A Value for Bi-cooperative Games

8/2/2019 A Value for Bi-cooperative Games

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Int J Game Theory (2008) 37:409438

DOI 10.1007/s00182-008-0126-5

ORIGINAL PAPER

A value for bi-cooperative games

Christophe Labreuche Michel Grabisch

Accepted: 12 March 2008 / Published online: 2 April 2008 Springer-Verlag 2008

Abstract Bi-cooperative games were introduced by Bilbao et al. as a generalization

of TU cooperative games, in which each player can participate positively, negatively,

or not at all. In this paper, we propose a definition of a share of the worth obtained by

some players after they decided on their participation in the game. It turns out that the

cost allocation rule does not look for a given player to her contribution at the opposite

participation option to the one she chooses. The relevance of the value is discussed on

several examples.

Keywords Bi-cooperative games Value Efficiency

1 Introduction

Acquisition and possession of a scarce resource is economically costly so that the

mutual use of such a resource by several agents usually implies positive externalities

on the users. Once agents have decided to cooperate, the question of how to allo-cate the resource, and how to share its costs and benefits among the users arises. The

primary concern of Cooperative Game Theory is how to deal with this issue. The most

classical model is the concept of a game in characteristic form. When side-payments

are possible among players, this results in TU games. This representation is used to

model the exchange of private goods under private ownership, or the collaborative

use of commons such as public good or common technology. This has been applied

C. Labreuche (B)

Thales Research and Technology France, RD 128, 91767 Palaiseau Cedex, France

e-mail: [email protected]

M. Grabisch

University of Paris I, CERMSEM, 106-112, Bd de lHpital, 75647 Paris Cedex 13, France

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410 C. Labreuche, M. Grabisch

Fig. 1 Linear order among the different options in a multi-choice game and in a bipolar situation

to very different areas such as water supply, electricity supply, distributive justice, or

the share of maintenance costs among the common users of a railway infrastructure

(Norde et al. 2003; Moulin 1995; Henriet and Moulin 1999).

A TU game is a function v : 2N

R where N is the set of players, which assignsto any coalition S N its worth v(S), namely the stand-alone worth of coalition S,

i.e. the cost for supplying all players of S with the resource. In a TU game, the players

have only two options: to participate or not to participate. This model is not sufficient

to represent all situations. As an example, a player may demand several items of a

good or a commodity in cost sharing problems (Sprumont 2000). A similar idea arises

in the concept of a multi-choice game in which there are several levels of participation

in the game, ranked from non-participation to full participation (Hsiao and Raghavan

1993). The more a player participates, the larger the cost or worth.

In this paper, we are interested in the situation in which there are two options ofparticipation, but these two options do not follow the above schema. The added value

on the cost or worth of playing is non-negative for one option (called the positive

option hereafter) of participation and non-positive for the other one (called the nega-

tive option hereafter). A player choosing the positive (resp. negative) option is called

a positive contributor (resp. negative contributor). Figure 1 below summarizes the

major difference between multi-choice games and the new situation.

In Sect. 2, we illustrate this situation in cost sharing problems with two examples

(see Examples 1 and 3). The positive participation option depicts the usual partici-

pation in cost sharing problems, that is, a player asks for the resource, and this new

demand results in a cost increase. By contrast, the negative participation option corre-

sponds also to the demand by a player to access the resource, but the player performs

an action resulting in a decrease in the overall cost. This action can be seen as altruistic

by the group of players and may require commitment by this player. However, she

chooses the negative option since she expects to be rewarded from her action.

Ternary voting games (Felsenthal and Machover 1997) are another example of this

bipolar situation (see Example 4 in Sect. 2), where positive and negative participa-

tions depict a player voting in favor and against a legislative bill respectively, while

non-participation corresponds to abstention.

It can be assumed that the option of participation chosen by each player is given

exogenously and can thus be frozen. Hence each player can either cooperate given her

exogenous choice of cooperation, or not participate. This situation can be translated

into a TU game. However, a positive contributor could become a negative contrib-

utor. Computing the share alloted to each player from previous TU game implies

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A value for bi-cooperative games 411

that the contribution of a player when she chooses the opposite option (positive vs.

negative) to that given exogenously, is ruled out in the share. Since players may hesitate

between the two options of participation, there is a priori no argument for discarding

the opposite option when computing the sharing rule. This shows that the TU model

is not sufficient. One needs to know the option chosen by the active players. Thustwo disjoint coalitions are necessary - one for the positive contributors and one for

the negative ones. The game we need is thus a function that assigns a worth to any

couple of disjoint coalitions a worth. This is a bi-cooperative game (Bilbao et al.

2000).

A central question in Cooperative Game Theory is how the players should share

the total worth v(N) of a TU game v if the grand coalition N forms. A solution is a

payoff vector x RN+ such that

i N xi = v(N). As an example, the Shapley value

is a solution concept that evaluates the players mean prospect.

Three different definitions of a value have been proposed for bi-cooperative games(Felsenthal and Machover 1997; Grabisch and Labreuche 2005; Bilbao et al. 2007).

These values satisfy the same efficiency axiom, which states that the value is a share

of the difference of worths between situations where all players are positive contribu-

tors and where all players are negative contributors. This axiom is technical and lacks

of clear interpretation. As a result, the solutions defined in Felsenthal and Machover

(1997), Grabisch and Labreuche (2005), and Bilbao et al. (2007) cannot be viewed

as a share of the worth of the grand coalition but instead as mean prospects over the

three possible choices of all players (i.e. participate positively, negatively or not at

all).The goal of this paper is to revisit this problem. We abandon the idea of a value

that represents a mean prospect whatever the type of participation of the players in

the game. What we aim to do is similar to what is done in cost allocation problems in

which the sought value is a share of the total cost corresponding to the actual demand

of each player. In our problem, the demand of the players is described by two disjoint

coalitions, and indicates which player chooses to participate either positively (for the

first coalition), negatively (for the second coalition) or not at all (for the remaining

players). It is thus natural to define efficiency, in our case, as the property that the

payoff is a share of the worth of this pair of coalitions (Sect. 3). Five other axioms

are used to characterize a value for bi-cooperative games: linearity, null player, mono-

tonicity and two symmetry axioms. Our main result (Sect. 3) is that under linearity,

null player, monotonicity and efficiency axioms, the payoff for a positive contribu-

tor depends only on her added value from non-participation to positive participation.

The information regarding her added value on the game when she becomes a nega-

tive contributor, is not relevant. The same argument holds symmetrically for negative

contributors. Hence for the computation of the value, one can restrict the bi-coopera-

tive game to a usual TU game. The Shapley value is then applied to the TU game so

obtained.

An overall value that is not conditional on the choice of the action of each player is

defined in Sect. 4. It is an expected value computed from the value defined earlier. The

expression we obtain is different from the existing ones (Felsenthal and Machover

1997; Grabisch and Labreuche 2005; Bilbao et al. 2007).

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Fig. 2 Situation of the three

fields and the well

Section 5 discusses the relevance of our value for several examples. Section 6

compares our proposal to previous ones.

2 Bi-cooperative games: definition and examples

Let N := {1, . . . , n} denote the finite set of players. Cardinalities of coalitions are

denoted by corresponding lower case letters, i.e. |S| =: s.

The following example illustrates the existence of positive and negative participa-

tion options.

Example 1 (Irrigation Network) Water is a critical resource in many agricultural areas.

Farmers have thus interest in constructing common irrigation networks. We assume

that there is only one possible location for a well in a given area. Hence the farmers

Fig. 3 The four different cases of irrigation networks

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Fig. 4 The case of field 1 being

cut

who possess a field close to that well must share this well and decide to construct

jointly an irrigation network that will convey water to all fields.

