A value for multichoice games

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

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

´ ´ Departamento de Analisis Economico , Universidad de Valencia, Campus dels Tarongers, Avinguda dels Tarongers s /n, Edificio Departamental Oriental, 46022 Valencia, Spain b ´ Aplicada IV, Universidad del Paıs ´ Vasco /E.H.U., Departamento de Economıa 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 has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and related solution 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 the Aumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.  2000 Elsevier Science B.V. All rights reserved. Keywords: Multichoice games; Shapley value; Aumann-Shapley value; Balanced contributions; Cost allocation

1. Introduction One of the most interesting applications of the Cooperative Game Theory has been done in the setting of allocating costs.1 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 be provided 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). 1 For 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. PII: S0165-4896( 99 )00054-2

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jointly, for each subset S # N. The function c is called a discrete cost function, or a cost-sharing problem (alternatively, c can be interpreted as a production function that gives the output for any coalition of agents, or factors). Modelled in this way, a cost allocation problem can be considered as a cooperative game, with c being its characteristic function. The Shapley (1953) value provides an efficient and fair cost allocation mechanism for sharing costs between products (or factors). Another framework is considered when the output can vary continuously. Here the problem can be modelled as a non-atomic game with a continuum of n types of players: each good i, produced at level qi , is represented by qi mass of players of type i. The Aumann and Shapley (1974) value for this non-atomic game gives a cost-sharing method for this type of continuum problems.2 In this setting it is assumed that commodities are totally divisible goods and then magnitudes of goods can be measured with real numbers. This is an appropriate approach for cases such as petroleum products, various agricultural products (cereals, wine, olive oil, fruits, etc.), chemical products, etc. Nevertheless, there are many others types of goods for which this is not possible (cars, machines, buildings, etc.). This family of indivisible goods are only available in finite integer amounts. This is the kind of situation that we want to cover in this paper: cost allocation problems in which products can be provided (or factors used) at a certain finite number of levels. A survey of this problem and different solutions for it can be found in Moulin (1995). In that paper cost sharing methods for these problems were compared, the Shapley-Shubik method (Shubik, 1962), the discrete Aumann-Shapley method (Moulin, 1995), the serial cost sharing method (Moulin and Shenker, 1992) and the pseudo-average cost (Moulin, 1995). Recently, Sprumont and Wang (1998) have characterized the discrete AumannShapley method using axioms that involves only economic terms. The appropriate game-theoretic tool for modelling this setting are the so called multichoice games. These are games in which each player has a certain finite number of activity levels at which he can play. In general, different players may have different possible levels, and the worth that a coalition can obtain depends on the level at which each 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. They defined extended Shapley values by using weights on activity levels, each level having the same weight for all players, and provided axiomatic characterizations of the corresponding values. Van Den Nouweland et al. (1995) considered the more general case with different numbers of activity levels, and extended the notions of core, dominant core and Weber set. Also they proposed an alternative extension for the Shapley value based on an extension of the probabilistic formula by orders; but they did not give additional support for this extension. In Van Den Nouweland (1993) an example is given of a multichoice game for which this value is not equal to any of the values of Hsiao and Raghavan; and other alternative proposals for the multichoice value are also shown. Recently, Klijn et al. (1998) have studied a new solution to multichoice games.

2

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

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This solution is based on the work of Derks and Peters (1993) on the extended Shapley value. 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 found asymptotically by means of the multichoice value using admissible sequences of discrete multichoice games which converge to the continuum game. Third, an axiomatic characterization is offered of the multichoice value which is consistent with the axiomatic characterization of the Aumann-Shapley value for continuum finite type games. Following this introduction, Section 2 is devoted to some preliminary definitions and notations. In Section 3 we present the solution for multichoice games. In Section 4, we state and prove the limit theorem. Section 5 is devoted to the axiomatic characterization of the multichoice value, and finally, in Section 6 we offer some concluding remarks.

