a value for multichoice games

Mathematical Social Sciences 40 (2000) 341–354www.elsevier.nl / locate /econbase

A value for multichoice gamesa b ,*Emilio Calvo , Juan Carlos Santos

a ´ ´Departamento de Analisis Economico, Universidad de Valencia, Campus dels Tarongers,Avinguda dels Tarongers s /n, Edificio Departamental Oriental, 46022 Valencia, Spain

b ´ ´Departamento de Economıa Aplicada IV, Universidad del Paıs Vasco /E.H.U.,Avda. Lehendakari Aguirre 83, 48015 Bilbao, Spain

Received 1 January 1999; received in revised form 1 October 1999; accepted 1 October 1999

Abstract

A multichoice game is a generalization of a cooperative TU game in which each player hasseveral activity levels. We study the solution for these games proposed by Van Den Nouweland etal. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and relatedsolution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41,289–311]. We show that this solution applied to the discrete cost sharing model coincides with theAumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods toshare joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that theAumann-Shapley value for continuum games can be obtained as the limit of multichoice valuesfor admissible convergence sequences of multichoice games. Finally, we characterize this solutionby using the axioms of balanced contributions and efficiency. 2000 Elsevier Science B.V. Allrights reserved.

Keywords: Multichoice games; Shapley value; Aumann-Shapley value; Balanced contributions; Cost alloca-tion

1. Introduction

One of the most interesting applications of the Cooperative Game Theory has been1done in the setting of allocating costs. This kind of problem can be formulated as

follows: let N 5 h1,2, . . . ,nj be a set of projects, products, or services that can beprovided jointly by some organization. Let c(S) be the cost of providing the items in S

*Corresponding author. Tel.: 134-94-601-3806; fax: 134-94-447-5154.E-mail address: [email protected] (J.C. Santos).1For comprehensive surveys about this topic the reader is referred to Tauman (1988) and Young (1994).

0165-4896/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PI I : S0165-4896( 99 )00054-2

342 E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 –354

jointly, for each subset S # N. The function c is called a discrete cost function, or acost-sharing problem (alternatively, c can be interpreted as a production function thatgives the output for any coalition of agents, or factors). Modelled in this way, a costallocation problem can be considered as a cooperative game, with c being itscharacteristic function. The Shapley (1953) value provides an efficient and fair costallocation mechanism for sharing costs between products (or factors).

Another framework is considered when the output can vary continuously. Here theproblem can be modelled as a non-atomic game with a continuum of n types of players:each good i, produced at level q , is represented by q mass of players of type i. Thei i

Aumann and Shapley (1974) value for this non-atomic game gives a cost-sharing2method for this type of continuum problems.

In this setting it is assumed that commodities are totally divisible goods and thenmagnitudes of goods can be measured with real numbers. This is an appropriateapproach for cases such as petroleum products, various agricultural products (cereals,wine, olive oil, fruits, etc.), chemical products, etc. Nevertheless, there are many otherstypes of goods for which this is not possible (cars, machines, buildings, etc.). This familyof indivisible goods are only available in finite integer amounts. This is the kind ofsituation that we want to cover in this paper: cost allocation problems in which productscan be provided (or factors used) at a certain finite number of levels. A survey of thisproblem and different solutions for it can be found in Moulin (1995). In that paper costsharing methods for these problems were compared, the Shapley-Shubik method(Shubik, 1962), the discrete Aumann-Shapley method (Moulin, 1995), the serial costsharing method (Moulin and Shenker, 1992) and the pseudo-average cost (Moulin,1995). Recently, Sprumont and Wang (1998) have characterized the discrete Aumann-Shapley method using axioms that involves only economic terms.

The appropriate game-theoretic tool for modelling this setting are the so calledmultichoice games. These are games in which each player has a certain finite number ofactivity levels at which he can play. In general, different players may have differentpossible levels, and the worth that a coalition can obtain depends on the level at whicheach player in the coalition has decided to participate. Hsiao and Raghavan (1992, 1993)introduced games in which all players have the same number of activity levels. Theydefined extended Shapley values by using weights on activity levels, each level havingthe same weight for all players, and provided axiomatic characterizations of thecorresponding values. Van Den Nouweland et al. (1995) considered the more generalcase with different numbers of activity levels, and extended the notions of core,dominant core and Weber set. Also they proposed an alternative extension for theShapley value based on an extension of the probabilistic formula by orders; but they didnot give additional support for this extension. In Van Den Nouweland (1993) an exampleis given of a multichoice game for which this value is not equal to any of the values ofHsiao and Raghavan; and other alternative proposals for the multichoice value are alsoshown. Recently, Klijn et al. (1998) have studied a new solution to multichoice games.