We consider an example with three farmers. The configuration of their fields is

given in Fig. 2.

The water network consists of pipes that are buried at the edges of the fields. The

field of the first farmer is close to the well, and is also the largest one. To supply water

to the two other fields, it is necessary to skirt farmer 1s field. This requires a long

length of pipes and leads thus a large cost. Figure 3 below presents the different paths

of the network in all possible cases: Case a corresponds to the situation where only

farmer 1 asks for water, case b to the situation where farmer 2 asks for water and not

farmer 3, case c to the situation where farmer 3 asks for water and not farmer 2, and

case d to the situation where farmers 2 and 3 both ask for water. Figure 3 also gives

the cost in each situation. The cost of digging the well is 1 and the cost of buying andburying the pipes is 1 per unit of length, where the union of the three fields corresponds

to a square of 10 10 in unit of length.

The cost for constructing the network supplying water to all possible coalitions of

farmers defines a usual TU game which is a tree game ( Norde et al. 2003).

Since field 1 covers a large area, the price of supplying water to fields 2 and 3

is large. An option to reduce cost is to have the pipes cut through farmer 1s field.

This leads to the situation depicted in Fig. 4. Yet, cutting farmer 1s field can only

be decided by farmer 1 since this causes annoyance to her. It will indeed result in a

decrease in the estate price and a more complex ploughing path. Therefore, farmer 1wants to quantify the consequence of the two possible options (allowing cutting her

field or not) before making her decision. Note that farmers 2 and 3 may also allow

the pipes to cut through their field but this will not help in reducing the length of the

network, and thus the overall cost.

In the previous example, when farmer 1 asks for water, she has two options: to

require that the pipes lie on the border of her field, or to allow the pipes to cut her

field. The cost of the irrigation network increases when a farmer switches from not

asking water to asking water while choosing the first option (e.g. from 11 up to 16 for

farmer 2 if the other two farmers already asked water). Moreover, the network cost

usually decreases when farmer 1 switches from not asking water to asking water while

adopting the second option (e.g. from 16 down to 6 if the other two farmers already

asked water).

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Formally, we introduce Q(N) := {(S, T) | S, T N, S T = }, the set of

pairs of disjoint coalitions. A bi-cooperative game is a function v : Q(N) Rwith

v(, ) = 0, where v(S, T) is the worth when players in S are positive contributors,

players in T are negative contributors, and the remaining players do not participate.

Since cooperative games focus on the added value of players when they participate,it is natural that the worth v(, ), depicting a situation in which no player partici-

pates, vanishes. In the context of cost sharing, condition v(, ) = 0 means that the

cost when no player demands the technology or the commons is zero. We denote by

G[2](N) the set of all bi-cooperative games on N.

A ternary voting game is a particular bi-cooperative game where v can take only

two values: value 1 when the bill is accepted, and value 0 when the bill is rejected

(Felsenthal and Machover 1997). Worth v(S, T) corresponds to the result of the vote

when S are the voters in favor of the bill, T are the voters against the bill, and the

remaining voters abstain. Abstention is an alternative to the usual yes and no opinions.

Example 2 (Irrigation Network continued) We are now in position to construct the

bi-cooperative game in Example 1 from the costs given in Figs. 3 and 4.

v(, ) = 0, v({1}, ) = v(, {1}) = 1 (case a),

v({2}, ) = v({3}, ) = v({1, 2}, ) = v({1, 3}, ) = 11 (cases b and c),

v({2, 3}, ) = v({1, 2, 3}, ) = 16 (case d),

v({2}, {1}) = v({3}, {1}) = v({2, 3}, {1}) = 6 (case e)

and for any (S, T) Q({2, 3})

v(S {1}, T) = v( S T {1}, ),

v(S, T) = v( S T, ),

v(S, T {1}) = v( S T, {1}),

which means choosing positive or negative roles for farmers 2 and 3 has no conse-

quence on the overall cost.

A player i is positively monotone if

(S, T) Q(N\{i }), v(S {i }, T) v(S, T).

In Example 1, all farmers are clearly positively monotone. A player i is negatively

monotone if

(S, T) Q(N\{i }), v(S, T {i }) v(S, T).

In Example 1, farmer 1 is negatively monotone, except in the bi-coalition (, ).

A bi-cooperative game is monotone if all players are both positively and negatively

monotone. Bi-cooperative games which are monotone were introduced independently

of the work of Bilbao et al. (2000) by the authors under the name of bi-capacities

(Grabisch and Labreuche 2002; Labreuche and Grabisch 2006).

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Fig. 5 Situation of the three cities (the vertical axis indicates elevation)

When

(S, T) Q(N\{i }), v(S {i }, T) = v( S, T {i })

the choice of a positive or negative role for player i has no impact on the added value

of this player to a coalition. Such a player is a symmetric contributor. In Example 1,

farmers 2 and 3 are symmetric contributors.

Here are two other examples of bi-cooperative games.

Example 3 (Urban Water Supply) Three settlements want to share the cost of a com-

mon water supply network. In the original project, the water is supplied by a unique

facility F which pumps from the groundwater aquifer. The geographical situation of

the cities, the facility and the water network (in dotted lines) is depicted in Fig. 5.The problem is to share the water supply costs. The cost of conveying water to a town

depends mainly on its distance to F and its elevation, since this has an impact on the

number of intermediate pumping stations that need to be put on the network. If the net

is private and constructed by the cities, the share of cost will be based on the effective

part of each settlement in the overall cost.

City C is located in a valley. The price for supplying only C is pC. Towns A and

B are located on a hill. Transporting water uphill is expensive. We denote by pA, pBand pA B the price for supplying only A, only B, and both A and B, respectively. One

assumes that

0 < pB < pA < pA B < pA + pB . (1)

Town A has several major clearwater springs, which makes this place a well-known

recreation area (trout fishing, kayak, . . .). On the other hand, the springs can be easily

collected. If A decides to allow the pumping of its springs, the overall supply price

will decrease dramatically. In this case, A is a negative contributor. We denote by qAand qA B the price for supplying water to only A, and to both A and B, respectively,

when A is a negative contributor. We assume the following conditions

0 < qA < pA, 0 < qA B < pA B . (2)

We obtain the following game, assuming that the water consumption of city C can

only be provided by facility F and that the marginal cost of C is independent of the

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choices of towns A and B.

v(, ) = 0, v(A, ) = pA, v(B, ) = pB , v(C, ) = pC,

v(AB, ) = pA B , v(AC, ) = pA + pC, v(BC, ) = pB + pC,

v(ABC, ) = pA B + pC, v(, A) = qA,

v(B, A) = qA B , v(C, A) = qA + pC, v(BC, A) = qA B + pC.

Since the cost of supplying towns A and B as well as city C is high, this extra cost

shall not be shared uniformly by all towns but shall be covered mainly by towns A

and B. This is a classical argument in cost sharing problems.

Now here we see that a new problem arises. If town A produces enough water for

itself and town B, then the cost of transporting water to A and B becomes negligible.

However, if town A decides to pump a great quantity of clear water from the springs,its recreational attractiveness could decrease. So A must analyze the two options of

being a positive or a negative contributor.

Example 4 (Multi-valued Ternary voting games) In ternary voting games, v(S, T)

can take only two values: 1 when the motion is accepted and 0 if it is rejected. Yet

in many voting situations, the motion is not just simply accepted or rejected. For a

strong winning coalition of yes voters, the initial motion is adopted. However, for a

weak winning coalition of yes voters, a negotiation between the yes and the no voters

usually yields a modified motion which can be seen as a weaker version of the initial

one. This situation corresponds to bills with amendment. In this case, since the initial

motion is not applied, an intermediate value between 0 and 1 is assigned to v(S, T).