2. Preliminaries We start by defining the general model. We say that a subset L of R N1 is full dimensional if h l [ L: li . 0 for all i [ Nj ± [. The zero vector (0, . . . ,0) will be denoted by u. Definition 2.1. A cooperative multilevel game is a triple (N,L,v), where N5h1, . . . ,nj is a finite set of players, L is a full dimensional subset of R N1 , u [ L, and v is a function from L into R, with v(u ) 5 0. The interpretation is the following: for each l [ L, li means the activity level at which player i participates in the game. The vector of zero levels is always possible; we also assume that all players can play the game simultaneously. Given l [ L, if li ± 0 we will say that i is an active player at l, and the set of all active players at l will be denoted by A( l). The function v: L → R, gives for every action l the worth that the players can obtain when each player i plays at level li . The function v itself will also be called a multilevel game, or a game, on (N,L). The set of all multilevel games on (N,L) is denoted by G (N,L ) . Definition 2.2. Given (N,L) and a subset Q (N,L ) of G (N,L ) , a solution on Q (N,L ) is a function c : Q (N,L ) 3 L\u →

<

S #N

R S , where c (v, l) [ R A( l) .

The number c (v, l)(i) represents the per unit payoff (prices) that player i receives, hence li ? c (v, l)(i) is the total payoff that player i receives at (v, l), with solution c. For S # N, let e S be the vector in R N satisfying e jS 5 1 if j [ S, and e jS 5 0 if j [ ⁄ S. Given a vector l we denote the vector l 1 e i by l 1i , and l 2 e i by l 2i . For every two vectors x, y [ R N , x # y means x i # y i for all i [ N. For S # N and x [ R N , we often write x(S) instead of o i [S x i .

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Two subfamilies of multilevel games have already been studied in literature. The first one arises when L 5 h0,1j N . 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 vectors l and active coalitions A( l); a level vector m belongs to L if and only if there exists S # N such that e S 5 m. Then, in this context, we can often write S # N instead of m [ L or e S [ L. We will denote this subfamily by G N . The well known Shapley (1953) value solution, is defined by

f (v,T )(i) 5

uSu! ?suT u 2 uSu 2 1d! O ]]]]]] ? f v(S < hij) 2 v(S) g , uT u!

S #T \hi j

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

f (v, l)(i) 5

O

u # m # l 2i

uA( m )u! ?suA( l)u 2 uA( m )u 2 1d! ]]]]]]]]] ? f v( m 1i ) 2 v( m ) g uA( l)u!

for (v, l) [ G N 3 h0,1j N \u and i [ A( l). The second subfamily arises when L is a comprehensive subset of R N1 (by 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: f 0,z i g , R 1 . We will call these multilevel games continuum games. In this context, for every z [ L\u, (v,z) can be represented by a non-atomic game (see Mirman et al., 1982)

<

on (I,C), where Ii 5 f i 2 1, i g for each i [ N, I 5 i [N Ii , and C is the s -field of Borel subsets of I. Given T [ C, let z(T ) [ R N1 such that zsTd i 5 z i ? jsT > Iid for each i [ N, where j denotes the Lebesgue measure. The non-atomic game f(v,z) is defined by fsv,zdsTd 5 vszsTdd for each T [ C. When v has continuous first partial derivatives on L (it is understood that the derivatives are one sided when z belongs to the boundary of L), it holds that f(v,z) belongs to the pNAD class of non-atomic games (see Mirman et al., 1982) and then, the Aumann and Shapley (1974) value, C, for every Ii , reduces to (C f(v,z) )(Ii ) 5 z i ? p(v,z)(i) , for each z [ L\u and i [ N, where 1

E

≠v p(v,z)(i) 5 ] (tz) dt . ≠x i 0

The number psv,zdsid is the per capita payoff (price) of the set of players Ii and is the Aumann-Shapley price of player i. In order to determine C it is sufficient to specify p because in the game f(v,z) the players of each type i (the set Ii ) are symmetric with respect to f(v,z) . Hence, we denote by CG (N,L ) the family of continuum games v on (N,L), where v has continuous first partial derivatives on L; and we define, for each (v,z) [ CG (N,L ) 3 L\u, the Aumann-Shapley value F at (v,z) as

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345

1

E

≠v Fsv,zdsid 5 ]stzd dt , (i [ A(z)) . ≠x i 0

N Multichoice games appear when L is a comprehensive subset of N 1 5 (h0j < N)N , i.e., N if l [ L then for all m [ N 1 such that u # m # l, it holds that m [ L.3 N The 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 game on N in the particular case L 5 h0,1j N .