2This application of the theory of values of non-atomic games stems from the work of Billera et al. (1978).

E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 –354 343

This solution is based on the work of Derks and Peters (1993) on the extended Shapleyvalue.

Our goal is to show, first, that the value notion of Van Den Nouweland et al. (1995)corresponds to the discrete Aumann-Shapley method proposed by Moulin (1995).Second, the Aumann-Shapley value for continuum finite type games can be foundasymptotically by means of the multichoice value using admissible sequences of discretemultichoice games which converge to the continuum game. Third, an axiomaticcharacterization is offered of the multichoice value which is consistent with theaxiomatic characterization of the Aumann-Shapley value for continuum finite typegames.

Following this introduction, Section 2 is devoted to some preliminary definitions andnotations. In Section 3 we present the solution for multichoice games. In Section 4, westate and prove the limit theorem. Section 5 is devoted to the axiomatic characterizationof the multichoice value, and finally, in Section 6 we offer some concluding remarks.

2. Preliminaries

NWe start by defining the general model. We say that a subset L of R is full1

dimensional if hl [ L: l . 0 for all i [ Nj ± [. The zero vector (0, . . . ,0) will bei

denoted by u.

Definition 2.1. A cooperative multilevel game is a triple (N,L,v), where N5h1, . . . ,nj isNa finite set of players, L is a full dimensional subset of R , u [ L, and v is a function1

from L into R, with v(u ) 5 0.

The interpretation is the following: for each l [ L, l means the activity level ati

which player i participates in the game. The vector of zero levels is always possible; wealso assume that all players can play the game simultaneously. Given l [ L, if l ± 0 wei

will say that i is an active player at l, and the set of all active players at l will bedenoted by A(l). The function v: L → R, gives for every action l the worth that theplayers can obtain when each player i plays at level l . The function v itself will also bei

called a multilevel game, or a game, on (N,L). The set of all multilevel games on (N,L)(N,L )is denoted by G .

(N,L ) (N,L ) (N,L )Definition 2.2. Given (N,L) and a subset Q of G , a solution on Q is a(N,L ) S A( l)function c : Q 3 L\u → < R , where c(v,l) [ R .S#N

The number c(v,l)(i) represents the per unit payoff (prices) that player i receives,hence l ? c(v,l)(i) is the total payoff that player i receives at (v,l), with solution c.i

S N S SFor S # N, let e be the vector in R satisfying e 5 1 if j [ S, and e 5 0 if j [⁄ S.j ji 1i i 2iGiven a vector l we denote the vector l 1 e by l , and l 2 e by l . For every two

N Nvectors x,y [ R , x # y means x # y for all i [ N. For S # N and x [ R , we ofteni i

write x(S) instead of o x .i[S i


Two subfamilies of multilevel games have already been studied in literature. The firstNone arises when L 5 h0,1j . Here, each player can choose to participate, 1, or not, 0, in

the game. In this classical setting, a bijection can be established between action vectorsl and active coalitions A(l); a level vector m belongs to L if and only if there exists

SS # N such that e 5 m. Then, in this context, we can often write S # N instead of m [ LS Nor e [ L. We will denote this subfamily by G . The well known Shapley (1953) value

solution, is defined by

uSu! ? uT u 2 uSu 2 1 !s d]]]]]]f(v,T )(i) 5 O ? v(S < hij) 2 v(S) ,f g

uT u!S#T \hi j

N Nfor (v,T ) [ G 3 2 \[ and i [ T.This value can be rewritten, using our notation, as follows:

uA(m)u! ? uA(l)u 2 uA(m)u 2 1 !s d 1i]]]]]]]]]f(v,l)(i) 5 O ? v(m ) 2 v(m)f guA(l)u!2iu #m #l