3 Definition of a value

Let us define the notion of a value for bi-cooperative games, in the spirit of what was

done by Shapley for cooperative games (Shapley 1953).

The set of cooperative games on N is denoted by G(N). In classical cooperative

game theory, an imputation (more precisely, a pre-imputation) is a vector x Rnsatisfying the efficiency principle, that is,

i N xi = v(N) (Driessen 1988). The

imputation represents the share of the total worth of the game, v(N), among the play-

ers, assuming that all players have decided to join the grand coalition N. A value

is a mapping : G(N) Rn which assigns to every game an imputation. The

well-known Shapley value (Shapley 1953) is defined by

Shi (v) :=

SN\is!(n s 1)!

n![v( S i ) v(S)]. (3)

In usual games, each player has only two options: either to join a coalition or to

stay aside. Cost allocation problems (Moulin 1995; Sprumont 2000) and multi-choice

games (Hsiao and Raghavan 1993) model situations in which each player has several

levels of participation in the games.

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In a cost allocation problem, each agent i N is interested in one type of a per-

sonalized good (each agent asks for a special good) and may demand several items of

a good. A demand vector q Nn describes the number of items asked by each agent.

A cost function is a non-decreasing function v : Nn R such that v(0, . . . , 0) = 0,

where v(q) is the overall cost for satisfying demand q. One is interested by cost-sharingmethods that define for each agent i N a non-negative cost xi satisfying the budget

balance condition

i N xi = v(q) (Moulin 1995).

Unlike usual games where in the end, all players join the grand coalition (unless

for games in coalition form), it is not assumed here that all players have decided to be

positive contributors. In multi-choice games, players do not necessarily participate in

the game at the highest possible level. We denote hereafter by S the set of players who

have decided to be positive contributors, by T the set of players who have decided to

be negative contributors. The remaining players N\(S T) are not participating. In the

situation depicted by bi-coalition (S, T), the worth that is obtained by the players isv( S, T). Let S,T : G[2](N) RN be the payoff vector in the situation of bi-coalition

(S, T) as described previously. We are interested in defining this share S,T(v).

In the Weber characterization of the Shapley value, four axioms are used: linear-

ity, null player, symmetry and efficiency (Weber 1988). Consider first linearity. This

axiom states that if several games are combined linearly then the values of each indi-

vidual game shall be combined in the same way to obtain the value of the resulting

game. This axiom is trivially extended to the case of bi-cooperative games.

Linearity (Lin): S,T is linear on G[2](N).

Proposition 1 Under Lin , for every i N, there exists real constants ai,S,TS,T

for all

(S, T) Q(N) such that for all games v G[2](N)

S,Ti (v) =

(S,T)Q(N)

ai,S,TS,T

v(S, T).

The proof of the proposition and all other results are given in the Appendix.The second axiom is called the null playeraxiom. A player i is null ifv( S {i }) =

v( S) for all S N\{i }. Since this player does not contribute to any coalition, the

payoff for this player shall be zero. For bi-cooperative games, a player is null if the

asset is the same if she joins the positive or the negative contributors.

Definition 1 The player i is null for the bi-cooperative game v if v(S, T {i }) =

v( S, T) = v(S {i }, T) for all (S, T) Q(N\{i }).

We propose the following axiom.

Null player (Null): If a player i is null for the bi-cooperative game v G[2](N)

then S,Ti (v) = 0.

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Proposition 2 Under Lin and Null , there exists for all i N and all (S, T)

Q(N\{i }), ai,S,TS,T

and bi,S,TS,T

such that

S,T

i(v) =

(S,T)Q(N\{i })

ai,S,T

S

,T v( S {i }, T) v(S, T)

+

(S,T)Q(N\{i })

bi,S,TS,T

v(S, T {i }) v(S, T)

. (4)

It is possible to replace Null by a stronger axiom, namely the dummy player axiom.

For a usual TU game v, a player i is dummy if v( S {i }) = v( S) + v({i }) for all

S N\ {i }. The contribution of this player to any coalition is constant. Thus the

payoff for this player shall by his solo worth v({i }). For a bi-cooperative game, a

dummy player is a player whose contribution to any bi-coalition is always the same,whatever the (positive or negative) level of contribution. This implies in particular that

this player is a symmetric contributor.

Definition 2 The player i is dummy for the bi-cooperative game v if there exists

R such that v(S {i }, T) v( S, T) = v(S, T {i }) v(S, T) = for all

(S, T) Q(N\{i }). Constant is the constant added value of player i to v.

Dummy player (Dum): If a player i is dummy for the bi-cooperative game

v G[2](N) then S,Ti (v) = , where R is the constant added value of

player i to v.

Proposition 3 Under Lin and Dum, there exists for every i N , ai,S,TS,T

for all (S,

T) Q(N\{i }) , bi,S,TS,T

for all (S, T) Q(N\{i }) such that(4) holds. Moreover,

(S,T)Q(N\{i })

ai,S,TS,T

+

(S,T)Q(N\{i })

bi,S,TS,T

= 1.

Replacing Null by Dum in Proposition 2, one sees that the coefficients become

normalized since they sum up to one.

The basic idea of monotonicity of solutions is that if the worths of all coalitions of

players increase, then the payoff of the players shall not decrease. Clearly solutions

are not monotone w.r.t. the game in the previous sense since, for instance, increasing

the game on only a coalition S cannot imply increasingness of the payoff vector for

players outside S. Hence, it is not a trivial matter to define monotonicity of solutions.

Property called Aggregate Monotonicity (AM) states that if the worth for the grand

coalition increases and the value of the game for sub-coalitions remains the same, then

the payoff vector shall not decrease (Megiddo 1974).

We define monotonicity in the following way. Let (N, v) and (N, v) two TU games

such that there exists i N with v(K) = v( K) and v(K {i }) v( K {i }) for all

K N\{i }. Then the contribution of player i to game v is larger than that of player

i to game v. Then the payoff ofi in game v shall not be greater to that ofi in v.

Generalizing that to bi-cooperative games, we obtain

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Monotonicity (Mon): Let v, v be two bi-cooperative games such that there

exists i N with

v(S, T) = v(S, T),

v

(S

{i }, T

) v( S

{i }, T

),v(S, T {i }) v( S, T {i }),

for all (S, T) Q(N\{i }), then S,Ti (v

) S,Ti (v).

When i joins the positive coalition S, the added value for i is larger for game v than

for v. When i joins negative coalition T, the negative added value for i is smaller in

absolute value for v than for v (the decrease of the value of the game when i joins

the negative contributors is less important for v than for v). Then the payoff ofi in v

shall not be greater to that of i in v.

Proposition 4 Under Lin, Null and Mon , for every i N, there exists ai,S,TS,T

0

for all (S, T) Q(N\{i }) , bi,S,TS,T

0 for all (S, T) Q(N\{i }) such that

S,Ti (v) =

(S,T)Q(N\{i })

ai,S,TS,T

v( S {i }, T) v(S, T)

+

(S,T)Q(N\{i })

bi,S,TS,T

v(S, T {i }) v(S, T)

.

Replacing Null by Dum in Proposition 4, one would obtain, as for Proposition 3,

that the non-negative coefficients ai,S,T and bi,S,T sum up to one. The probabilistic

values (Monderer and Samet 2001) for a bi-cooperative game are thus defined by Lin,

Dum and Mon, and result from a variant of Proposition 4.