3. The multichoice value In this section we show that the solution for multichoice games proposed by Van Den Nouweland et al. (1995) applied to discrete cost problems coincides with the AumannShapley method proposed by Moulin (1995). The definition by Nouweland et al. is based on a generalization of the probabilistic formula by orders of the Shapley value. To define this solution, let (v, l) [ MG (N,L ) 3 L\u and assume that level l forms step by step, starting from level zero, and that at each step the level of one of the players is increased by 1, up to l. There are l(N) steps in this procedure. Suppose that at each step the player that increases his level receives the marginal contributions of this step and suppose that all orders from level u up to l have the same probability.4 Then the per unit expected marginal contribution of each player is the value proposed by Nouweland et al. We call this solution the multichoice value and we denote it by w (v, l)(i).5 Now we summarize the Aumann-Shapley method proposed by Moulin (1995). In N order to do this, let q 5s q1 , . . . ,qnd [ N 1 and C a cost function defined on the interval N of N 1 , f 0,q g . Given the demand profile q 5s q1 , . . . ,qnd consider the cooperative game with q1 1 ? ? ? 1 qn players where each player is a particular unit of a particular good. Then the cost sharing of a particular good is the sum of the Shapley value of all units of this particular good. Obviously, this method is adaptable to multichoice games and hence it determines a solution for these games. Here, we formalize this procedure since it will be used in this work. Given a set N5h1, . . . ,nj, L # N N1 , a multichoice game v [ MG (N,L ) and l [ L\u, let l D be a set of replica players defined as: 3

N denotes the set of positive integers. This procedure can be interpreted as follows: Consider the process of picking (without replacing) l(N) coloured balls from a box; li balls of the same colour for each player i on A( l), and with different colours for 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 remain in 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. 5 Actually, the solution proposed by Van Den Nouweland et al. is li ? w (v, l)(i). 4

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346

Dl 5


i [ A( l )

l i

,

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

li (B) 5

H

uB > D li u , if i [ A( l) , if i [ N\A( l) .

0,

l

Then, we define the replica game R l v [ G D by R l v(B) 5 v( l(B)), for every B # D l . 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 the same strategy as described in Section 2 to obtain the Aumann-Shapley value F on CG (N,L ) . Proposition 3.1. For every multichoice game v [ MG (N,L ) and l [ L\u it holds that 1 w (v, l)(i) 5 f (R l v,D l )(ij ) 5 ] ? f (R l v,D l )(D il ) , (i [ A( l)) , li where w is the multichoice value and f is the Shapley value. Proof. It is straightforward taking into account the probabilistic formula with orders of the Shapley value. h Remark 3.2. An alternative formula for the multichoice value is given by

O

PS D

a (N)! ?s l(N) 2 a (N) 2 1d! l j2i ]]]]]]]] ? ? f v(a 1i ) 2 v(a ) g a l (N)! j [ A( l ) 2i j u #a #l

w (v, l)(i) 5

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

4. Going to the limit As we have seen in the preliminaries, the Aumann-Shapley value F on CG (N,L ) is defined for each (v,z) [ CG (N,L ) 3 L\u by 1

E

≠v F (v,z)(i) 5 ](tz) dt , for every i [ A(z) . ≠x i 0

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

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347

a discrete version of the original one, allowing only a finite number, instead of a continuum, of activity levels for players. To see this, we start by identifying levels li with admissible amounts of z i . An admissible sequence of partitioning vectors h l t j, where l t [ N N , is defined by

l ti 11 5 at i ? l it ,

(i)

with at i [ N ,

(t [ N, i [ N) .