N Nfor (v,l) [ G 3 h0,1j \u and i [ A(l).NThe second subfamily arises when L is a comprehensive subset of R (by1

comprehensive, we mean that if z [ L, then for all u # x # z, it holds that x [ L). Now,for every pair (v,z), each active player i has a continuum of admissible actions:0,z , R . We will call these multilevel games continuum games. In this context, forf gi 1

every z [ L\u, (v,z) can be represented by a non-atomic game (see Mirman et al., 1982)

on (I,C), where I 5 i 2 1, i for each i [ N, I 5 < I , and C is the s-field of Borelf gi i[N iNsubsets of I. Given T [ C, let z(T ) [ R such that z T 5 z ? j T > I for each i [ N,s d s di1 i i

where j denotes the Lebesgue measure. The non-atomic game f is defined by(v,z)

f T 5 v z T for each T [ C. When v has continuous first partial derivatives on L (its d s s ddv,zs d

is understood that the derivatives are one sided when z belongs to the boundary of L), itholds that f belongs to the pNAD class of non-atomic games (see Mirman et al.,(v,z)

1982) and then, the Aumann and Shapley (1974) value, C, for every I , reduces toi

(Cf )(I ) 5 z ? p(v,z)(i) ,(v,z) i i

for each z [ L\u and i [ N, where

1

≠v]p(v,z)(i) 5E (tz) dt .≠xi

0

The number p v,z i is the per capita payoff (price) of the set of players I and is thes ds d i

Aumann-Shapley price of player i. In order to determine C it is sufficient to specify pbecause in the game f the players of each type i (the set I ) are symmetric with(v,z) i

(N,L )respect to f . Hence, we denote by CG the family of continuum games v on (N,L),(v,z)

where v has continuous first partial derivatives on L; and we define, for each (v,z) [(N,L )CG 3 L\u, the Aumann-Shapley value F at (v,z) as


1

≠v]F v,z i 5E tz dt , (i [ A(z)) .s ds d s d≠xi

0

N NMultichoice games appear when L is a comprehensive subset of N 5 (h0j < N) , i.e.,1N 3if l [ L then for all m [ N such that u # m # l, it holds that m [ L.1

NThe set of all multichoice games on (N,L), where L is a comprehensive subset of N ,1(N,L )is denoted by MG . Note that a multichoice game on (N,L) is a finite cooperative TU

Ngame on N in the particular case L 5 h0,1j .

3. The multichoice value

In this section we show that the solution for multichoice games proposed by Van DenNouweland et al. (1995) applied to discrete cost problems coincides with the Aumann-Shapley method proposed by Moulin (1995).

The definition by Nouweland et al. is based on a generalization of the probabilistic(N,L )formula by orders of the Shapley value. To define this solution, let (v,l) [ MG 3

L\u and assume that level l forms step by step, starting from level zero, and that at eachstep the level of one of the players is increased by 1, up to l. There are l(N) steps in thisprocedure. Suppose that at each step the player that increases his level receives themarginal contributions of this step and suppose that all orders from level u up to l have

4the same probability. Then the per unit expected marginal contribution of each player isthe value proposed by Nouweland et al. We call this solution the multichoice value and

5we denote it by w(v,l)(i).Now we summarize the Aumann-Shapley method proposed by Moulin (1995). In

Norder to do this, let q 5 q , . . . ,q [ N and C a cost function defined on the intervals d1 n 1Nof N , 0,q . Given the demand profile q 5 q , . . . ,q consider the cooperative gamef g s d1 1 n

with q 1 ? ? ? 1 q players where each player is a particular unit of a particular good.1 n

Then the cost sharing of a particular good is the sum of the Shapley value of all units ofthis particular good.

Obviously, this method is adaptable to multichoice games and hence it determines asolution for these games. Here, we formalize this procedure since it will be used in thiswork.

N (N,L )Given a set N5h1, . . . ,nj, L # N , a multichoice game v [ MG and l [ L\u, let1l

D be a set of replica players defined as:

3N denotes the set of positive integers.

