ui1b (∅, N)

+n−1Xj=1

¡uij − uij+1

¢ " jXk=1

yik +

jXk=1

zik + b (∅, N)#

=nX

j=1

uij¡yij + zij

¢=

nXj=1

uijxij = x · u

which is in contradiction with the inequality 4. We conclude that a core distributionhas to be an element of the Weber set. ¤


5. Bisupermodular gamesNow we introduce a special class of bicooperative games.

Definition 8. A bicooperative game b ∈ BGN is called bisupermodular if, for all(S1, T1) and (S2, T2) it holds

b((S1, T1) ∨ (S2, T2)) + b ((S1, T1) ∧ (S2, T2)) ≥ b (S1, T1) + b (S2, T2) ,

or equivalently

b(S1 ∪ S2, T1 ∩ T2) + b (S1 ∩ S2, T1 ∪ T2) ≥ b (S1, T1) + b (S2, T2) .

The next proposition characterizes the bisupermodular games as those bicooper-ative games for which the marginal contributions of a player to one signed coalitionis never less that the marginal contribution of this player to any signed coalitioncontained in it. This characterization will be used in the proves of the followingresults.

Proposition 12. Let b ∈ BGN . The bicooperative game b is bisupermodular if andonly if for all i ∈ N and (S1, T1), (S2, T2) ∈ 3N\i such that (S1, T1) v (S2, T2) , itholds

b (S2 ∪ i, T2)− b (S2, T2) ≥ b (S1 ∪ i, T1)− b (S1, T1) ,

andb (S2, T2)− b (S2, T2 ∪ i) ≥ b (S1, T1)− b (S1, T1 ∪ i) .

Proof. Necessary condition. Let (S1, T1), (S2, T2) ∈ 3N\i with (S1, T1) v (S2, T2).If S01 = S1 ∪ i and we apply the definition of bisupermodularity to (S01, T1) and(S2, T2) , it follows

b¡S01 ∪ S2, T1 ∩ T2

¢+ b

¡S01 ∩ S2, T1 ∪ T2

¢≥ b (S1 ∪ i, T1) + b (S2, T2) ,

and henceb (S2 ∪ i, T2) + b (S1, T1) ≥ b (S1 ∪ i, T1) + b (S2, T2) .

In analogous form, taking T 02 = T2 ∪ i and applying the definition of supermodularityto (S1, T1) and (S2, T 02) , it follows

b (S1, T1 ∪ i) + b (S2, T2) ≥ b (S1, T1) + b (S2, T2 ∪ i) .

Sufficient condition. Let (S1, T1), (S2, T2) ∈ 3N . If (S1, T1) v (S2, T2) or (S2, T2) v(S1, T1), the equality trivially holds. So, we consider the case (S1, T1) ∧ (S2, T2) 6=(S1, T1) and (S1, T1) ∧ (S2, T2) 6= (S2, T2).

Let θ ∈ Θ¡3N¢be a maximal chain that contains the signed coalitions (S2, T2)

and (S1, T1)∨(S2, T2) .As Λ (S1, T1)\Λ (S2, T2) 6= ∅,we assume that |Λ (S1, T1) \ Λ (S2, T2)| =k and so, we write Λ (S1, T1) \ Λ (S2, T2) = {i1, i2, . . . , ik} , where the ij are in thesame order that appear in the order θ, i.e.,

α [θ (i1)] @ α [θ (i2)] @ · · · @ α [θ (ik)] .


Then, the chain θ is given by

∅ ⊂ · · · ⊂ Λ (S2, T2) ⊂ Λ (S2, T2) ∪ {i1} ⊂ · · · ⊂ Λ (S2, T2) ∪ {i1, . . . , ik} ⊂ · · · ⊂ N

or equivalently

(∅, N) @ · · · @ (S2, T2) @ · · · @ (S1, T1) ∨ (S2, T2) @ · · · @ (N, ∅) .