We finally introduce the efficiency axiom. Value S,T(v) is a share of the worth

obtained by the bi-coalition (S, T).

Efficiency (Eff): For all games in G[2](N),

i N

S,Ti (v) = v(S, T).

For K S T, we set

V(K) := v(S K, T K). (5)

Proposition 5 Under Lin, Null, Mon and Eff, for every i S T there exist coef-

ficients ai,S,TK 0 for all K (S T)\{i } such that

S,Ti (v) =

K(ST)\{i }

ai,S,TK [V(K {i }) V(K)].

Moreover, if i N\(S T), S,Ti (v) = 0

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Replacing Null by Dum in Proposition 5, one would obtain, as for Proposition

3, that the non-negative coefficients ai,S,T sum up to one. Quasi-values, which are

defined as efficient probabilistic values (Monderer and Samet 2001), are obtained from

Lin, Dum, Mon and Eff, and result from a variant of Proposition 5.

Proposition 5 is essential since it states that the only terms of a bi-cooperative gamev that are used to determine S,T(v) belong to

QS,T(N) =

(S, T) Q(N), S S , T T

.

From Proposition 5, the allocation rule does not take into consideration the added value

to the game of a positive (resp. negative) contributor when she becomes a negative

(resp. positive) contributor. In other words, the payoff for a positive (resp. negative)

contributor depends only on her contribution from non-participation to positive (resp.

negative) participation. Hence for the computation of the value, one can restrict the

bi-cooperative game to a usual game (namely V). Even though a bi-cooperative game

contains much more information than V, the remaining terms cannot be used in a

coherent way. This argument justifies the use of classical Cooperative Game Theory.

This property holds for cost allocation problems. For such games, the value for

a demand q depends only on the restriction of the cost function on lower demands

{q Nn , i N q i qi } (Sprumont 2000).

From Proposition 5, one easily gets the following corollary.

Corollary 1 We have

i N\(S T), S,Ti (v) = 0, (6)

i S with i positively monotone, S,Ti (v) 0, (7)

i T with i negatively monotone, S,Ti (v) 0. (8)

Ifi T is positively monotone and a symmetric contributor, then S,Ti (v) 0.

Let us give an interpretation of Corollary 1. For classical values, it is assumed that

all players finally participate in the game so that the grand coalition is formed. If it

is not the case, a coalition structure has formed. Then the players of each coalition

share the worth obtained by that coalition (Aumann and Drze 1974). In this case, the

players who do not belong to the coalition get zero from this coalition. Hence it is

natural that the payoff for a player who does not participate vanishes. Hence (6).

Inequality (7) comes from the fact that players ofS contribute positively to the game

(provided i is positively monotone) and thus deserve a non-negative payoff. Likewise,

inequality (8) holds since players in T contribute negatively in the game (provided i is

negatively monotone) and thus deserve a non-positive payoff. In Example 1, (8) means

that farmer 1 should be rewarded for having accepted that the pipes cut her field. In

cost sharing problems, this condition is interpreted as a compensation for players who

have accepted to be negative contributors, since this option of participation causes

them annoyance. The trouble caused by the choice to be a negative contributor is clear

in all three Examples 1 and 3.

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In the Weber characterization of the Shapley value (Weber 1988), the last axiom

that we have not considered yet is symmetry with respect to the players. It states that

the rule for computing the share does not depend on the numbering of the players. It

means that nothing special is done to one player compared to another one. In other

words, the players are anonymous. This can be interpreted as fairness.Let be a permutation on N. With some abuse of notation, we denote (K) :=

{ (i )}i K. The roles of the players in S, T and N\(S T) are different. Hence, for

bi-cooperative games, symmetry holds only among the players in S, the players in T

and the players in N\(S T).

Intra-Coalition Symmetry axiom (IntraSym): For all permutations of N

such that (S) = S and (T) = T,

S,T

(i )(v

1

) =

S,T

i (v).

Proposition 6 Under Lin, Null, Mon, Eff and IntraSym , there exist coefficients

p+,S,Ta,b 0 for all a {0, . . . , s 1}, b {0, . . . , t}, and there exist coefficients

p,S,Ta,b 0 for all a {0, . . . , s}, b {0, . . . , t 1} such that for all i S T

S,Ti (v) =

K(ST)\{i }

pi ,S,T|KS|,|KT|[V(K {i }) V(K)],

where i = + if i S andi = if i T .

Inter-Coalition Symmetry (InterSym): Let i S and j T, and vi , vj be

two bi-cooperative games such that for all (S, T) Q((S T)\{i, j })

vi (S {i }, T) vi (S

, T) = vj (S, T) vj (S

, T {j }),

vi (S {i }, T {j }) vi (S

, T {j })

= vj (S {i }, T) vj (S

{i }, T {j }).

Then

S,Ti (vi ) =

S,Tj (vj ).

The axiom says that when the contribution of a player i S to a game vi is the negative

of that of a player j T to a game vj , then the incentive payoff for i shall be the

negative of the payoff for j .

Proposition 7 Under axioms Lin, Null, Mon, Eff, IntraSym and InterSym, thereexist coefficients p

S,Tk 0 for all k {0, . . . , s + t 1},

S,Ti (v) =

K(ST)\{i }

pS,Tk [V(K {i }) V(K)].

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Theorem 1 S,T satisfies axioms Lin, Null, Mon, IntraSym, InterSym and Eff if

and only if for all i N ,

S,T

i(v) =

K(ST)\{i }

k!(s + t k 1)!

(s + t)![V(K {i }) V(K)].

The vector of indices S,T(v) coincides with the Shapley value of cooperative game

V (see (3)).

Let us give now some properties of this value.

Property 1 If all players of T are null, then for all i S, S,Ti (v) equals the usual

Shapley value Si (w) of player i for game (S, w), where w(S) := v(S, ), S S.

Property 2 If all players ofS are null, then for i T, S,Ti (v) equals the usual Shapley

value Ti (w) of player i for game (T, w), where w(T) := v(, T), T T.

Combining previous results with Lin, we obtain the following property.

Property 3 Ifv is decomposable (i.e. v( S, T) = w+(S) w(T

) for all (S, T)

QS,T(N), where w+ and w are TU games) then

S,Ti (v) =

Si (w+) ifi S,

Ti (w) ifi T,

0 otherwise.

4 Mean prospect to a player

One interesting problem is to know the mean prospect of player i without the prior

knowledge of the other players intentions. In other words, one does not know in

advance coalitions S and T. Two options are basically possible. The first one is to

define the mean prospect to player i considering all possible choices (S, T) of the

other players. If player i chooses to be a positive contributor, her mean payoff is

+i (v) =1

3n1

(S,T)Q(N\{i })

S{i },Ti (v)

and, if she chooses to be a negative contributor, it is

i (v) =1

3n1

(S,T)Q(N\{i })

S,T{i }i (v).

The mean payoff of a player who chooses to be absent is always zero by definition.

Finally the mean prospect to player i if she has not made up her mind yet is

i (v) =1

3

+i (v) +

i (v)

.

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The second option is to compute the mean prospect not over all bi-coalitions (S, T)

but over the bi-coalitions that form a partition of N (i.e. S T = N). The idea is that

the only relevant decisions we are interested in are positive and negative contributors.

So each player is either a positive or a negative contributor. In the previous approach,

none of the three decisions is ruled out. The reason is that non-participation is a veryspecial situation, often not considered as a relevant option. In usual cooperative game

theory, all players are supposed to participate in the end. In our context, this means

that all players will have a non-zero level of participation.