4This procedure can be interpreted as follows: Consider the process of picking (without replacing) l(N)coloured balls from a box; l balls of the same colour for each player i on A(l), and with different colours fori

different players. When a ball is picked, it increases the level of the player associated with its colour. Then,every order in which balls are chosen yields an order in which levels are increased. When all balls that remainin the box are equally likely to be chosen, all orders have the same probability to happen. We would like to

`thank Herve Moulin for pointing out this interpretation to us.5Actually, the solution proposed by Van Den Nouweland et al. is l ? w(v,l)(i).i


l lD 5 < D ,i

i[A( l)

lwhere D 5 hi , . . . ,i j for any i [ A(l).i 1 lilNow, for any B # D define the level vector l(B) [ L as follows:

luB > D u , if i [ A(l) ,il (B) 5Hi 0, if i [ N\A(l) .ll DThen, we define the replica game R v [ G by

lR v(B) 5 v(l(B)),

lfor every B # D .The next result shows that the multichoice value w (proposed by Nouweland et al.,

1995) coincides with the solution proposed by Moulin (1995). Notice that this is thesame strategy as described in Section 2 to obtain the Aumann-Shapley value F on

(N,L )CG .

(N,L )Proposition 3.1. For every multichoice game v [ MG and l [ L\u it holds that

1l l l l l]w(v,l)(i) 5 f(R v,D )(i ) 5 ? f(R v,D )(D ) , (i [ A(l)) ,j ili

where w is the multichoice value and f is the Shapley value.

Proof. It is straightforward taking into account the probabilistic formula with orders ofthe Shapley value. h

Remark 3.2. An alternative formula for the multichoice value is given by

2ia(N)! ? l(N) 2 a(N) 2 1 !s d l 1ij]]]]]]]]w(v,l)(i) 5 O ? P ? v(a ) 2 v(a)f gS Dal(N)! j[A( l)2i ju #a #l

(N,L )for any v [ MG , l [ L\u and i [ A(l).

4. Going to the limit

(N,L )As we have seen in the preliminaries, the Aumann-Shapley value F on CG is(N,L )defined for each (v,z) [ CG 3 L\u by

1

≠v]F(v,z)(i) 5E (tz) dt , for every i [ A(z) .≠xi

0

We will show in this section that F(v,z) can be obtained by taking an asymptotict t tapproach by means of a sequence w(v , l ) of multichoice values. These games, v , arez z


a discrete version of the original one, allowing only a finite number, instead of acontinuum, of activity levels for players.

To see this, we start by identifying levels l with admissible amounts of z . Ani it t Nadmissible sequence of partitioning vectors hl j, where l [ N , is defined by

t 11 t(i) l 5 a ? l , with a [ N , (t [ N, i [ N) .i t i i t i

t(ii) hl j → ` , when t → ` , (i [ N) .i

t t t NGiven an admissible sequence hl j, for every l we denote by L the subset of N1t tsuch that m [ L if and only if m # l .

(N,L ) tNow, given a pair (v,z) [ CG 3 L\u and an admissible sequence hl j, for everytt t (N,L )

l we define the multichoice game v [ MG asz

z z1 nt t] ]v (m) 5 v m ? , . . . ,m ? , (m [ L ) .t tz S 1 n Dl l1 n

(N,L ) tTheorem 4.1. For all (v,z) [ CG 3 L\u and all admissible sequences hl j, it holdsthat

t t tlim l ? w(v , l )(i) 5 z ? F(v,z)(i) , (i [ A(z)) ,i z it →`

where w is the multichoice value and F is the Aumann-Shapley value.

Proof. The proof has four steps.

(N,L ) tSTEP 1: Given (v,z) [ CG 3 L\u and an admissible sequence hl j, for every pairtt t l t(v ,l ) we build the replicated game R v as in Proposition 3.1. Therefore we knowz z

thatt t tt t t l t l l

l ? w(v ,l )(i) 5 f(R v ,D )(D ) , (i [ N,t [ N) , (1)i z z i

tlwhere D 5 hi , . . . ,i j and f is the Shapley value.ti 1 l i(N,L )STEP 2: Given (v,z) [ CG 3 L\u we saw, in Section 2, that