(ii) h l ti j → ` , when t → ` , (i [ N) . Given an admissible sequence h l t j, for every l t we denote by L t the subset of N N1 such that m [ L t if and only if m # l t. Now, given a pair (v,z) [ CG (N,L ) 3 L\u and an admissible sequence h l t j, for every t t (N,L t ) l we define the multichoice game vz [ MG as

S

D

z1 zn v tz ( m ) 5 v m1 ? ]t , . . . , mn ? ]t , ( m [ L t ) . l1 ln Theorem 4.1. For all (v,z) [ CG that

(N,L )

3 L\u and all admissible sequences h l t j, it holds

lim l ti ? w (v tz , l t )(i) 5 z i ? F (v,z)(i) , (i [ A(z)) ,

t →`

where w is the multichoice value and F is the Aumann-Shapley value. Proof. The proof has four steps. STEP 1: Given (v,z) [ CG (N,L ) 3 L\u and an admissible sequence h l t j, for every pair t lt t (v , l ) we build the replicated game R v z as in Proposition 3.1. Therefore we know that t z

t

t

t

l ti ? w (v tz , l t )(i) 5 f (R l vzt ,D l )(D il ) , (i [ N,t [ N) ,

(1)

t

where D li 5 hi 1 , . . . ,il ti j and f is the Shapley value. STEP 2: Given (v,z) [ CG (N,L ) 3 L\u we saw, in Section 2, that (C f(v,z) )(Ii ) 5 z i ? F (v,z)(i) , (i [ A(z)) ,

(2)

<

where f(v,z) is the non-atomic game on (I,C), with Ii 5 f i 2 1,i g and I 5 i [N Ii , as defined in Section 2. STEP 3: The admissible sequence h l t j induces a partition of I into a finite collection t t lt P 5 hP i k : i k [ D i , i [ Nj of disjoint measurable sets, where P ti k # Ii , jsP ti kd 5 1 /l ti and P t 11 refines to P t (i.e., each member of P t is a union of members of P t 11 ), for lt every i k [ D i , i [ N and t [ N. This allows us to build a finite ‘quotient’ game t vP t [ G P defined by 6

6

t

Note that uP t u 5 uD l u.

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348

vP t (T ) 5 f(v,z)

S< D P tj [T

P jt , for each T # P t . t

Then the quotient game vP t coincides with the replicated game R l v tz of STEP 1, because vP t (T ) 5 f(v,z)

S< D S S< P tj [T

t

P jt 5 v z 1 ? j

P tj [T

D

P jt > I1 , . . . ,zn ? j

S<

P tj [T

P jt > In

DD

t

5 R l v tz sh j [ D l : P tj [ T jd . Hence, t

t

t

t

f (R l v tz ,D l )(D li ) 5 f (vP t ,P t )(hP itk :ik [ D il j) , (i [ N) .

(3)

STEP 4: When (v,z) [ CG (N,L ) 3 L\u, its associated non-atomic game f(v,z) belongs to the space ASYMP ( f(v,z) [ pNAD and Proposition 43.13 in Aumann and Shapley, 1974). This means that under a suitable sequence of partitions of I, the Shapley value for the quotient games associated with f(v,z) gives, at the limit, the Aumann-Shapley value for f(v,z) . In our case, the sequence of partitions hP t j built in STEP 3 from h l t j satisfies the conditions for belonging to that family; this means that

S

t

D

lim f (vP t ,P t ) hP itk :i k [ D li j 5 (C f(v,z) )(Ii ) , (i [ N) .

t →`

(4)

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 a piecewise 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 not coincide with the Aumann-Shapley value, and in that paper the author does not prove that the solution for continuous games can be regarded as the limit of values for admissible convergence sequences of multichoice games. The solution for multichoice values 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 TU games to multichoice games. In that paper, they proved that the Shapley value and the AS prices can be obtained as the gradient of a potential function on G N 3 h0,1j N and CG (N,L ) 3 L, respectively.

