GAMES AND ECONOMIC BEHAVIOR
ARTICLE NO.
24, 25]46 Ž1998.
GA980657
Vector Measure Games Based on Measures with
Values in an Infinite Dimensional Vector Space
Igal Milchtaich*
Department of Mathematics, The Hebrew Uni¨ ersity of Jerusalem, Jerusalem 91904,
Israel
Received December 11, 1995
The definition of vector measure game is generalized in this paper to include all
cooperative games of the form f ( m , where m is a nonatomic vector measure of
bounded variation that takes values in a Banach space. It is shown that if f is
weakly continuously differentiable on the closed convex hull of the range of m then
the vector measure game f ( m is in pNA` and its value is given by the diagonal
formula. Moreover, every game in pNA` has a representation that satisfies this
condition. These results yield a characterization of pNA` as the set of all differentiable games whose derivative satisfies a certain continuity condition. Journal of
Economic Literature Classification Numbers: C71, D46, D51. Q 1998 Academic Press
INTRODUCTION
A list of axioms, adapted from those which uniquely characterize the
Shapley value for finite-player cooperative games, determines a unique
value on certain classes of nonatomic cooperative games}games involving
an infinite number of players, each of which is individually insignificant
ŽAumann and Shapley, 1974.. Concrete criteria for identifying a given
nonatomic game as belonging to such a class of games, and a formula for
computing the value, are known for certain kinds of vector measure games
ŽAumann and Shapley, 1974.. In a vector measure game, the worth of a
coalition S depends only on the value that a particular vector measure on
the space of players takes in S. In this context, the term ‘‘vector measure’’
usually refers to an R n-valued measure, that is, to a vector of n scalar
*This work is based on Chapter Four of my Ph.D. dissertation, written at the Center for
Rationality and Interactive Decision Theory of the Hebrew University of Jerusalem under the
supervision of Prof. B. Peleg, Prof. U. Motro, and Prof. S. Hart. I thank two referees and an
associate editor for remarks that led to the improvement of this paper. Current address:
Managerial Economics and Decision Sciences, J. L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208. E-mail: [email protected].
25
0899-8256r98 $25.00
Copyright Q 1998 by Academic Press
All rights of reproduction in any form reserved.
26
IGAL MILCHTAICH
measures. However, certain nonatomic games are more naturally described in terms of measures that take values in an infinite dimensional
Banach space.
Consider, for example, a Žtransferable utility. market game ¨ , where the
worth of a coalition S is
¨ Ž S . s max
½H
S
u Ž x Ž i . , i . dm Ž i .
HS x Ž i . dmŽ i . s HS aŽ i . dmŽ i .
5
, Ž 1.
the maximum aggregate utility that S can guarantee to itself by an
allocation x of its aggregate endowment HS a dm among its members
wAumann, 1964. In this formula, uŽ j , i . is the utility that player i gets from
the bundle j and m, the population measure, is a nonatomic probability
measure.x Since ¨ Ž S . depends on S only through certain integrals over S,
if mŽ S _ T . s mŽT _ S . s 0 then ¨ ŽT . s ¨ Ž S .. Thus, ¨ Ž S . can be expressed as a function of the characteristic function xS of the coalition S,
seen as an element of L1Ž m.. The market game under consideration can
therefore be viewed as a vector measure game based on the L1Ž m.-valued
vector measure defined by S ¬ xS .
Only one previous work known to me deals with vector measure games
based on vector measures with values in an infinite dimensional vector
space. Sroka Ž1993. studied games based on vector measures of bounded
variation with values in a relatively compact subset of a Banach space with
a shrinking Schauder basis. The vector measure considered above in
connection with the market game is not of this kind: its range is not a
relatively compact subset of L1Ž m., and L1Ž m. itself does not have a
shrinking Schauder basis. ŽOnly a separable Banach space with a separable
dual space can have such a basis. Considering the range of the measure in
question to be a subset of L2 , say, rather then L1 , would not help, for the
measure would not then be of bounded variation. Note that if the range
space were taken to be L` then the above set function would not even be a
measure; specifically, it would not be countably additive.. In the first part
of this paper, these limitations on the range of the vector measure and on
the space in which it lies are dispensed with. Thus, the results of Aumann
and Shapley are generalized to a much larger class of vector measure
games.