If player i chooses to be a positive contributor, her mean payoff is

+i (v) =1

2n1

SN\{i }

S{i },N\(S{i })i (v).

By Theorem 1,

+i (v) =1

2n1

SN\{i }

TN\{i }

t!(n t 1)!

n!

[v((T S) {i }, T (N\ S)) v(T S, T (N\ S))]

=1

2n1 (K,L)Q(N\{i })dK,L [v( K {i }, L) v( K, L)],

where

dK,L =

SN\{i }

TN\{i }

K=ST , L=T\S

t!(n t 1)!

n!

=(k + l)!(n k l 1)!

n! AN\(KL{i })S=KA , T=KL1

=(k + l)!(n k l 1)!

n! 2n1kl .

Hence

+i (v) =

(K,L)Q(N\{i })

(k + l)!(n k l 1)!

2k+l n!(v( K {i }, L) v( K, L)). (9)

Similarly the mean prospect when i decides to be a negative contributor is

i (v) =

(K,L)Q(N\{i })

(k + l)!(n k l 1)!

2k+l n!(v( K, L {i }) v( K, L)). (10)

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Finally, the mean prospect when i has not yet made a decision (but will decide

between positive and negative contributors) is

i(v) =

1

2 +i (v) + i (v) . (11)5 Discussion

In this section, we show the relevance of our proposal to our previous examples.

Example 5 (Irrigation Network continued) Let us consider the bi-cooperative game

given in Example 2. When farmer 1 does not allow the pipes to cut through her field

(positive contributor), the cost share of the irrigation network is :

{1,2,3},1 (v) =

1

3,

{1,2,3},2 (v) =

{1,2,3},3 (v) =

47

6 7.83.

The construction of the well is necessary for any of the farmers to be able to get water.

Hence it is fair that the cost 1 of the well be divided equally among the farmers, hence

a cost 1/3 per farmer. Farmer 1 shall not pay more than her solo contribution to the

well, since she does not need any pipe for her own usage of water. Hence the price for

farmer 1 is 1/3.

When farmer 1 decides to allow the pipes to cut through her field (negative con-tributor), the cost share becomes

{2,3},{1}1 (v) =

14

3 4.66,

{2,3},{1}2 (v) =

{2,3},{1}3 (v) =

16

3 5.33.

We see that farmer 1 is given a significant amount of money as a compensation

for allowing her field to be cut by the pipes. Hence farmer 1 earns {1,2,3},1 (v)

{2,3},{1}1 (v) = 5 by switching from positive to negative option. One also notices that

this switch results in positive externalities on the two other players since the price theyhave to pay decreases from 476

downto 163

.

In Example 3, town A is actually the only settlement that has to make up its mind

about which role to adopt. Settlements B and C cannot become suppliers. For these

settlements, assigning a negative role to them is purely artificial. This example shows

that in practice there are players for whom one of the two roles is forbidden. Let N+

be the set of players who are allowed to be positive contributors, and N be the set of

players who are allowed to be positive contributors. We assume that every player can

choose to be absent. In Example 3, N+ = {A, B, C} and N = {A}. Let

QN+,N (N) :=

(S, T) Q(N) , S N+ and T N

.

Let us denote by vN+,N a bi-cooperative game restricted to QN+,N (N). By Prop-

osition 5, the value S,T(v) where S N+ and T N, requires information of v

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only in QN+,N (N) and thus ofvN+,N . Hence our value is well suited to the case of

forbidden roles for players.

Example 6 (Urban Water Supply continued) Example 3 is typical of the situation

where settlements B and C can only be positive contributors. Hence

ABC,A (v) =

1

2(pA + pA B pB ) ,

BC,AA (v) =

1

2(qA + qA B pB ) ,

ABC,B (v) =

1

2(pB + pA B pA) ,

BC,AB (v) =

1

2(pB + qA B qA) ,

ABC,C (v) =

BC,AC (v) = pC.

The last relation comes from the fact that the contribution of city C to the cost of

any coalition is always pC. By virtue of (1) and (2), ABC,A (v) >

ABC,B (v) and

BC,AA (v) <

ABC,A (v). The benefit for A when becoming a negative contributor is

12 (pA + pA B qA qA B ) > 0. The decision by A to allow pumping implies positive

externalities on B (and thus a decrease in the cost share for B)iffqA B qA pAB pA.

This condition means that the marginal cost for B in the presence ofA is smaller when

A decides to be a negative contributor instead of a positive contributor.

Example 7 (Multi-valued Ternary voting games continued) For any (S, T) Q(N),

v( S, T) is the degree to which the initial motion has been adopted. It seems reasonable

that v is monotone. By Corollary 1, S,Ti (v) 0 for all i S,

S,Ti (v) 0 for all

i T, S,Ti (v) = 0 for all i N\(S T). Value

S,Ti (v) can be interpreted as a power

index. However, classical power indices are non-negative numbers. Here a negative

value S,Ti (v) 0 corresponds to a voter who defeats the motion, and can quantify

the harmful power of that voter against the motion. One can define a power index in

situation (S, T) from S,Ti (v) as

S,Ti (v) =

S,Ti (v).Reproducing what has been said in Sect. 4, the mean power index of voter i when

she defends the motion is

+i (v) =1

2n1

SN\{i }

S{i },N\(S{i })

i (v)

=

(K,L)Q(N\{i })

(k + l)!(n k l 1)!

2k+l n!(v( K {i }, L) v( K, L)).

Similarly, the mean power index of voter i when she defeats the motion is

i (v) =1

2n1

SN\{i }

S,N\S

i (v)

=

(K,L)Q(N\{i })

(k + l)!(n k l 1)!

2k+l n!(v( K, L) v( K, L {i })).

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Hence the mean power index of voter i is

i (v) =1

2 +i (v) +

i (v)

=

(K,L)Q(N\{i })

(k + l)!(n k l 1)!

2k+l n!(v( K {i }, L) v( K, L {i })).

Note that these indices do not sum up to 1 in the general case, which is also the case

of some well-known power indices such as the Banzhaf one for usual voting games.

6 Comparison with previous works

Let us compare these solutions to previous proposals.

The first proposal is due to Felsenthal and Machover for ternary voting games

(Felsenthal and Machover 1997). We introduce it briefly, and for this we need some

definitions. A ternary roll-call R is a triplet R = (R , DR , ER ) composed of an order-

ing R of the voters, a coalition DR which contains all voters that are in favor of the

bill, and a coalition ER which contains all voters who are against the bill. The voters

in N\(DR ER ) are abstentionist. The set of all ternary roll-calls for the voters N is

denoted by TN. When a voter i is called he/she tells his/her opinion, that is to say in

favorifi DR , againstifi ER or abstention otherwise. The pivotPiv(v, R) is the

voter R (j ) called at position j , where j is the smallest index such that the result of

the vote does not change after. The following definition is then proposed ( Felsenthal

and Machover 1997):

FMi (v) =|{R TN, i = Pi v(v, R)}|

|TN|,

where |TN| = 3n n!. After some tedious computations, it can be shown that (Labreuche

and Grabisch 2006)

FMi (v) =

(S,T)Q(N\{i })

(s +

t

2)(v(S {i }, T) v(S, T))

(t +s

2)(v(S, T {i }) v(S, T))

with

t = 13n n!

tt=0

t

!(t t

)!t!

(n 1 t + t)!(t t)! 3tt .