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Definition 5.1. Let P be a function P:MG (N,L ) 3 L → 5. For all l [ L\u and active players i [ A( l), we define the marginal contribution of player i with respect to P at (v, l) as D i P(v, l) 5 P(v, l) 2 P(v, l 2i ) . Definition 5.2. The function P is said to be a potential function on MG (N,L ) if it satisfies

O

li ? D i P(v, l) 5 v( l),

s(v, l) [ MG (N,L ) 3 L\ud ,

(PM.1)

i [ A( l )

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

(PM.2)

Theorem 5.3. There is a unique potential function on MG (N,L ) . Proof. For l ± u, formula PM.1 can be rewritten as

O

1 P(v, l) 5 ]] ? v( l) 1 li ? P(v, l 2i ) i [ A( l ) li

O F

G.

(5)

i [ A( l )

Taking a game v [ MG (N,L ) 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 uniquely determined 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: Corollary 5.4. The multichoice value coincides with the solution w on MG (N,L ) defined by

w (v, l)(i) 5 D i P(v, l) ,

s(v, l) [ MG (N,L ) 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 P(v, l) 5

sa (N) 2 1d! ?s l(N) 2 a (N)d! l O ]]]]]]]] ? P Sa D ? v(a ) . l(N)!

u #a #l a ±u

i

i [ A( l )

i

for any v [ MG (N,L ) and l [ L\u. Now, we will give an axiomatic characterization of this solution. In essence, it says that the value is an efficient rule which equalizes the marginal contributions between the players in the game. Let w be a solution on MG (N,L ) . We say that w satisfies efficiency if, for every (v, l) [ MG (N,L ) 3 L\u, it holds that

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

350

O l ? c(v,l)(i) 5 v( l) . i

i [ A( l )

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

c (v, l)(i) 2 c (v, l 2j )(i) 5 c (v, l)( j) 2 c (v, l 2i )( j) , for each hi, jj # A( l), i ± j. The efficiency axiom is the translation to MG (N,L ) of the cost-sharing principle. The balanced contributions axiom is a fair-marginal rule. For a better understanding of its meaning we refer back to the cost allocation framework. Assume we have a rule w in order to allocate the production cost of a bundle of n goods. In this case c (v, l)(i) 2 c (v, l 2j )(i) is the per unit cost variation in the production of li units of i when the level of production of j diminishes in one unit. In other words, this term can be interpreted as j’ s marginal contribution to i’ s unit cost at level l of production. c will be a fair rule when these marginal cost contributions are equal for every pair of goods in A( l). For a generic game this rule implies that the marginal per capita value contributions between pairs of players must be equal. This axiom was introduced in Myerson (1980), and with efficiency characterizes the Shapley value on G N (see also Hart and Mas-Colell (1989), Theorem 3.4). The next theorem extends this result to MG (N,L ) . Theorem 5.6. A solution c on MG (N,L ) satisfies efficiency and balanced contributions if and only if c5w, where w is the multichoice value. Proof. It is straightforward to check that w satisfies efficiency. To see balanced contributions, note that from Corollary 5.4, we have

w (v, l)(i) 2 w (v, l 2j )(i) 5 D i P(v, l) 2 D i P(v, l 2j ) 5 P(v, l) 2 P(v, l 2i ) 2 P(v, l 2j ) 1 P(v, l 2hi, j j ) 5 D j P(v, l) 2 D j P(v, l 2i ) 5 w (v, l)( j) 2 w (v, l 2i )( j) , where l 2hi, j j 5 l 2 e i 2 e j . Hence, w satisfies the axioms. Now let c be a solution on MG (N,L ) that satisfies balanced contributions and efficiency. We define the function Q:MG (N,L ) 3 L → R as follows: (i) Q(v,u ) 5 0, (v [ MG (N,L )) , (ii) Q(v, l) 5 Q(v, l 2i ) 1 c (v, l)

((v, l) [ MG (N,L ) 3 L\u, i [ A( l)) .