As the above market game example demonstrates, the present interpretation of ‘‘vector measure games’’ is broad enough to include all games in
which the worth of a coalition is not affected by the addition or subtraction
of a set of players of measure zero}the measure in question being a fixed
nonatomic scalar measure on the space of players. All the games that
belong to one of the spaces of games on which Aumann and Shapley have
proved the existence of a unique value have this property, and can
27
VECTOR MEASURE GAMES
therefore be represented as vector measure games. This representation is,
however, not unique. It is therefore desirable to reformulate the conditions for a vector measure game to belong to one of these spaces in a
language that does not make an explicit reference to vector measures.
Such an alternative formulation is presented in the second part of the
paper, where the above conditions are stated as differentiability and
continuity conditions on a suitable extension of the game, an ideal game,
that assigns a worth to every ideal, or ‘‘fuzzy’’, coalition, in which some
players are only partial members.
The last part of the paper contains an example that shows how these
general results can be applied to market games. Another application,
involving cooperative games derived from a particular class of nonatomic
noncooperative congestion games, is given in a separate paper ŽMilchtaich,
1995.. Two rather technical lemmas, which are of some independent
interest, are given in the Appendix.
PRELIMINARIES
The player space is a measurable space Ž I, C .. A member of the s-field
C is called a coalition. A set function is a function from C into a real
Banach space X. The ¨ ariation of a set function ¨ is the extended
real-valued function < ¨ < defined by
n
< ¨ < Ž S . s sup Ý 5 ¨ Ž Si . y ¨ Ž Siy1 . 5
ŽS g C.,
is1
where the supremum is taken over all finite nondecreasing sequences of
coalitions of the form S0 : S1 : ??? : S n s S. A set function ¨ is of
bounded ¨ ariation if < ¨ <Ž I . - `. A game is a real-valued set function ¨
such that ¨ ŽB. s 0. A game ¨ is monotonic if T : S implies ¨ ŽT . F ¨ Ž S ..
BV denotes the normed linear space of all games of bounded variation
endowed with the operations of pointwise addition and multiplication by a
Žreal. scalar and with the ¨ ariation norm Žcalled the ‘‘variation’’ in Aumann
and Shapley, 1974. 5 ¨ 5 BV s < ¨ <Ž I .. The monotonic games span this space.
The subspace of BV that consists of all finitely additive real-valued set
functions of bounded variation is denoted FA.
A ¨ ector measure is a countably additive set function. A Žfinite, signed.
measure is a real-valued vector measure. A vector measure m is nonatomic
if for every S g C such that m Ž S . / 0 there is a subset T : S such that
m ŽT ., m Ž S _ T . / 0. The variation < m < Žalso called the total variation
measure. of a vector measure of bounded variation m is a measure
ŽDiestel and Uhl, 1977, p. 3.. < m < is nonatomic if and only if m is
28
IGAL MILCHTAICH
nonatomic. The subspace of BV that consists of all nonatomic Žfinite,
signed real-valued. measures is denoted NA.
An ideal coalition is a measurable function h: I ª w0, 1x. For a vector
measure m and an ideal coalition Žor a linear combination of ideal
coalitions. h, m Ž h. denotes the integral H h dm. See Dunford and Schwartz
Ž1958, Section IV.10. for a definition of integration with respect to a vector
measure. The space I of ideal coalitions is topologized by the NA-topology, defined as the smallest topology on I with respect to which all
functions of the form m Ž?., m g NA, are continuous. As a base for the
neighborhood system of an ideal coalition h one can take the collection of
all open sets of the form g g I < max 1 F iF k < m i Ž g y h.< - « 4 , with « ) 0
and m1 , m 2 , . . . , m k g NA.
