Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol40.Issue3.Nov2000:
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
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 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).
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 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 Sj 5 1 if j [ S, and e Sj 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 .
344
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
345
1
E
≠v
Fsv,zdsid 5 ]stzd dt , (i [ A(z)) .
≠x i
0
Multichoice games appear when L is a comprehensive subset of N N1 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
The set of all multichoice games on (N,L), where L is a comprehensive subset of N N1 ,
is denoted by MG (N,L ) . 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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
346
Dl 5
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 zt , are
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 zt , 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 zt , 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 itk # Ii , jsP itkd 5 1 /l it
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.
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 jt [ T jd .
Hence,
t
t
t
t
f (R l v tz ,D l )(D il ) 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.
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
349
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
O F
1
P(v, l) 5 ]] ? v( l) 1
li ? P(v, l 2i )
i [ A( l )
li
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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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).
352
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 zt , 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
w
N
i
with (t * z) [ R being such that (t w * z) i 5 t w ? z i , for each i [ N.
References
Aumann, R.J., Shapley, L.S., 1974. Values of Non-Atomic Games. Princeton University Press, Princeton, N.J.
Billera, L.J., Heath, D.C., 1982. Allocation of shared costs: a set of axioms yielding a unique procedure.
Mathematics of Operations Research 7, 32–39.
Billera, L.J., Heath, D.C., Raanan, J., 1978. International telephone billing rates: a novel application of
non-atomic game theory. Operations Research 26, 956–965.
Calvo, E., Santos, J.C., 1997. Potentials in cooperative TU-games. Mathematical Social Sciences 34, 175–190.
Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game
Theory 21, 351–360.
Hart, S., Mas-Colell, A., 1989. Potential, value and consistency. Econometrica 57, 589–614.
Hart, S., Monderer, D., 1997. Potentials and weighted values of nonatomic games. Mathematics of Operations
Research 22, 619–630.
Hsiao, C.R., 1995. A value for continuously-many-choice cooperative games. International Journal of Game
Theory 24, 273–292.
Hsiao, C.-R., Raghavan, T.E.S., 1992. Monotonicity and dummy free property for multi-choice cooperatives
games. International Journal of Game Theory 21, 301–312.
Hsiao, C.-R., Raghavan, T.E.S., 1993. Shapley value for multichoice cooperatives games I. Games and
Economic Behavior 5, 240–256.
Klijn, F., Slikker, M., Zarzuelo, J.M., 1998. Characterizations of a Multi-Choice Value. International Journal of
Game Theory, forthcoming.
Mirman, L.J., Raanan, J., Tauman, Y., 1982. A sufficient condition on f for f o m to be in pNAD. Journal of
Mathematical Economics 9, 251–257.
Mirman, L.J., Tauman, Y., 1982. Demand compatible equitable cost sharing prices. Mathematics of Operations
Research 7, 40–56.
Myerson, R., 1980. Conference structures and fair allocation rules. International Journal of Game Theory 9,
169–182.
Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332.
Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037.
Van Den Nouweland, A., 1993. Games and graphs in economic situations. Ph.D. Thesis, Tilburg University,
Tilburg, The Netherlands.
Owen, G., 1972. Multilinear extensions of games. Management Science 18, 64–79.
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.
Ortmann K.M., 1995. Conservation of energy in nonatomic games. Working Paper 237, Inst. of Math. Ec.,
University of Bielefeld.
Ortmann, K.M., 1998. Conservation of energy in value theory. Mathematical methods of Operations Research
47, 423–450.
354
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
Samet, D., Tauman, Y., Zang, I., 1984. An application of the Aumann-Shapley prices to transportation problem.
Mathematics of Operations Research 10, 25–42.
Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.). Contributions to
Theory of Games, Vol II. Princeton University Press, Princeton, pp. 307–317, Annals of Mathematics
Studies, 28.
Shubik, M., 1962. Incentives, decentralized control, the assignment of joint costs and internal pricing.
Management Science 8, 325–343.
Sprumont Y., Wang Y.T., 1998. A characterization of the Aumann-Shapley method in the discrete cost sharing
mode. Mimeo, Montreal University.
Tauman, Y., 1988. The Aumann-Shapley prices: a survey. In: Roth, A. (Ed.), The Shapley Value. Cambridge
University Press, New York, pp. 279–304.
Young, H.P., 1994. Cost allocation. In: Aumann, R.J., Hart, S. (Eds.). Handbook of Game Theory with
Economic Applications, Vol 2. Elsevier Science, Amsterdam, pp. 1193–1235.
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
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 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).