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

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351

Q(v, l 2i ) 1 c (v, l)(i) 2 Q(v, l 2j ) 2 c (v, l)( j) 5 Q(v, l 2i ) 2 Q(v, l 2j ) 1 c (v, l)(i) 2 c (v, l)( j) 5 Q(v, l2i ) 2 Q(v, l 2hi, j j ) 1 Q(v, l 2hi, j j ) 2 Q(v, l 2j ) 1 c (v, l)(i) 2 c (v, l)( j) 2i

2j

5 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 well defined. Furthermore, by definition Q(v,u )5 0, for all v [ MG (N,L ) , and

O l ? D Q(v,l) 5 O l ? fQ(v,l) 2 Q(v,l i

i

2i

i

i [ A( l )

i [ A( l )

)g 5

O l ? c(v,l)(i) 5 v( l) i

i [ A( l )

because c is efficient. From Theorem 5.3 we conclude that Q is the potential function on MG (N,L ) . Furthermore, c (v, l)(i) 5 D i Q(v, l) and then c 5 w. h Remark 5.7. The translation of these two properties to a solution c on CG (N,L ) is as follows: 7 Efficiency: For any sv,zd [ CG (N,L ) 3 L\u it holds that

O

z i ? c (v,z)(i) 5 v(z) .

i [ A(z)

Balanced contributions: For any sv,zd [ CG (N,L ) 3 L\u, all hi, j j # A(z) and a continuously differentiable solution C it holds: ≠c (v, ? )(i) ≠c (v, ? )( j) ]]](z) 5 ]]](z) . ≠x j ≠x i These two properties also characterize the Aumann-Shapley value on CG (N,L ) (see Calvo and Santos, 1997, or Ortmann, 1995 and Ortmann, 1998), that is, a continuously differentiable solution c on CG (N,L ) satisfies efficiency and balanced contributions if, and only if c 5 F.

6. Concluding remarks We will show here that the multichoice value coincides with the Aumann-Shapley value of a continuum game that is a sort of multilinear extension of the initial game. Formally: Given (v, l) [ MG (N,L ) 3 L\u, let E l v be the function defined by: E l v(x) 5

O F P Sla D ? x i

u #a #l

i [ A( l )

i

ai i

? (1 2 x i ) l i 2 ai

G

? v(a ) ,

for every x [ f 0,1 g N . 7

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

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Note that if li 5 1 for all i [ N, then v is a classic finite TU game and E l v is the multilinear extension of v (see Owen, 1972). Theorem 6.1. For all (v, l) [ MG (N,L ) 3 L\u and i [ A( l), it holds: 1 w (v, l)(i) 5 ]FsE l 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 with the Aumann-Shapley value of their multilinear extensions, the result follows easily by applying Proposition 3.1. h Corollary 6.2. For all (v, l) [ MG (N,L ) 3 L\u it holds that, 1

E

1 Psv, ld 5 ]E l vst, . . . ,td dt , t 0

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 Pw , if we want weighted multichoice values we need to add weights in condition PM.1 and in the definition of balanced contributions. Formally, a system of weights is a function w:N → R 11 , where w(i)5w i is the weight of player i. A solution c on MG (N,L ) satisfies w-balanced contributions if 1 1 ]i ? f c (v, l)(i) 2 c (v, l 2j )(i) g 5 ]j ? f c (v, l)( j) 2 c (v, l 2i )( j) g w w holds for all hi, jj # A( l), v [ MG (N,L ) and l [ L\u. A w-potential on MG (N,L ) is a function Pw :MG (N,L ) 3 L → R satisfying the following conditions (w-PM.1 )

O w ? l ? D P (v,l) 5 v( l) , i

i

i

w

(v [ MG (N,L ) , l [ L) ,

i [ A( l )

(w-PM.2 ) Pw (v,u ) 5 0 , (v [ MG (N,L )) . It can be checked that for every weight system w, there exists a unique w-potential Pw . Then we can define the w-multichoice value ww as

ww (v, l)(i) 5 w i ? D i Pw (v, l)(i) , ((v, l) [ MG (N,L ) 3 L\u, i [ A( l)) . Furthermore, this is the unique solution that satisfies w-balanced contributions and efficiency.

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

353

Theorem 4.1 also works here, and we obtain the weighted Aumann-Shapley value (see Hart and Monderer, 1997), i.e., lim l ti ? ww (v tz , l t )(i) 5 z i ? Fw (v,z)(i) , (i [ A(z)) ,

t →`

where 1

E

≠v Fw (v,z)(i) 5 w i ? ](t w * z) dt , (i [ A(z)) , ≠x i 0

i

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

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