The range of a vector measure m is the set m Ž C .. If the range of m Žor,
more precisely, the subspace it spans. is finite dimensional then m is
automatically of bounded variation. If m is also nonatomic then its range is
compact and convex ŽLyapunoff theorem.. The range of a general
nonatomic vector measure need not be compact nor convex. However, for
every vector measure m the set m Ž I . s m Ž h. N h g I 4 is convex and
weakly compact ŽDiestel and Uhl, 1977, p. 263.. This set coincides with the
closed convex hull of m Ž C ., and if m is nonatomic then it is also the weak
closure of m Ž C . ŽDiestel and Uhl, 1977, p. 264.. We will call m Ž I . the
extended range of m.
EXAMPLE. Let m be a probability measure on Ž I, C ., and define
m : C ª L1Ž m. by m Ž S . s xS . Then m is a vector measure of bounded
variation whose variation is m. Therefore, m is nonatomic if and only if m
is nonatomic. The range of m consists of all Žequivalence classes of.
characteristic functions of measurable subsets of I. This is a closed, but
not convex, subset of L1Ž m., and if m is nonatomic then it is also not
compact. The extended range of m is the set of all Žequivalence classes of.
measurable functions from I into the unit interval. Indeed, for every
h g I , m Ž h. s h. Note that in this example the weak compactness of the
extended range of m follows immediately from Alaoglu’s theorem and
from the fact that the relative weak topology on this set coincides with the
relative weak* topology on it when seen as a subset of L`Ž m..
We will say that a real-valued function f defined on a convex subset C
of a Banach space X is differentiable at x g C if there exists a continuous
linear functional Df Ž x . g Y *, where Y is the subspace of X spanned by
C y C s y y z N y, z g C 4 and Y * is its dual space, such that for every
ygC
f Ž x q u Ž y y x . . s f Ž x . q u ² y y x, Df Ž x . : q o Ž u .
as u ª 0q. ŽThe angled brackets ² ? , ? : denote the operation of applying
an element of Y * to an element of Y.. This continuous linear functional,
VECTOR MEASURE GAMES
29
which is necessarily unique, will be called the deri¨ ati¨ e of f at x. The
function f is Ž weakly . continuously differentiable at x if it is differentiable
in a Žrespectively, weak. neighborhood of x in C and Df is Žrespectively,
weakly. continuous at x. The function f is Ž weakly . continuously differentiable if it is Žrespectively, weakly. continuously differentiable in the whole
of its domain. The restriction of a Žweakly. continuously differentiable
function to a convex subset of its domain is Žrespectively, weakly. continuously differentiable. Continuous differentiability and weak continuous
differentiability are equivalent for functions with compact domain. This
follows from the fact that the relativization of the weak topology to a
compact subset of a Banach space coincides with the relative norm
topology Žbecause every set which is closed, and hence compact, with
respect to the relative norm topology is compact, and hence closed, also
with respect to the relative weak topology.. A real-valued function f
defined on a bounded convex subset C of a Banach space X is Žweakly.
continuous at every point x at which it is Žrespectively, weakly. continuously differentiable. wProof: If U is a convex neighborhood of x in C in
which f is differentiable then it follows from the mean value theorem that
< f Ž y . y f Ž x .< F sup 0 - u - 1 < ² y y x, Df Ž x q u Ž y y x ..: < F < ² y y
x, Df Ž x .: < q 5 y y x 5 sup z g U 5 Df Ž z . y Df Ž x .5 for every y g U. By a suitable choice of U the last two terms can be made arbitrarily small.x If X is
a Euclidean space and C is compact then f is continuously differentiable
if and only if it can be extended to a continuous function on X with
continuous first-order partial derivatives.
