www.elsevier.nl / locate / econbase
A value for multichoice games
a 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 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 alloca-tion
1. Introduction
One of the most interesting applications of the Cooperative Game Theory has been 1
done in the setting of allocating costs. This kind of problem can be formulated as follows: let N5h1,2, . . . ,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.
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 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-sharing 2
method for this type of continuum problems.
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 Aumann-Shapley 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 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
N
We start by defining the general model. We say that a subset L of R is full
1 dimensional if hl[L: l .i 0 for all i[Nj±[. The zero vector (0, . . . ,0) will be denoted byu.
Definition 2.1. A cooperative multilevel game is a triple (N,L,v), where N5h1, . . . ,njis
N
a finite set of players, L is a full dimensional subset ofR ,u[L, andv is a function 1
from L into R, with v(u)50.
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. Givenl[L, ifl ±0 we
i
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 levell. The functionvitself will also be
i
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 3L\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 S
For S#N, let e be the vector in R satisfying ej51 if j[S, and ej50 if j[⁄ S.
i 1i i 2i
Given a vectorl we denote the vectorl 1e by l , andl 2e byl . For every two
N N
vectors x,y[R , x#y means x #y for all i[N. For S#N and x[R , we often
i i
Two subfamilies of multilevel games have already been studied in literature. The first
N
one arises when L5h0,1j . 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 vectors l and active coalitions A(l); a level vector m belongs to L if and only if there exists
S
S#N such that e 5m. Then, in this context, we can often write S#N instead ofm[L
S N
or e [L. We will denote this subfamily by G . The well known Shapley (1953) value
solution, is defined by
uSu!?
s
uTu2uSu21 !d
]]]]]]f(v,T )(i )5
O
?fv(S<hij)2v(S ) ,g uTu!S#T \hij
N N
for (v,T )[G 32 \[ and i[T.
This value can be rewritten, using our notation, as follows:
uA(m)u!?
s
uA(l)u2uA(m)u21 !d
1i]]]]]]]]]
f(v,l)(i )5
O
?f
v(m )2v(m)g
uA(l)u!2i
u #m #l
N N
for (v,l)[G 3h0,1j \u and i[A(l).
N
The second subfamily arises when L is a comprehensive subset of R1 (by
comprehensive, we mean that if z[L, then for allu #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, for
f ig 1
every z[L\u, (v,z) can be represented by a non-atomic game (see Mirman et al., 1982) on (I,C), where Ii5fi21, i for each ig [N, I5
<
i[N iI , and C is thes-field of BorelN
subsets of I. Given T[C, let z(T )[R1 such that z Ts di5zi?jsT>I for each iid [N,
where j denotes the Lebesgue measure. The non-atomic game f(v,z) is defined by
f s dT 5vs s ddz T for each T[C. Whenvhas continuous first partial derivatives on L (it v,z
s d
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 I , reduces toi
(Cf )(I )5z ?p(v,z)(i ) , (v,z) i i
for each z[L\u and i[N, where
1 ≠v ]
p(v,z)(i )5
E
(tz) dt . ≠xi0
The number psv,z i is the per capita payoff (price) of the set of players I and is theds d
i
Aumann-Shapley price of player i. In order to determineC it is sufficient to specify p because in the game f(v,z) the players of each type i (the set I ) are symmetric withi
(N,L )
respect to f . Hence, we denote by CG the family of continuum gamesvon (N,L), (v,z)
where v has continuous first partial derivatives on L; and we define, for each (v,z)[ (N,L )
1 ≠v ]
Fsv,z ids d5
E
s dtz dt , (i[A(z)) . ≠xi0
N N
Multichoice games appear when L is a comprehensive subset ofN15(h0j<N) , i.e.,
N 3
if l[L then for allm[N1 such thatu # m # l, it holds that m[L.
N
The set of all multichoice games on (N,L ), where L is a comprehensive subset ofN1, (N,L )
is denoted by MG . Note that a multichoice game on (N,L ) is a finite cooperative TU
N
game on N in the particular case L5h0,1j .
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 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\uand assume that levellforms step by step, starting from level zero, and that at each step the level of one of the players is increased by 1, up tol. There arel(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 levelu up tolhave
4
the same probability. 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
5 we denote it by w(v,l)(i ).
Now we summarize the Aumann-Shapley method proposed by Moulin (1995). In
N
order to do this, let q5sq , . . . ,q1 nd[N1 and C a cost function defined on the interval
N
of N1, 0,q . Given the demand profile qf g 5sq , . . . ,q1 ndconsider the cooperative game with q11 ? ? ? 1q players where each player is a particular unit of a particular good.n
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.