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 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 Sj 5 1 if j [ S, and e Sj 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 .
344
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
345
1
E
≠v
Fsv,zdsid 5 ]stzd dt , (i [ A(z)) .
≠x i
0
Multichoice games appear when L is a comprehensive subset of N N1 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
The set of all multichoice games on (N,L), where L is a comprehensive subset of N N1 ,
is denoted by MG (N,L ) . 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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
346
Dl 5
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 zt , are
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 zt , 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 zt , 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 itk # Ii , jsP itkd 5 1 /l it
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.
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 jt [ T jd .
Hence,
t
t
t
t
f (R l v tz ,D l )(D il ) 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.
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
349
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
O F
1
P(v, l) 5 ]] ? v( l) 1
li ? P(v, l 2i )
i [ A( l )
li
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
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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).
352
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
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 zt , 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
w
N
i
with (t * z) [ R being such that (t w * z) i 5 t w ? z i , for each i [ N.
References
Aumann, R.J., Shapley, L.S., 1974. Values of Non-Atomic Games. Princeton University Press, Princeton, N.J.
Billera, L.J., Heath, D.C., 1982. Allocation of shared costs: a set of axioms yielding a unique procedure.
Mathematics of Operations Research 7, 32–39.
Billera, L.J., Heath, D.C., Raanan, J., 1978. International telephone billing rates: a novel application of
non-atomic game theory. Operations Research 26, 956–965.
Calvo, E., Santos, J.C., 1997. Potentials in cooperative TU-games. Mathematical Social Sciences 34, 175–190.
Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game
Theory 21, 351–360.
Hart, S., Mas-Colell, A., 1989. Potential, value and consistency. Econometrica 57, 589–614.
Hart, S., Monderer, D., 1997. Potentials and weighted values of nonatomic games. Mathematics of Operations
Research 22, 619–630.
Hsiao, C.R., 1995. A value for continuously-many-choice cooperative games. International Journal of Game
Theory 24, 273–292.
Hsiao, C.-R., Raghavan, T.E.S., 1992. Monotonicity and dummy free property for multi-choice cooperatives
games. International Journal of Game Theory 21, 301–312.
Hsiao, C.-R., Raghavan, T.E.S., 1993. Shapley value for multichoice cooperatives games I. Games and
Economic Behavior 5, 240–256.
Klijn, F., Slikker, M., Zarzuelo, J.M., 1998. Characterizations of a Multi-Choice Value. International Journal of
Game Theory, forthcoming.
Mirman, L.J., Raanan, J., Tauman, Y., 1982. A sufficient condition on f for f o m to be in pNAD. Journal of
Mathematical Economics 9, 251–257.
Mirman, L.J., Tauman, Y., 1982. Demand compatible equitable cost sharing prices. Mathematics of Operations
Research 7, 40–56.
Myerson, R., 1980. Conference structures and fair allocation rules. International Journal of Game Theory 9,
169–182.
Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332.
Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037.
Van Den Nouweland, A., 1993. Games and graphs in economic situations. Ph.D. Thesis, Tilburg University,
Tilburg, The Netherlands.
Owen, G., 1972. Multilinear extensions of games. Management Science 18, 64–79.
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.
Ortmann K.M., 1995. Conservation of energy in nonatomic games. Working Paper 237, Inst. of Math. Ec.,
University of Bielefeld.
Ortmann, K.M., 1998. Conservation of energy in value theory. Mathematical methods of Operations Research
47, 423–450.
354
E. Calvo, J.C. Santos / Mathematical Social Sciences 40 (2000) 341 – 354
Samet, D., Tauman, Y., Zang, I., 1984. An application of the Aumann-Shapley prices to transportation problem.
Mathematics of Operations Research 10, 25–42.
Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.). Contributions to
Theory of Games, Vol II. Princeton University Press, Princeton, pp. 307–317, Annals of Mathematics
Studies, 28.
Shubik, M., 1962. Incentives, decentralized control, the assignment of joint costs and internal pricing.
Management Science 8, 325–343.
Sprumont Y., Wang Y.T., 1998. A characterization of the Aumann-Shapley method in the discrete cost sharing
mode. Mimeo, Montreal University.
Tauman, Y., 1988. The Aumann-Shapley prices: a survey. In: Roth, A. (Ed.), The Shapley Value. Cambridge
University Press, New York, pp. 279–304.
Young, H.P., 1994. Cost allocation. In: Aumann, R.J., Hart, S. (Eds.). Handbook of Game Theory with
Economic Applications, Vol 2. Elsevier Science, Amsterdam, pp. 1193–1235.