The closed linear subspace of BV that is generated by all powers Žwith
respect to pointwise multiplication. of nonatomic probability measures is
denoted pNA. There exists a unique continuous linear operator w : pNA
ª FA that satisfies w Ž m k . s m for every nonatomic probability measure
m and positive integer k, called the ŽAumann]Shapley. ¨ alue on pNA. See
Aumann and Shapley Ž1974. for an axiomatic characterization of the value.
For a game ¨ , define 5 ¨ 5 ` s inf mŽ I . N m g NA, and < ¨ Ž S . y ¨ ŽT .< F
Ž
m S _ T . for every S, T g C , T : S4 Žinf B s `.. The collection of all
games ¨ such that 5 ¨ 5 ` - ` is a linear subspace of BV, denoted AC` , and
5 ? 5 ` is a norm on this space. The 5 ? 5 ` -closed linear subspace of AC` that
is generated by all powers of nonatomic probability measures is denoted
pNA` . This space is a proper subset of pNA.
VECTOR MEASURE GAMES
A composed set function of the form f ( m , where m is a nonatomic
vector measure of bounded variation and f is a real-valued function
defined on m Ž I . such that f Ž0. s 0, will be called a ¨ ector measure game.
30
IGAL MILCHTAICH
Aumann and Shapley Ž1974. proved that if the range of m is finite
dimensional then a sufficient condition for a vector measure game f ( m to
be in pNA Žactually in pNA` . is that f be continuously differentiable. The
value of such a vector measure game is given by the so-called diagonal
formula. Sroka Ž1993. generalized this result to the case where the range
of m is a relatively compact subset of a Banach space with a shrinking
Schauder basis. These results are generalized further in the following
theorem.
THEOREM 1. Let m be a nonatomic ¨ ector measure of bounded ¨ ariation
with ¨ alues in a Banach space X. If f is a weakly continuously differentiable
real-¨ alued function defined on the extended range of m such that f Ž0. s 0,
then f ( m is in pNA` and its ¨ alue is gi¨ en by the Ž diagonal . formula
w Ž f ( m. Ž S. s
1
H0
² m Ž S . , Df Ž t m Ž I . . : dt
ŽS g C..
Ž 2.
If X is finite dimensional then the con¨ erse is also true: a ¨ ector measure game
f ( m is in pNA` only if f is continuously differentiable on the range of m.
The restriction that X is finite dimensional cannot be removed. For
example, if m is Lebesgue measure on the unit interval and m is as in the
Example in the previous section then the function f : m Ž I . Ž: L1Ž m.. ª R
defined by f Ž h. s H01 ty1r2 hŽ t . dt is not differentiable according to the
present definition. Nevertheless, f ( m g NA. The question of what conditions on f, if any, are both necessary and sufficient for a general vector
measure game f ( m to be in pNA` , or in pNA, remains open Žcf.
Kohlberg, 1973; Aumann and Shapley, 1974, Theorem C; Tauman, 1982..
Note that if the range of m is relatively compact Žthis is automatically the
case if X is a reflexive space or a separable dual space; see Diestel and
Uhl, 1977, p. 266. then by Mazur theorem ŽDunford and Schwartz, 1958, p.
416. the extended range of m is compact. Therefore, in such a case f is
weakly continuously differentiable if and only if it is continuously differentiable.
If a vector measure game f ( m is monotonic, then for it to be in pNA it
suffices that f be continuous, rather than differentiable, at 0 and m Ž I ..
PROPOSITION 1. Let m be a nonatomic ¨ ector measure of bounded
¨ ariation, and let f : m Ž I . ª R be weakly continuously differentiable in
m Ž h g I < 0 - < m <Ž h. - < m <Ž I .4. and continuous at 0 and at m Ž I .. If f ( m is
a monotonic game then it is in pNA and its ¨ alue is gi¨ en by Ž2..
The following lemma, which is of some independent interest, is used in
the proofs of Theorem 1 and Proposition 1.