A second proposal was made by the authors in Grabisch and Labreuche (2005),

which distinguishes the cases where i is a positive contributor (denoted by +) or a

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negative contributor (denoted by ):

+i (v) = SN\i(n s 1)!s!

n![v( S i, N\(S i )) v(S, N\(S i ))],

i (v) =

SN\i

(n s 1)!s!

n![v( S, N\(S i )) v(S, N\ S)].

A single value can be obtained putting LGi (v) := +i (v) +

i (v).

The most recent proposal seems to be due to Bilbao et al. (2007). It reads

Bi (v) = (S,T)Q(N\i ) pi(S,T)(v(S i, T)v(S, T)+p

i

(S,T)(v(S, T)v( S, T i )

with pi(S,T), pi(S,T)

given by

ps,t =(n + s t)!(n + t s 1)!

(2n)!2nst,

ps,t

=(n + t s)!(n + s t 1)!

(2n)!2nst.

Comparing with the mean prospect to a player (formulas (9) to (11)), all these

values differ from our proposal. The meaning of the power index of Felsenthal and

Machover is deeply rooted in voting theory, and its explicit expression in terms of v

is not simple. It nevertheless satisfies natural axioms such as linearity Lin, null player

Null, the classical symmetry axiom (which is stronger than IntraSym), and 2FMisatisfies an efficiency axiom Eff1 which is very similar to the classical one:

Efficiency 1 (Eff1):

i N i (v) = v(N, ) v(, N).

Imposing in addition to Lin, Null, the classical symmetry axiom, and Eff1, a fifthone expressing a symmetry between the positive and the negative parts, one obtains

the value LG (Labreuche and Grabisch 2006). An interesting feature of this value is

that only a small subset ofQ(N\i ) is used in the computation.

The proposal of Bilbao et al. also relies on Eff1. It is based on the following well

known formula of the classical Shapley value:

i (v) =

SN,Sic(S \i ) c([S, N])

n![v(S) v(S \i )] (12)

with c([T, S]) the number of maximal chains from T to S, c(S) := c([, S]),

c([S, S]) = c() = 1. In fact, the coefficients ps,t, ps,tare the relative numbers of max-

imal chains from (, N) to (N, ) passing by (S, T), (S i, T) and (S, T i ), (S, T)

respectively. Hence, B is a transposition of (12) to Q(N).

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An important remark is that Eff1 is satisfied by all previous proposals, while we

use Eff instead. It turns out that the crucial point lies here. Axiom Eff1 implies that

Q(N) has two remarkable elements which are (N, ) and (, N). These are the top and

bottom elements ofQ(N) when the order relation is the one implied by monotonicity:

(S, T) (S, T) iff S S and T T. This situation makes bi-cooperative gamesisomorphic to multi-choice games with three levels of participation (say 0, 1 and

2), since 3N endowed with the usual order on vectors is isomorphic to (Q(N), ). To

our opinion, bi-cooperative games are very different from 3-choice games.

By contrast, axiom Eff considers (, ) as a remarkable element in Q(N), as well

as all elements (S, N\S), S N. Then the suitable order relation is now the product

order, i.e. (S, T) (S, T) iffS S and T T, making (, ) the bottom element,

and all (S, N\ S), S N becoming top elements. We call this a bipolar extension

of 2N, a concept which is studied in detail in Grabisch and Labreuche (2005, sub-

mitted). This structure is much more in accordance with the meaning conveyed bybi-cooperative games.

7 Conclusion

We have given examples that justify the use of an extension of TU games where each

player who decides to participate in the game can participate positively or negatively.

Our concern is then how to fairly share the worth obtained by the players given their

choice. Several axioms are defined to characterize a value. This value is based on theShapley value. The positive contributors will receive a positive payoff whereas the

negative contributors will receive a negative payoff.

The value is the sharing rule that the benevolent dictator will apply on the players.

So if the game is a common knowledge, it can help the players in understanding what

is their best choice, i.e. being a positive or a negative contributor. In cost sharing prob-

lems, non-participation is not an option since all players want access to the resource

or the commons. In Example 1, farmer 1 has pros and cons for the two options. The

difference {1,2,3},1 (v)

{2,3},{1}1 (v) represents the financial gain for farmer 1 when

switching from positive contribution to negative contribution. If this figure is largerthan the estimated financial harm of allowing her field to be cut by a pipe, farmer 1

will accept being a negative contributor.

More generally, each player chooses her best response, given the strategy adopted

by the other players. Hence rational players will certainly adopt a Nash-like equilib-

rium. This suggests that the coalitions S and T can be determined endogenously rather

than given exogenously.

Appendix

Proof of Proposition 1: Consider S,T satisfying Lin. Let UK,L be defined by

UK,L (S, T) = 1 if S = K and T = L, UK,L (S

, T) = 0 otherwise. Then,

v =

(K,L)Q(N) v( K, L) UK,L . By Lin,

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S,Ti (v) =

(K,L)Q(N)

v( K, L) S,Ti (UK,L ).

Setting ai,S,TK,L :=

S,Ti (UK,L ), we obtain the wished result.

Proof of Proposition 2: Consider

S,Ti (v)

i N

satisfying Lin and Null. By Propo-

sition 1, there exists ci,S,TS,T

for (S, T) Q(N) such that

S,Ti (v) =

(S,T)Q(N)

ci,S,TS,T

v( S, T).

We write

S,Ti (v) =

(S,T)Q(N\{i })

ci,S,TS,T

v(S, T) +

(S,T)Q(N\{i })

ci,S,TS{i },T

v(S {i }, T)

+

(S,T)Q(N\{i })

ci,S,TS,T{i }

v(S, T {i }). (13)

Assume now that i is null for the bi-cooperative game v. Hence

S,Ti (v) = (S,T)Q(N\{i })

v( S, T) ci,S,TS,T

+ ci,S,TS{i },T

+ ci,S,TS,T{i } .

This relation holds for any bi-cooperative game v such that i is null for v, in particular

for all games satisfying for a given (K, L) Q(N\i ):

(S, T) Q(N\i ),

v(S, T) = UK,L (S, T),

v(S i, T) = v(S, T),

v(S, T i ) = v(S, T).

By Null, this gives for all (S, T

) Q(N\{i })

ci,S,TS,T

+ ci,S,TS{i },T

+ ci,S,TS,T{i }

= 0.

Consequently, formula (13) can be rearranged in such a way to give the wished form.

Yet ci,S,TS{i },T

is denoted by ai,S,TS,T

and ci,S,TS,T{i }

by bi,S,TS,T

.

Proof of Proposition 3: Consider

S,Ti (v)

i N

satisfying Lin and Dum. Axiom

Dum implies Null. Hence by Proposition 2, there exists for all i N, ai,S,TS,T

for

all (S, T) Q(N\{i }) , bi,S,TS,T

for (S, T) Q(N\{i }) such that (4) holds. Assume

that player i is dummy. Then by Dum(S,T)Q(N\{i })

ai,S,TS,T

+

(S,T)Q(N\{i })

bi,S,TS,T

= .

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This concludes the proof since one can take any R.

Proof of Proposition 4: Let i N. Under Lin and Null,

S,T

i(v) =

(S,T)Q(N\{i })

i,S,T

S,T v( S {i }, T) v(S, T)+

(S,T)Q(N\{i })

i,S,TS,T

v(S, T {i }) v(S, T)

.

Let v be a monotone bi-cooperative game such that

min(S,T)Q(N\{i })

min

v( S {i }, T)v( S, T),v(S, T)v(S, T {i })

=: > 0

Let (K, L) Q(N\{i }). Define v as follows: For all (S, T) Q(N\{i })

v(S, T) = v( S, T),

v(S {i }, T) =

v( K {i }, L) +

2if S = K and T = L ,

v( S {i }, T) otherwise,

v(S, T {i }) = v( S, T {i }).