N (N,L )
Given a set N5h1, . . . ,nj, L#N , a multichoice gamev[MG andl[L\u, let 1
l
D be a set of replica players defined as:
3
Ndenotes the set of positive integers. 4
This procedure can be interpreted as follows: Consider the process of picking (without replacing) l(N ) coloured balls from a box;liballs 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. isl ? w(v,l)(i ).
l l D 5
<
Di ,i[A(l) l
whereD 5i hi , . . . ,i1 lij for any i[A(l). l
Now, for any B#D define the level vector l(B )[L as follows:
l
uB>Diu, if i[A(l) , li(B )5
H
0, if i[N\A(l) .
l
l D Then, we define the replica game R v[G by
l
R v(B )5v(l(B )),
l for 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 the same 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 andl[L\u it holds that 1
l l l l l
]
w(v,l)(i )5f(R v,D )(i )5 ?f(R v,D )(D ) , (i[A(l)) ,
j l i
i
wherew 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
2i
a(N )!?sl(N )2a(N )21 !d lj 1i
]]]]]]]]
w(v,l)(i )5
O
?P
S D
?f
v(a )2v(a)g
al(N )! j[A(l)
2i j
u #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 3L\u by
1 ≠v ]
F(v,z)(i )5
E
(tz) dt , for every i[A(z) . ≠xi0
We will show in this section that F(v,z) can be obtained by taking an asymptotic
t t t
approach by means of a sequencew(v ,l ) of multichoice values. These games,v, are
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 . Ani
t t N
admissible sequence of partitioning vectors hl j, where l [N , is defined by
t 11 t
(i) li 5ati?li , with ati[N, (t[N, i[N ) .
t
(ii) hlij→`, whent→`, (i[N ) .
t t t N
Given an admissible sequence hl j, for every l we denote by L the subset ofN1
t t
such that m[L if and only if m # l .
(N,L ) t
Now, given a pair (v,z)[CG 3L\u and an admissible sequence hl j, for every
t
t t (N,L )
l we define the multichoice game v [MG as
z
z1 zn
t ] ] t
v(m)5v m ? t, . . . ,m ? t , (m[L ) .
z
S
1 l n lD
1 n
(N,L ) t
Theorem 4.1. For all (v,z)[CG 3L\u and all admissible sequenceshl j, it holds
that
t t t
lim l ? w(v ,l )(i )5z ?F(v,z)(i ) , (i[A(z)) ,
i z i
t→`
wherew is the multichoice value and F is the Aumann-Shapley value.
Proof. The proof has four steps.
(N,L ) t
STEP 1: Given (v,z)[CG 3L\u and an admissible sequencehl j, for every pair
t
t t l t
(v ,l ) we build the replicated game R v as in Proposition 3.1. Therefore we know
z z
that
t t t
t t t l t l l
l ? w(v,l )(i )5f(R v ,D )(D ) , (i[N,t[N) , (1)
i z z i
t
l
whereDi 5hi , . . . ,i1 ltjand f is the Shapley value.
i
(N,L )
STEP 2: Given (v,z)[CG 3L\u we saw, in Section 2, that
(Cf )(I )5z ?F(v,z)(i ) , (i[A(z)) , (2) (v,z) i i
where f(v,z) is the non-atomic game on (I,C), with Ii5fi21,ig and I5
<
i[N iI , asdefined in Section 2.
t
STEP 3: The admissible sequencet hl jinduces a partition of I into a finite collection
t t l t t t
P 5hPik: ik[Di , i[Nj of disjoint measurable sets, where Pik#I ,i j P
s d
ik 51 /lit 11 t t t 11
andP refines toP (i.e., each member ofP is a union of members ofP ), for
t
l
every ik[Di , i[N and t[N. This allows us to build a finite ‘quotient’ game
t
P 6
vPt[G defined by
t
6 t l
t t vPt(T )5f(v,z)
S
<
t PjD
, for each T#P .Pj[T
t
l t
Then the quotient game vPt coincides with the replicated game R vz of STEP 1, because
t t t
vPt(T )5f(v,z)
S
<
t PjD S S
5v z1?j<
t Pj>I1D
, . . . ,zn?jS
<
t Pj>InDD
Pj[T Pj[T Pj[T
t t
l t l t 5R v
s
hj[D :P [Tjd
.z j
Hence,
t t t t
l t l l t t l
f(R vz,D )(Di )5f(vPt,P )(hPi:ik[Di j) , (i[N ) . (3)
k
(N,L )
STEP 4: When (v,z)[CG 3L\u, its associated non-atomic game f belongs to (v,z)
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
t t
f(v,z). In our case, the sequence of partitionshP jbuilt in STEP 3 fromhl jsatisfies the conditions for belonging to that family; this means that
t
t t l
limf(vPt,P )
S
hPik:ik[Di jD
5(Cf(v,z))(I ) ,i (i[N ) . (4)t→`
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
N N
AS prices can be obtained as the gradient of a potential function on G 3h0,1j and (N,L )
(N,L )
Definition 5.1. Let P be a function P:MG 3L →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
i 2i
D P(v,l)5P(v,l)2P(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)5v(l),s(
v,l)[MG 3L\ud, (PM.1)i i[A(l)
(N,L )
P(v,u)50, (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)i
F
i[A(l)G
O
lii[A(l)
(N,L )
Taking a game v[MG and starting from P(v,u)50, (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 allu # 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 defined by
i (N,L )
w(v,l)(i )5D P(v,l) ,
s(
v,l)[MG 3L\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 )21 !? l(N )2a(N ) !
s d s d li
]]]]]]]]
P(v,l)5
O
?P
S D
?v(a) . al(N )! i[A(l) i
u #a #l a±u
(N,L )
for any v[MG andl[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.