31
VECTOR MEASURE GAMES
LEMMA 1. Let m : C ª X be a nonatomic ¨ ector measure of bounded
¨ ariation, and let f be a real-¨ alued function defined on m Ž I .. Define
m
ˆ : C ª L1Ž< m <. by m
ˆ Ž S . s xS . Ž Note that < m
ˆ < s < m <.. Then there exists a
ˆ defined on m
unique real-¨ alued function f,
ˆ Ž I ., such that
fˆŽ m
ˆ Ž h. . s f Ž m Ž h. .
ŽhgI ..
Ž 3.
For e¨ ery h g I , if f is Ž weakly . continuous at m Ž h. then fˆ is Ž respecti¨ ely,
weakly . continuous at m
ˆ Ž h., and if f is weakly continuously differentiable at
m Ž h. then fˆ is weakly continuously differentiable at m
ˆ Ž h. and DfˆŽ m
ˆ Ž h..
satisfies
²m
ˆ Ž g . , DfˆŽ m
ˆ Ž h . . : s ² m Ž g . , Df Ž m Ž h . . :
Ž ggI ..
Ž 4.
It follows from Lemma 1 that, conceptually, there is only one kind of
vector measures that needs to be considered in the present context,
namely, vector measures that map coalitions into their characteristic
functions. One may thus wonder whether vector measures need to be
considered at all. An alternative approach might be to express the above
conditions for a game to be in pNA` or in pNA directly in terms of a
particular ‘‘extension’’ of the game into a function on I . We will see in the
next section that these results can indeed be reformulated in such a
manner.
DIFFERENTIABLE IDEAL GAMES
An ideal game is a real-valued function on I that vanishes at Žthe
constant function. 0. We will say that an ideal game ¨ * is monotonic if
h F g implies ¨ *Ž h. F ¨ *Ž g ., and that ¨ * is differentiable at h g I if there
exists a Žnecessarily unique. nonatomic measure D¨ *Ž h., called the deri¨ ati¨ e of ¨ * at h, such that for every g g I
¨ * Ž h q u Ž g y h . . s ¨ * Ž h . q u D¨ * Ž h . Ž g y h . q o Ž u .
as u ª 0q. An ideal game is differentiable if it is differentiable at every
point in I . An ideal game ¨ * is a continuous extension of a game ¨ if
¨ *Ž xS . s ¨ Ž S . for every S g C and ¨ * is continuous Žwith respect to the
NA-topology.. Aumann and Shapley Ž1974, Proposition 22.16. showed that
a continuous extension is always unique, and that a sufficient condition for
a game to have such an extension is that there exists a sequence in pNA
that converges to that game in the supremum norm 5 ¨ 59 s sup S g C < ¨ Ž S .<.
The set of all games that satisfy this condition is closed under pointwise
addition and multiplication by a real scalar, and is denoted pNA9.
32
IGAL MILCHTAICH
THEOREM 2. An ideal game ¨ * is the continuous extension of some game
¨ in pNA` if and only if there is a nonatomic probability measure m such that,
for e¨ ery h g I , the deri¨ ati¨ e D¨ *Ž h. exists and is absolutely continuous with
respect to m, dŽ D¨ *Ž h..rdm is essentially bounded, and dŽ D¨ *Ž?..rdm is
continuous at h as a function into L`Ž m.. The ¨ alue of ¨ is then gi¨ en by
1
Ž w ¨ . Ž S . s H D¨ * Ž t . Ž S . dt
ŽS g C..
Ž 5.
0
The necessary and sufficient condition for an ideal game ¨ * to be the
continuous extension of a game in pNA` that is given in Theorem 2 is
apparently stronger then the sufficient condition obtained by Hart and
Monderer Ž1997.. In fact, Hart and Monderer’s condition is equivalent to
the requirement that dŽ D¨ *Ž?..rdm be continuous as a function into
L1Ž m.. The above condition is equivalent to the requirement that there is a
representation of the game as a vector measure game that satisfies the
conditions of Theorem 1. Thus, we have the following result.