By definition of, v is monotone. Moreover

S,Ti (v

) S,Ti (v) =

i,S,TK,L

2

.

By Mon, i,S,TK,L 0 for all (K, L) Q(N\{i }).

Define now v by for all (S, T) Q(N\{i })

v(S, T) = v(S, T),

v(S {i }, T) = v(S {i }, T),

v(S, T {i }) = v( K, L {i }) +2

if S = K and T = L ,

v(S, T {i }) otherwise,

v is monotone. Hence

S,Ti (v

) S,Ti (v) =

i,S,TK,L

2.

By Mon, i,S,TK,L 0 for all (K, L) Q(N\{i }).

The proof of Proposition 5 is based on the following lemmas

Lemma 1 Under Lin, Null, Mon and Eff , for all (K, L) Q(N) with (K, L)

{(S, T), (, )}i K

ai,S,TK\{i },L

i L

bi,S,TK,L\{i }

i N\(KL)

a

i,S,TK,L b

i,S,TK,L

= 0, (14)

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24/30


+

LN\K , i N\(KL)

bi,S,TK,L

=

LN\K , i Ka

i,S,TK\{i },L .

By Proposition 4, all terms ai,S,TK\{i },L are non-negative. Hence

i K, L N\ K, ai,S,TK\{i },L = 0.

Hence (14) becomes i L

bi,S,TK,L\{i }

i N\(KL)

bi,S,TK,L = 0 (16)

for all L N\ K. Let us prove by induction on l that for all l {0, . . . , n k 1}

and all L N\ K with |L| = l,

bi,S,TK,L = 0, i N\(K L).

By (16) applied to L = N\ K, we obtain

i N\K bi,S,TK,N\(K{i })

= 0. From Prop-

osition 4, all terms bi,S,TK,N\(K{i })

are non-positive. Hence for all L N \ K with

|L| = n k 1, bi,S,TK,L = 0 for i N\(K L). The induction assumption is thus

satisfied for l = n k 1.Assume that the induction assumption holds till index l. Let L N\K with |L| = l.

By (16) and the induction assumption

i L

bi,S,TK,L\{i } = 0.

From Proposition 4, we conclude that the induction assumption also holds for index

l 1.

Lemma 3 Assume that Lin, Null, Mon and Eff hold. Consider coefficients ai,S,TK,L

and bi,S,TK,L given by Proposition 4. Assume that(S, T) = (N, ). Then for all i N

and all (K, L) Q(N\{i })

|K| s ai,S,TK,L = 0, |K| > s b

i,S,TK,L = 0.

Moreover, for all i N and all (K, L) Q(N\{i })

(|K| < s a n d K {i } S) ai,S,T

K,L = 0,

(|K| s a n d K S) bi,S,TK,L = 0.

Proof Assume that (S, T) = (N, ). Let us show by induction on |K| that for all

K N with |K| > s, (15) holds. By Lemma 1, condition (i) of Lemma 2 is satisfied

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for all K N with |K| > s. Consider first the case K = N. Then condition (ii) in

Lemma 2 is vacously satisfied. By Lemma 2, then (15) holds for K = N. Assume

now by induction that (15) is satisfied for all K N with |K| = p {s + 2, . . . , n}.

Consider thus K N with |K| = p 1. By the induction assumption, condition (ii)

holds. Hence by Lemma 2, (15) is fulfilled for K. We have thus shown the first partof the lemma.

We now show by induction on p that for all p {s, . . . , 0}, all (K, L) Q(N)

and all i N\(K L)

(|K| = p 1 and K {i } S) ai,S,TK,L = 0.

(|K| = p and K S) bi,S,TK,L = 0.

For K = S, condition (i) of Lemma 2 is not fulfilled for L = T. Consider thus K

with |K| = s and K = S. By the first part of the lemma, condition (ii) holds. By

Lemma 1, (i) is also satisfied. Hence Lemma 2 can be applied, leading to the induction

assumption for p = s. Assume now that the induction assumption holds at index p.

Let K with |K| = p and K S. Since K S, there does not exists L N\ K and

i N\(K L) such that K {i } S. Thus condition (ii) in Lemma 2 is satisfied by

the induction assumption at p. By this lemma, we obtain the induction assumption at

index p 1. This proves the second part of the lemma.

Lemma 4 Consider coefficients ai,S,TK,L and b

i,S,TK,L given by Proposition 4. For L N

fixed, if the following two conditions are fulfilled

(i) (14) holds for all K N\L;

(ii) bi,S,TK,L = 0 for all K N\L and all i N\(K L);

then i L , K N\L , b

i,S,TK,L\{i } = 0,

K N\L , i N\(K L), ai,S,TK,L = 0.

(17)

Proof Similar to that of Lemma 2.

Lemma 5 Assume that Lin, Null, Mon and Eff hold. Consider coefficients ai,S,TK,L

and bi,S,TK,L given by Proposition 4. Assume that (S, T) = (, N). Then for all i N

and all (K, L) Q(N\{i })

|L| t bi,S,TK,L = 0, |L| > t a

i,S,TK,L = 0.

Moreover, for all i N and all (K, L) Q(N\{i })

(|L| < t a n d L {i } T) bi,S,TK,L = 0,

(|L| t a n d L T) ai,S,TK,L = 0.

Proof Similar to that of Lemma 3.

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We are now in position to show Proposition 5.

Proof of Proposition 5: S,T can be expressed as in Proposition 4. Let i N.

Let us show that for (K, L) Q(N\{i })

(K, L)

(K, L ) Q(N\{i }) , K {i } S and L T

ai,S,TK,L = 0, (18)

(K, L)

(K, L ) Q(N\{i }) , K S and L {i } T

bi,S,TK,L = 0. (19)

First of all, if (S, T) {(N, ), (, N)}, then by Lemmas 3 and 5, for (K, L)

Q(N\{i }), one has ai,S,TK,L = 0 if K {i } S or L T, and b

i,S,TK,L = 0 if K S or

L {i } T. Hence (18) and (19) are proved.Now when (S, T) = (N, ), (18) and (19) follow from Lemma 5. When (S, T) =

(, N), (18) and (19) follow from Lemma 3.

From (18) and (19), ifi N\(S T), all terms ai,S,TK,L and b

i,S,TK,L vanish. Hence

S,Ti (v) = 0.

Let i S. Then by (18) and (19), all the bi,S,TK,L terms vanish, and the only non-zero

terms of ai,S,TK,L are such that K S \{i } and L T. For R (S \{i }) T, we set

ci,S,TR := a

i,S,TRS,RT. Then we obtain

S,Ti (v) =

R(ST)\{i }

ci,S,TR [v((R S) {i }, R T) v(R S, R T)]

=

R(ST)\{i }

ci,S,TR [V(R {i }) V(R)].

By Proposition 4, ci,S,TR 0.

Let i T. Then by (18) and (19), all the ai,S,TK,L terms vanish, and the only non-zero

terms of bi,S,T

K,L are such that K S and L T\{i }. For R S (T\{i }), we setc

i,S,TR := b

i,S,TRS,RT. Then we obtain

S,Ti (v) =

R(ST)\{i }

ci,S,TR [v(R S, R T) v(R S, (R T) {i })]

=

R(ST)\{i }

ci,S,TR [V(R {i }) V(R)].

By Proposition 4, ci,S,T

R 0.

Proof of Proposition 6: By Lin, Null, Mon and Eff,

S,Ti (v) =

K(ST)\{i }

ai,S,TK [V(K {i }) V(K)].