(N,L )
Let w be a solution on MG . We say that w satisfies efficiency if, for every (N,L )
O
l ? c(v,l)(i )5v(l) .i i[A(l)
(N,L ) We say that c satisfies balanced contributions if, for every (v,l)[MG 3L\u, with A(l)u u$2, it holds that
2j 2i
c(v,l)(i )2c(v,l )(i )5c(v,l)( j )2c(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 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
2j
c(v,l )(i ) is the per unit cost variation in the production ofl units of i when the level
i
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 levell 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
N
efficiency 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 solutioncon MG 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
2j i i 2j
w(v,l)(i )2w(v,l )(i )5D P(v,l)2D P(v,l )
2i 2j 2hi, jj 5P(v,l)2P(v,l )2P(v,l )1P(v,l )
j j 2i 2i
5D P(v,l)2D P(v,l )5w(v,l)( j )2w(v,l )( j ) ,
2hi, jj i j
wherel 5l 2e 2e . 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 3L →R as follows:
(N,L ) (i) Q(v,u)50, (v[MG ) ,
2i (N,L )
(ii) Q(v,l)5Q(v,l )1c(v,l) ((v,l)[MG 3L\u, i[A(l)) .
2i 2j
Q(v,l )1c(v,l)(i )2Q(v,l )2c(v,l)( j ) 2i 2j
5Q(v,l )2Q(v,l )1c(v,l)(i )2c(v,l)( j ) 2hi, jj 2hi, jj 2j
5Q(v,l )2Q(v,l )1Q(v,l )2Q(v,l )1c(v,l)(i )2c(v,l)( j ) 2i
2i 2j
5c(v,l )( j )2c(v,l )(i )1c(v,l)(i )2c(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 2i
O
l ?D Q(v,l)5O
l ?f
Q(v,l)2Q(v,l )g
5O
l ? c(v,l)(i )5v(l)i i i
i[A(l) i[A(l) i[A(l)
becausecis efficient. From Theorem 5.3 we conclude that Q is the potential function on
(N,L ) i
MG . Furthermore, c(v,l)(i )5D 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 as
7
follows:
(N,L )
Efficiency: For any sv,zd[CG 3L\u it holds that
O
z ?c(v,z)(i )5v(z) .i i[A(z)
(N,L )
Balanced contributions: For anysv,zd[CG 3L\u, all i, jh j#A(z) and a
continu-ously differentiable solutionC it holds:
≠c(v,?)(i ) ≠c(v,?)( j ) ]]](z)5]]](z) .
≠xj ≠xi
(N,L ) These two properties also characterize the Aumann-Shapley value on CG (see Calvo and Santos, 1997, or Ortmann, 1995 and Ortmann, 1998), that is, a continuously
(N,L )
differentiable solutioncon CG 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:
(N,L ) l
Given (v,l)[MG 3L\u, let E v be the function defined by:
l
l i ai l 2ai i
E v(x)5
O P
F
S D
?x ?(12x )G
?v(a) ,i i
ai i[A(l) u #a #l
N
for every x[f0,1g .
7
l Note that if l 51 for all i[N, then v is a classic finite TU game and E v is the
i
multilinear extension of v (see Owen, 1972).
(N,L )
Theorem 6.1. For all (v,l)[MG 3L\u and i[A(l), it holds:
1 l ]
w(v,l)(i )5 FsE v,1d(i ) , li
wherew 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
(N,L )
Corollary 6.2. For all (v,l)[MG 3L\u it holds that, 1
1 l ]
Psv,l 5d
E
E vst, . . . ,t dt ,dt
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 P , if we want weightedw
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 , where
11
i (N,L )
w(i )5w is the weight of player i. A solution c on MG satisfies w-balanced contributions if
1 2j 1 2i
]?
f
c(v,l)(i )2c(v,l )(i )g
5]?f
c(v,l)( j )2c(v,l )( j )g
i j
w w
(N,L )
holds for all hi,jj#A(l), v[MG andl[L\u.
(N,L ) (N,L )
A w-potential on MG is a function P :MG 3L →Rsatisfying the following
w
conditions
i i (N,L )
(w-PM.1 )
O
w ?l ?D P (v,l)5v(l) , (v[MG ,l[L ) ,i w i[A(l)
(N,L ) (w-PM.2 ) P (v,u)50 , (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 ww as
i i (N,L )
w (v,l)(i )5w ?D P (v,l)(i ) , ((v,l)[MG 3L\u, i[A(l)) .
w w
Theorem 4.1 also works here, and we obtain the weighted Aumann-Shapley value (see Hart and Monderer, 1997), i.e.,
t t t
lim l ? w (v,l )(i )5z ?F (v,z)(i ) , (i[A(z)) ,
i w z i w
t→`
where
1 ≠v
i w
] *
F (v,z)(i )5
E
w ? (t z) dt , (i[A(z)) ,w ≠x i
0
i
w N w w
* *
with (t z)[R being such that (t z)i5t ?z , for each ii [N.
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