LEMMA 2. A game ¨ can be represented as a ¨ ector measure game f ( m ,
with f weakly continuously differentiable on the extended range of m , if and
only if ¨ g pNA` . A game ¨ can be represented as a ¨ ector measure game
f ( m , with f weakly continuous on the extended range of m , if and only if
¨ g pNA9.
The following sufficient condition for a monotonic ideal game to be the
continuous extension of a game in pNA is derived from Proposition 1.
PROPOSITION 2. If ¨ * is a monotonic ideal game such that lim t ª 0q ¨ *Ž t .
s 0 and lim t ª 1y ¨ *Ž t . s ¨ *Ž1., and there exists a nonatomic probability
measure m such that, for e¨ ery h g I such that 0 - mŽ h. - 1, the deri¨ ati¨ e
D¨ *Ž h. exists and is absolutely continuous with respect to m, dŽ D¨ *Ž h..rdm
is essentially bounded, and dŽ D¨ *Ž?..rdm is continuous at h as a function
into L`Ž m., then the game ¨ defined by ¨ Ž S . s ¨ *Ž xS . is in pNA and its ¨ alue
is gi¨ en by Ž5..
PROOFS
Proof of Lemma 1. For every h g I y I and every continuous linear
functional x* g X *, x*Ž m Ž h.. s H h dŽ x*( m . s H h dŽ x*( m .rd < m < d < m <
by Theorem IV.10.8 of Dunford and Schwartz Ž1958.. Taking the maximum
over the unit sphere in X *, we get 5 m Ž h.5 X F H < h < d < m < s 5 m
ˆ Ž h.5 L 1 Ž < m <. ,
since 5 dŽ x*( m .rd < m < 5 L`Ž < m <. F 5 x* 5 X * for every x*. Therefore, m
ˆ Ž h. ¬
Ž
.
Ž
.
m Ž h. is a well-defined continuous function from m
I
onto
m
I
that is
ˆ
continuous also with respect to the relative weak topologies on these
spaces. It follows that fˆ is well defined by Ž3. and that it is Žweakly.
33
VECTOR MEASURE GAMES
continuous at m
ˆ Ž h. if f is Žrespectively, weakly. continuous at mŽ h.. Also,
if f is weakly continuously differentiable at m Ž h. then Eq. Ž4. well defines
a continuous linear functional DfˆŽ m
ˆ Ž h.. g L1Ž< m <.* wwhich is in fact equal
to, or rather identifiable with, d Ž Df Ž m Ž h ..( m .rd < m < g L`Ž < m <.x.
5 DfˆŽ m
ˆ Ž g .. y DfˆŽ m
ˆ Ž h..5 L 1 Ž < m <.* F 5 Df Ž mŽ g .. y Df Ž mŽ h..5 SpŽ m Ž I ..* holds
for every m
ˆ Ž g . in a weak neighborhood of m
ˆ Ž h., and therefore DfˆŽ?. is
weakly continuous at m
ˆ Ž h.. By Ž3. and Ž4., and the definition of Df Ž mŽ h..,
for every m
ˆŽ g . g m
ˆŽ I .
fˆŽ m
ˆ Ž h. q u Ž m
ˆŽ g. y m
ˆ Ž h. . .
s fˆŽ m
ˆ Ž h. . q u ² m
ˆŽ g. y m
ˆ Ž h . , DfˆŽ m
ˆ Ž h. . : q oŽ u .
as u ª 0q. This proves that fˆ is differentiable at m
ˆ Ž h. and that the
Ž
..
continuous linear functional DfˆŽ m
h
is
indeed
its
derivative
there. B
ˆ
Proof of Theorem 1. Suppose that f satisfies the condition of the
theorem. In light of Lemma 1, it can be assumed without loss of generality
that X s L1Ž m., where m is a nonatomic probability measure, and that
m Ž S . s xS Ž S g C .. For a sub-s-field F of C , define an L1Ž m.-valued
nonatomic vector measure m F by mF Ž S . s EŽ xS N F ., where EŽ?N F . denotes conditional expectation. If F is finite then the range of m F , which is
a subset of the convex hull of the range of m , clearly spans a finite
dimensional subspace of L1Ž m.. Therefore, by an immediate extension of
Proposition 7.1 of Aumann and Shapley Ž1974., f ( mF g pNA` . To prove
that f ( m is in pNA` it suffices to show that
lim 5 f ( mF y f ( m 5 ` s 0
Ž 6.