(Cf )(I ) 5 z ? F(v,z)(i) , (i [ A(z)) , (2)(v,z) i i

where f is the non-atomic game on (I,C), with I 5 i 2 1,i and I 5 < I , asf g(v,z) i i[N i

defined in Section 2.tSTEP 3: The admissible sequence hl j induces a partition of I into a finite collection

tt t l t t tP 5 hP : i [ D , i [ Nj of disjoint measurable sets, where P # I , j P 5 1/ls di k i i i i ik k k

t 11 t t t 11and P refines to P (i.e., each member of P is a union of members of P ), fortlevery i [ D , i [ N and t [ N. This allows us to build a finite ‘quotient’ gamek i

tP 6v [ G defined bytP

t6 t lNote that uP u 5 uD u.


t tv (T ) 5 f < P , for each T # P .tP (v,z) jS DtP [Tj

tl tThen the quotient game v coincides with the replicated game R v of STEP 1,tP z

because

t t tv (T ) 5 f < P 5 v z ? j <P > I , . . . ,z ? j <P > ItP (v,z) j 1 j 1 n j nS D S S D S DDt t tP [T P [T P [Tj j j

t tl t l t5 R v h j [ D : P [ T j .s dz j

Hence,t t t tl t l l t t l

f(R v ,D )(D ) 5 f(v ,P )(hP :i [ D j) , (i [ N) . (3)tz i P i k ik

(N,L )STEP 4: When (v,z) [ CG 3 L\u, its associated non-atomic game f belongs to(v,z)

the space ASYMP ( f [ pNAD and Proposition 43.13 in Aumann and Shapley, 1974).(v,z)

This means that under a suitable sequence of partitions of I, the Shapley value for thequotient games associated with f gives, at the limit, the Aumann-Shapley value for(v,z)

t tf . In our case, the sequence of partitions hP j built in STEP 3 from hl j satisfies the(v,z)

conditions for belonging to that family; this means thattt t llim f(v ,P ) hP :i [ D j 5 (Cf )(I ) , (i [ N) . (4)t S DP i k i (v,z) ikt →`

The proof follows from (1), (2), (3) and (4). h

Remark 4.2. The result of Theorem 4.1 is also true under the condition that f [(v,z)

pNAD (see Mirman et al., 1982). This includes, for example, the case in which v is apiecewise continuously differentiable function (see Samet et al., 1984).

Remark 4.3. The solution for multichoice games proposed by Hsiao and Raghavan(1993) was extended to continuous games by Hsiao (1995). This extension does notcoincide with the Aumann-Shapley value, and in that paper the author does not provethat the solution for continuous games can be regarded as the limit of values foradmissible convergence sequences of multichoice games. The solution for multichoicevalues proposed by Klijn et al. (1998) has not been extended to continuous games.

5. Axiomatic characterization

In this section we offer an axiomatic characterization of the multichoice value. First,we extend the potential approach started by Hart and Mas-Colell (1989) for finite TUgames to multichoice games. In that paper, they proved that the Shapley value and the

N NAS prices can be obtained as the gradient of a potential function on G 3 h0,1j and(N,L )CG 3 L, respectively.


(N,L )Definition 5.1. Let P be a function P:MG 3 L → 5. For all l [ L\u and activeplayers i [ A(l), we define the marginal contribution of player i with respect to P at(v,l) as

i 2iD P(v,l) 5 P(v,l) 2 P(v,l ) .

(N,L )Definition 5.2. The function P is said to be a potential function on MG if it satisfies

i (N,L )O l ? D P(v,l) 5 v(l), s(v,l) [ MG 3 L\ud , (PM.1)ii[A( l)

(N,L )P(v,u ) 5 0, (v [ MG ) . (PM.2)

(N,L )Theorem 5.3. There is a unique potential function on MG .

Proof. For l ±u, formula PM.1 can be rewritten as

1 2i]]P(v,l) 5 ? v(l) 1 O l ? P(v,l ) . (5)iF Gi[A( l)O li

i[A( l)

(N,L )Taking a game v [ MG and starting from P(v,u ) 5 0, (5) determines P(v,l)recursively. This proves the existence of P, and moreover that P(v,l) is uniquelydetermined by PM.1, or (5), applied to (v,m) for all u # m # l. h

Taking into account Proposition 3.1, it follows immediately that:

(N,L )Corollary 5.4. The multichoice value coincides with the solution w on MG definedby

i (N,L )w(v,l)(i) 5 D P(v,l) , s(v,l) [ MG 3 L\u, i [ A(l)d ,

where P is the potential function.