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Moreover,

S,T (i )(v

1) =

K(ST)\{ (i )}a

(i ),S,TK [V(

1(K { (i )})) V(1(K)].

But V(1(K { (i )})) = V(1(K) {i }). Hence

S,T (i )(v

1) =

K(ST)\{i }

a (i ),S,T

(K)[V(K {i }) V(K)],

where K = 1(K). By IntraSym, one obtains

K (S T)\{i }, a (i ),S,T (K)

= ai,S,TK .

Consider first the case when (i ) = i . Since S and T are invariant by , we conclude,

for i fixed, that ai,S,TK depends only on the cardinality of K S and K T. Hence

ai,S,TK =: p

i,S,T|KS|,|KT|.

Now when (i ) = i , we obtain thus

pi,S,T|KS|,|KT| = a

i,S,TK = a

(i ),S,T (K)

= p (i ),S,T|KS|,|KT|.

We conclude that mapping i pi,S,T|KS|,|KT| depends only on whether i S or i T.

Proof of Proposition 7: Under Lin, Null, Mon, Eff and IntraSym

S,Ti (v) =

K(ST)\{i }

pi ,S,T|KS|,|KT|[V(K {i }) V(K)].

Let v0 and v1 be any two bi-cooperative games on (S T)\{i, j }. Let vi and vj be

two bi-cooperative games on N. Assume that vi can take any value onQ((S T)\{i }).

Then define vi on Q(S T)\Q((S T)\{i }) by

vi (S {i }, T) = vi (S

, T) + v|T{j }|(S, T \{j })

for all (S, T) Q((S T)\{i }). Similarly vj can take any value on Q((S T)\{j }).

Then define vj on Q(S T)\Q((S T)\{j }) by

vj (S, T {j }) = vj (S

, T) v|S{i }|(S \{i }, T)

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for all (S, T) Q((S T)\{j }). Then, for all (S, T) Q((S T)\{i, j }),

vi (S {i }, T) vi (S

, T) = v0(S, T) = vj (S

, T) vj (S, T {j }),

vi (S {i }, T {j }) vi (S

, T {j })

= v1(S, T) = vj (S

{i }, T) vj (S {i }, T {j }).

Hence

S,Ti (vi ) =

K(ST)\{i,j }

p+,S,T|KS|,|KT|v0(K S, K T)

+

K(ST)\{i,j }

p+,S,T|KS|,|KT|+1v1(K S, K T)

and

S,Tj (vj ) =

K(ST)\{i,j }

p,S,T|KS|,|KT|v0(K S, K T)

K(ST)\{i,j }

p,S,T|KS|+1,|KT|v1(K S, K T).

Since this holds for all v0 and v1, we have, by InterSym, for all a {0, . . . , s 1},

b {0, . . . , t 1}

p+,S,Ta,b = p,S,Ta,b ,

p+,S,Ta,b+1 = p

,S,Ta+1,b .

Thus ifa + b s + t 2

p+,S,Ta,b = p

,S,Ta,b = p

+,S,Ta1,b+1 = = p

,S,T1,a+b1 = p

+,S,T0,a+b .

We set thus

pS,Ta+b := p+,S,Ta,b = p

,S,Ta,b .

Moreover, we set for a + b = s + t 1

pS,Ts+t1 := p

+,S,Ts1,t = p

,S,Ts,t1 .

Hence for any v,

S,Ti (v) = K(ST)\{i } p

S,T|KS|+|KT|[V(K {i }) V(K)]

=

K(ST)\{i }

pS,Tk [V(K {i }) V(K)].

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Proof of Theorem 1: The if part is straightforward and left to the reader.

Under the above axioms,

i N S,Ti (v) = i N K(ST)\{i } p

S,Tk [V(K {i }) V(K)]

=

LST

V(L)

i N, K(ST)\{i }

L=K{i },i L

pS,Tk

i N, K(ST)\{i }

L=K,i L

pS,Tk

=

LST

V(L)

i L

pS,Tl1

i (ST)\L

pS,Tl

=

LST

l pS,Tl1 (s + t l) pS,Tl V(L).Axiom (E) implies that the coefficient of V(S T) should be 1, that of V() should

be -1, and all others being 0. This yields the following relations:

(s + t) pS,Ts+t1 = 1

l pS,Tl1 (s + t l) p

S,Tl = 0 for all l {1, . . . , s + t 1}.

Hence

pS,Tl =

l!(s + t l 1)!

(s + t)!.

Proof of Property 1: If all players of T are null, then for (S, T) QS,T(N), T =

{i1, . . . , it }

v(S, T) = v(S, {i1, . . . , it }) = v(S, {i1, . . . , it1})

= = v(S, {i1}) = v(S, ) =: w(S).

References

Aumann RJ, Drze JH (1974) Cooperative games with coalition structures. Int J Game Theory 3:217237Bilbao JM, Fernndez JR, Jimnez N, Lpez JJ (2007) The value for bi-cooperative games. Ann Oper Res

(to appear)

Bilbao JM, Fernandez JR, Jimnez Losada A, Lebrn E (2000) Bicooperative games. In: Bilbao JM (ed)

Cooperative games on combinatorial structures. Kluwer, Dordrecht

Driessen T (1988) Cooperative Games, solutions and applications. Kluwer, Dordrecht

Felsenthal D, Machover M (1997) Ternary voting games. Int J Game Theory 26:335351

123


30/30


Grabisch M, Labreuche Ch (2002) Bi-capacities for decision making on bipolar scales. In: EUROFUSE

Workshop on Informations Systems, pp 185190, Varenna, Italy

Grabisch M, Labreuche Ch (2005) Bi-capacities. Part I: definition, Mbius transform and interaction. Fuzzy

Sets Syst 151:211236

Grabisch M, Labreuche Ch (2005) A general construction for unipolar and bipolar interpolative aggrega-

tion. In: 4th Conf. of the Eur. Soc. for Fuzzy Logic And Technology (EUSFLAT) and 11th LFA Conf.,pp 916921, Barcelona, Spain

Grabisch M, Labreuche Ch (submitted) Bi-capacities: towards a generalization of cumulative prospect

theory. J Math Psychol

Norde HW, Hamers H, Miquel S, van Velzen S (2003) Fixed tree game with repeated players. Discussion

Paper Tilburg University, 87

Henriet D, Moulin H (1999) Traffic-based cost allocation in a network. Rand J Econ 87:275312

Hsiao CR, Raghavan TES (1993) Shapley value for multichoice cooperative games, I. Games Econ Behav

5:240256

Labreuche Ch, Grabisch M (2006) Axiomatization of the Shapley value and power index for bi-cooperative

games. Technical report, CERMSEM, Universit Paris I

Labreuche Ch, Grabisch M (2006) Generalized Choquet-like aggregation functions for handling bipolar

scales. Eur J Oper Res 172:931955

Megiddo N (1974) On the monotonicity of the bargaining set, the kernel and the nucleolus of a game. SIAM

J Appl Math 27:355358

Monderer D, Samet D (2001) variations on the shapley value. In: Aumann RJ, Hart S (eds) HandBook of

Game Theory No. III, Chap 54

Moulin H (1995) Cooperative microeconomics. Princeton University Press, Princeton

Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the Theory

of Games, vol II, number 28 in Annals of Mathematics Studies. Princeton University Press, Princeton,

pp 307317

Sprumont Y (2000) Coherent cost-sharing rules. Games Econ Behav 33:126144

Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The shapley value. Essays in Honor of

Lloyd S. Shapley, pp 101119. Cambridge University Press, Cambridge

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