F
as F varies over the finite subfields of C , directed by inclusion. ŽThat is,
for every « ) 0 there exists a finite measurable partition of I such that, if
F is the field generated by some finer finite measurable partition, 5 f ( mF
y f ( m 5 ` - « ..
Let F be a sub-s-field of C , and let S, T g C be such that T : S. For
0 F t F 1, define h t s x T q t xS _T Žg I .. By the fundamental theorem of
calculus, applied to the function t ¬ f Ž mF Ž h t .. Žs f Ž mF ŽT . q t mF Ž S _
T ...,
1
Ž f ( mF . Ž S . y Ž f ( mF . Ž T . s H ² mF Ž S _ T . , Df Ž mF Ž h t . . : dt
0
1
s
H0 HI m
s
H0 HS_T E Ž Df Ž m
F
Ž S _ T . Df Ž mF Ž h t . . dm dt Ž 7 .
1
F
Ž h t . . N F . dm dt,
34
IGAL MILCHTAICH
where the last equality follows from the identity
HI E Ž h N F . g dm s HI hE Ž g N F . dm
Ž h g L1Ž m . , g g L`Ž m . . ,
applied to the functions h s xS _T and g s Df Ž mF Ž h t ... Note that the
same notation is used in Ž7. for the derivative of f at a point and for the
representation of that derivative as an element of L`Ž m.. In the special
case F s C we get
1
Ž f ( m. Ž S. y Ž f ( m. ŽT . s H
0
HS_T Df Ž m Ž h . . dm dt.
t
Ž 8.
It follows from Ž7. and Ž8. that
5 f ( mF y f ( m 5 ` F sup 5 E Ž Df Ž E Ž h N F . . N F . y Df Ž m Ž h . . 5 L`Ž m. .
hg I
Hence, in order to complete the proof of Ž6. if suffices to show that
lim 5 E Ž Df Ž E Ž h N F . . N F . y Df Ž E Ž h N F . . 5 L`Ž m. s 0
Ž 9.
lim 5 Df Ž E Ž h N F . . y Df Ž m Ž h . . 5 L`Ž m. s 0
Ž 10 .
F
and
F
uniformly in h g I as F varies over the finite subfields of C , directed by
inclusion. Df is a weakly continuous function defined on a weakly compact
subset of L1Ž m.. Therefore, it is uniformly weakly continuous and its range
is compact. Theorem IV.8.18 of Dunford and Schwartz Ž1958. asserts that,
for every compact subset K of L`Ž m., lim F 5 EŽ g N F . y g 5 L`Ž m. s 0 uniformly in g g K. This proves Ž9.. The same theorem, together with the
above identity, implies that, for every g g L`Ž m., lim F H EŽ h N F . g dm s
lim F H hEŽ g N F . dm s H hg dm uniformly in h g I . Thus, with respect to
the weak topology on L1Ž m., lim F EŽ h N F . s m Ž h. uniformly in h g I .
This, and the uniform weak continuity of Df, together imply Ž10..
Since f is weakly continuous, it follows from Lemma 5 in the Appendix
that the ideal game defined by h ¬ f Ž m Ž h.. is a continuous extension of
f ( m. Therefore, by Theorem H of Aumann and Shapley Ž1974., the value
of f ( m is given by
w Ž f ( m. Ž S. s
1
H0
d
du
us0
f Ž t m Ž I . q um Ž S . . dt
ŽS g C.