Remark 5.5. An alternative expression for the potential is given by

a(N) 2 1 ! ? l(N) 2 a(N) !s d s d li]]]]]]]]P(v,l) 5 O ? P ? v(a) .S Dal(N)! ii[A( l)u #a #l

a ±u

(N,L )for any v [ MG and l [ L\u.

Now, we will give an axiomatic characterization of this solution. In essence, it saysthat the value is an efficient rule which equalizes the marginal contributions between theplayers in the game.

(N,L )Let w be a solution on MG . We say that w satisfies efficiency if, for every(N,L )(v,l) [ MG 3 L\u, it holds that


O l ? c(v,l)(i) 5 v(l) .ii[A( l)

(N,L )We say that c satisfies balanced contributions if, for every (v,l) [ MG 3 L\u,with A(l) $ 2, it holds thatu u

2j 2ic(v,l)(i) 2 c(v,l )(i) 5 c(v,l)( j) 2 c(v,l )( j) ,

for each hi, jj # A(l), i ± j.(N,L )The efficiency axiom is the translation to MG of the cost-sharing principle. The

balanced contributions axiom is a fair-marginal rule. For a better understanding of itsmeaning we refer back to the cost allocation framework. Assume we have a rule w inorder to allocate the production cost of a bundle of n goods. In this case c(v,l)(i) 2

2jc(v,l )(i) is the per unit cost variation in the production of l units of i when the leveli

of production of j diminishes in one unit. In other words, this term can be interpreted asj’s marginal contribution to i’s unit cost at level l of production. c will be a fair rulewhen these marginal cost contributions are equal for every pair of goods in A(l). For ageneric game this rule implies that the marginal per capita value contributions betweenpairs of players must be equal. This axiom was introduced in Myerson (1980), and with

Nefficiency characterizes the Shapley value on G (see also Hart and Mas-Colell (1989),(N,L )Theorem 3.4). The next theorem extends this result to MG .

(N,L )Theorem 5.6. A solution c on MG satisfies efficiency and balanced contributions ifand only if c5w, where w is the multichoice value.

Proof. It is straightforward to check that w satisfies efficiency. To see balancedcontributions, note that from Corollary 5.4, we have

2j i i 2jw(v,l)(i) 2 w(v,l )(i) 5 D P(v,l) 2 D P(v,l )

2i 2j 2hi, j j5 P(v,l) 2 P(v,l ) 2 P(v,l ) 1 P(v,l )

j j 2i 2i5 D P(v,l) 2 D P(v,l ) 5 w(v,l)( j) 2 w(v,l )( j) ,

2hi, j j i jwhere l 5 l 2 e 2 e . Hence, w satisfies the axioms.

(N,L )Now let c be a solution on MG that satisfies balanced contributions and(N,L )efficiency. We define the function Q:MG 3 L → R as follows:

(N,L )(i) Q(v,u ) 5 0, (v [ MG ) ,

2i (N,L )(ii) Q(v,l) 5 Q(v,l ) 1 c(v,l) ((v,l) [ MG 3 L\u, i [ A(l)) .

To prove that Q is well defined, let hi, jj # A(l); by induction hypothesis, we havethat


2i 2jQ(v,l ) 1 c(v,l)(i) 2 Q(v,l ) 2 c(v,l)( j)2i 2j

5 Q(v,l ) 2 Q(v,l ) 1 c(v,l)(i) 2 c(v,l)( j)2hi, j j 2hi, j j 2j

5 Q(v,l ) 2 Q(v,l ) 1 Q(v,l ) 2 Q(v,l ) 1 c(v,l)(i) 2 c(v,l)( j)2i

2i 2j5 c(v,l )( j) 2 c(v,l )(i) 1 c(v,l)(i) 2 c(v,l)( j) ,

and the last expression is zero because c satisfies balanced contributions. Then, Q is(N,L )well defined. Furthermore, by definition Q(v,u )50, for all v [ MG , and

i 2iO l ? D Q(v,l) 5 O l ? Q(v,l) 2 Q(v,l ) 5 O l ? c(v,l)(i) 5 v(l)f gi i ii[A( l) i[A( l) i[A( l)

because c is efficient. From Theorem 5.3 we conclude that Q is the potential function on(N,L ) iMG . Furthermore, c(v,l)(i) 5 D Q(v,l) and then c 5 w. h

(N,L )Remark 5.7. The translation of these two properties to a solution c on CG is as7follows:

(N,L )Efficiency: For any v,z [ CG 3 L\u it holds thats d

O z ? c(v,z)(i) 5 v(z) .ii[A(z)

(N,L )Balanced contributions: For any v,z [ CG 3 L\u, all i, j # A(z) and a continu-s d h jously differentiable solution C it holds:

≠c(v, ? )(i) ≠c(v, ? )( j)]]] ]]](z) 5 (z) .