If we denote Aj = {i1, i2, . . . , ij} , for all 1 ≤ j ≤ k, A0 = ∅ and (P,Q) =(S1, T1) ∧ (S2, T2), it holds thatΛ−1 [Λ (P,Q) ∪Aj ] @ Λ−1 [Λ (S2, T2) ∪Aj ] for all 1 ≤ j ≤ k. We can apply the

hypothesis to Λ−1 [Λ (P,Q) ∪Aj ] and Λ−1 [Λ (S2, T2) ∪Aj ], and we obtain

b¡Λ−1 (Λ (P,Q) ∪Aj)

¢− b

¡Λ−1 (Λ (P,Q) ∪Aj−1)

¢≤ b

¡Λ−1 (Λ (S2, T2) ∪Aj)

¢− b

¡Λ−1 (Λ (S2, T2) ∪Aj−1)

¢for all 1 ≤ j ≤ k. Hence,

b ((S1, T1))− b ((S1, T1) ∧ (S2, T2)) = b¡Λ−1 (Λ (P,Q) ∪Ak)

¢− b (P,Q)

=kX

j=1

£b¡Λ−1 (Λ (P,Q) ∪Aj)

¢− b

¡Λ−1 (Λ (P,Q) ∪Aj−1)

¢¤≤

kXj=1

£b¡Λ−1 (Λ (S2, T2) ∪Aj)

¢− b

¡Λ−1 (Λ (S2, T2) ∪Aj−1)

¢¤= b ((S1, T1) ∨ (S2, T2))− b (S2, T2) . ¤

The following result permits the identification of the games for which the marginalworth vectors are distributions of the core.

Theorem 13. A necessary and sufficient condition so that all marginal worth vectorsof a bicooperative game b ∈ BGN are vectors of the core is that the game b isbisupermodular

Proof. Sufficient condition. Let θ ∈ Θ¡3N¢. We know that the marginal worth

vectors are efficient, we prove that the marginal worth vector aθi (b) = mθi (b)+Mθ

i (b)satisfiesX

j∈SMθ

j (b) +X

j∈N\Tmθ

j (b) ≥ b (S, T )− b (∅, N) , for all (S, T ) ∈ 3N .

By Proposition 8, for every (S, T ) in the chain θ, it holdsXj∈S

Mθj (b) +

Xj∈N\T

mθj (b) = b (S, T )− b (∅, N) .


We prove that, for every coalition (S, T ) , not in the chain θ,Xj∈S

Mθj (b) +

Xj∈N\T

mθj (b) ≥ b (S, T )− b (∅, N) .

Indeed, let (S, T ) be a signed coalition that does not belong to the chain θ, such thatΛ (S, T ) = {i1,i2, . . . , ik} , k = n+ s− t, where the elements are written following theorder of θ; that is, α [θ (i1)] @ α [θ (i2)] @ · · · @ α [θ (ik)] .

If we denote Aj = {i1, i2, . . . , ij} , for all 1 ≤ j ≤ k, and A0 = ∅, note that,for all 1 ≤ j ≤ k, we have that Aj = Λ (S, T ) ∩ Λ (α [θ (ij)]) , that is Λ−1 (Aj) =(S, T ) ∧ α [θ (ij)] . As b is a bisupermodular game, the Proposition 12 implies that,for all 1 ≤ j ≤ k,

b (α [θ (ij)])− b (α [θ (ij) \ ij ]) ≥ b¡Λ−1 (Aj)

¢− b

¡Λ−1 (Aj−1)

¢,

and we obtainXj∈S

Mθj (b) +

Xj∈N\T

mθj (b) =

X{ij∈Λ(S,T ):ij>0}

Mθij (b) +

X{ij∈Λ(S,T ):ij<0}

mθj (b)

=X

ij∈Λ(S,T )[b (α [θ (ij)])− b (α [θ (ij) \ ij ])]

=n+s−tXj=1

[b (α [θ (ij)])− b (α [θ (ij) \ ij ])]

≥n+s−tXj=1

£b¡Λ−1 (Aj)

¢− b

¡Λ−1 (Aj−1)

¢¤= b (S, T )− b (∅, N) .

Necessary condition. For all (S1, T1) , (S2, T2) ∈ 3N , consider a maximal chain θ ∈Θ¡3N¢which contains (S1, T1)∧(S2, T2) = (S1 ∩ S2, T1 ∪ T2) and (S1, T1)∨(S2, T2) =

(S1 ∪ S2, T1 ∩ T2) . As the marginal worth vectors are elements of C (N, b) , we havethat X

j∈S1Mθ

j (b) +X

j∈N\T1

mθj (b) ≥ b (S1, T1)− b (∅,N) ,

Xj∈S2

Mθj (b) +

Xj∈N\T2

mθj (b) ≥ b (S2, T2)− b (∅,N) ,

By the election of the maximal chain θ and Proposition 8, it is also verifiedXj∈S1∩S2

Mθj (b) +

Xj∈N\(T1∪T2)

mθj (b) = b ((S1, T1) ∧ (S2, T2))− b (∅, N) .