≠x ≠xj i

(N,L )These two properties also characterize the Aumann-Shapley value on CG (seeCalvo and Santos, 1997, or Ortmann, 1995 and Ortmann, 1998), that is, a continuously

(N,L )differentiable solution c on CG satisfies efficiency and balanced contributions if, andonly if c 5 F.

6. Concluding remarks

We will show here that the multichoice value coincides with the Aumann-Shapleyvalue of a continuum game that is a sort of multilinear extension of the initial game.Formally:

(N,L ) lGiven (v,l) [ MG 3 L\u, let E v be the function defined by:

ll a l 2ai i i iE v(x) 5 O P ? x ? (1 2 x ) ? v(a) ,S DF Gi iaii[A( l)u #a #l

Nfor every x [ 0,1 .f g

7An axiomatic characterization of the Aumann-Shapley value in the context of cost allocation problems isgiven in the works of Billera and Heath (1982) and Mirman and Tauman (1982).


lNote that if l 5 1 for all i [ N, then v is a classic finite TU game and E v is thei

multilinear extension of v (see Owen, 1972).

(N,L )Theorem 6.1. For all (v,l) [ MG 3 L\u and i [ A(l), it holds:

1 l]w(v,l)(i) 5 FsE v,1d(i) ,li

where w is the multichoice value and F is the Aumann-Shapley value.

Proof. Taking into account that, for finite TU games, the Shapley value coincides withthe Aumann-Shapley value of their multilinear extensions, the result follows easily byapplying Proposition 3.1. h

(N,L )Corollary 6.2. For all (v,l) [ MG 3 L\u it holds that,

1

1 l]P v,l 5E E v t, . . . ,t dt ,s d s dt0

where P is the potential function.

Remark 6.3. Although at first glance condition PM.1 in Section 5 resembles condition(5) of Hart and Mas-Colell (1989) for the weighted potentials P , if we want weightedw

multichoice values we need to add weights in condition PM.1 and in the definition ofbalanced contributions. Formally, a system of weights is a function w:N → R , where11

i (N,L )w(i)5w is the weight of player i. A solution c on MG satisfies w-balancedcontributions if

1 12j 2i] ]? c(v,l)(i) 2 c(v,l )(i) 5 ? c(v,l)( j) 2 c(v,l )( j)f g f gi jw w(N,L )holds for all hi, jj # A(l), v [ MG and l [ L\u.

(N,L ) (N,L )A w-potential on MG is a function P :MG 3 L → R satisfying the followingw

conditions

i i (N,L )(w-PM.1 ) O w ? l ? D P (v,l) 5 v(l) , (v [ MG , l [ L) ,i wi[A( l)

(N,L )(w-PM.2 ) P (v,u ) 5 0 , (v [ MG ) .w

It can be checked that for every weight system w, there exists a unique w-potential P .w

Then we can define the w-multichoice value w asw

i i (N,L )w (v,l)(i) 5 w ? D P (v,l)(i) , ((v,l) [ MG 3 L\u, i [ A(l)) .w w

Furthermore, this is the unique solution that satisfies w-balanced contributions andefficiency.


Theorem 4.1 also works here, and we obtain the weighted Aumann-Shapley value (seeHart and Monderer, 1997), i.e.,

t t tlim l ? w (v , l )(i) 5 z ? F (v,z)(i) , (i [ A(z)) ,i w z i wt →`

where

1

≠vi w] *F (v,z)(i) 5Ew ? (t z) dt , (i [ A(z)) ,w ≠xi0

iw N w w* *with (t z) [ R being such that (t z) 5 t ? z , for each i [ N.i i

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