Xj∈S1∪S2

Mθj (b) +

Xj∈N\(T1∩T2)

mθj (b) = b ((S1, T1) ∨ (S2, T2))− b (∅, N) .

Therefore,b (S1, T1) + b (S2, T2)− 2b (∅, N)


≤Xj∈S1

Mθj (b) +

Xj∈N\T1

mθj (b) +

Xj∈S2

Mθj (b) +

Xj∈N\T2

mθj (b)

=X

j∈S1∪S2Mθ

j (b) +X

j∈S1∩S2Mθ

j (b) +X

j∈N\(T1∪T2)mθ

j (b) +X

j∈N\(T1∩T2)mθ

j (b)

= b ((S1, T1) ∧ (S2, T2)) + b ((S1, T1) ∨ (S2, T2))− 2b (∅,N) .

Hence

b (S1, T1) + b (S2, T2) ≤ b ((S1, T1) ∧ (S2, T2)) + b ((S1, T1) ∨ (S2, T2)) . ¤

As the core of a bicooperative game b ∈ BGN is a convex set, an immediateconsequence of this theorem is the following result.

Corollary 14. Let b ∈ BGN . A necessary and sufficient condition so thatW (N, b) =C (N, b) is that the bicooperative game b is bisupermodular.

Note that the Shapley value of a bicooperative game b is given by

Φi (N, b) =1

c(3N)

Xθ∈Θ(3N )

hmθ

i (b) +Mθi (b)

i,

for all i ∈ N. Then the Shapley value of a bisupermodular game b is in C (N, b) andhence, the core of a bisupermodular game is non-empty.

AcknowledgementsThis research has been partially supported by the Spanish Ministry of Science andTechnology and the European Regional Development Fund, under grant SEC2003—00573, and by the FQM237 grant of the Andalusia Government.

References[1] Bilbao JM (2000) Cooperative games on combinatorial structures. Kluwer Aca-

demic Publishers, Boston

[2] Bilbao JM, Fernández JR, Jiménez N, López JJ (2006) Probabilistic values forbicooperative games. Working paper, University of Seville

[3] Chua VCH, Huang HC (2003) The Shapley-Shubik index, the donation paradoxand ternary games. Social Choice and Welfare 20:387—403

[4] Derks J (1992) A short proof of the inclusion of the core in the Weber set.International Journal of Game Theory 21:149—150

[5] Felsenthal D, Machover M (1997) Ternary voting games. International Journalof Game Theory 26:335—351

[6] Freixas J (2005) The Shapley-Shubik power index for games with several levelsof approval in the input and output. Decision Support Systems 39:185—195


[7] Freixas J (2005) Banzhaf measures for games with several levels of approval inthe input and output. Annals of Operations Research 137:45—66

[8] Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levelsof approval. Social Choice and Welfare 21:399—431

[9] Grabisch M, Labreuche Ch (2005) Bi-capacities I: definition, Möbius transformand interaction. Fuzzy Sets and Systems 151:211—236

[10] Grabisch M, Lange F (2006) Games on lattices, multichoice games and the Shap-ley value: a new approach. To appear in Mathematical Methods of OperationsResearch

[11] Myerson RB (1991) Game Theory: analysis of conflict. Harvard University Press,Cambridge

[12] Rockafellar RT (1970) Convex Analysis. Princeton University Press, Princeton,New Jersey

[13] Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW(eds) Contributions to the Theory of Games, Vol II. Princeton University Press,Princeton, New Jersey, 307—317

[14] Shapley LS (1953) Cores of convex games. International Journal of Game Theory1:11—26

[15] Stanley RP (1986) Enumerative Combinatorics, Vol I. Wadsworth, Monterey

[16] Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shap-ley Value: Essays in Honor of Lloyd S. Shapley. Cambridge University Press,Cambridge, 101—119

Indexcooperative game, 1

bicooperative game, 1core, 18imputation set, 18marginal worth vector, 19preimputation set, 18Shapley value, 8Weber set , 21zero-monotonic, 21

bisupermodular game, 23

cooperative gamecore, 3marginal worth vector, 3Shapley value, 2Weber set, 3

dummy axiom, 9dummy player, 9

efficiency axiom, 10

identity game, 7inferior unanimity game, 7

linearity axiom, 9

structural axiom, 16superior unanimity game, 7symmetry axiom, 9

ternary voting game, 2

28

matemática aplicada ii, escuela superior de ingenierosmbilbao/pdffiles/bicogames.pdf ·...

Documents