Žand the derivative on the right-hand side of this equation exists for almost
every 0 - t - 1.. By definition of Df, this gives Ž2..
35
VECTOR MEASURE GAMES
The proof of the second part of the theorem will be given after the proof
of Theorem 2. B
Proof of Proposition 1. We prove Proposition 1 by making the following
two modifications to the proof of the first part of Theorem 1.
First, since f is continuous at 0, and since m Ž h. ª 0 is equivalent to
mŽ h. ª 0, f is actually weakly continuous at 0. Similarly, f is weakly
continuous at m Ž I .. Since f is weakly continuously differentiable, and
hence weakly continuous, in m Ž h g I N 0 - mŽ h. - 14. s m Ž I . _
0, m Ž I .4 , the ideal game h ¬ f Ž m Ž h.. is continuous. Aumann and Shapley
Ž1974, p.150. showed that a continuous extension of a monotonic game is a
monotonic ideal game. It follows that, for every finite subfield F of C , the
restriction of f to mF Ž I . can be viewed as a nondecreasing continuous
function on the unit cube in R n, where n is the dimension of the subspace
of L1Ž m. that is spanned by mF Ž I . Žwhich is equal to the number of atoms
of nonzero m-measure of the field F .. This function is continuously
differentiable outside of the origin and Ž1, 1, . . . , 1.. Therefore, by an
extension of Proposition 10.17 of Aumann and Shapley Ž1974, proposition
and extension in p. 92., f ( mF g pNA.
Second, for fixed « ) 0, let 0 - d - 1r2 be such that f Ž m Ž h.. - « and
Ž
f m Ž1 y h.. ) f Ž m Ž I .. y « for every h g I that satisfies mŽ h. F d . If
S0 : S1 : ??? : Si 0 : ??? : Si1 : ??? : I is a finite nondecreasing sequence of coalitions such that mŽ Si 0 . s d and mŽ Si1 . s 1 y d , and if F is
a finite subfield of C , then for every i 0 - i F i1 Eqs. Ž7. and Ž8. hold for
S s Si and T s Siy1. The monotonicity of f ( m and of f ( mF implies that
0
<Ž f ( m F .Ž S i . y Ž f ( m .Ž S i . y Ž f ( m F .Ž S iy1 . q Ž f ( m .Ž S iy1 .< - 2 « ,
Ý iis1
and a similar inequality holds for the sum over i1 q 1, i1 q 2, . . . . It
follows that 5 f ( mF y f ( m 5 BV - supd F mŽ h.F 1y d 5 EŽ Df Ž EŽ h N F .. N F . y
Df Ž m Ž h..5 L`Ž m. q 4« . Since the set m Ž h. g m Ž I . N d F mŽ h. F 1 y d 4 is
weakly compact, an argument similar to that given in the proof of Theorem 1 shows that the limit of the last supremum as F varies over the finite
subfields of C , directed by inclusion, is zero. Since « is arbitrary, this
proves that lim F 5 f ( m F y f ( m 5 BV s 0, and therefore f ( m g pNA. B
Proof of Theorem 2. If ¨ * satisfies the condition of the theorem then
¨ *Ž g . s ¨ *Ž h. for every g and h in I which are equal m-almost every-
where. Indeed, by the mean value theorem, there exists some 0 - u - 1
such that
¨ * Ž g . y ¨ * Ž h . s D¨ * Ž h q u Ž g y h . . Ž g y h .
s
H Ž g y h.
d Ž D¨ * Ž h q u Ž g y h . . .
dm
dm s 0.
36
IGAL MILCHTAICH
Therefore, there exists a unique function f : m Ž I . ª R, where m is the
L1Ž m.-valued nonatomic vector measure of bounded variation defined by
m Ž S . s xS , such that ¨ *Ž h. s f Ž m Ž h.. Ž h g I .. This function is differentiable. Indeed, its derivative at m Ž h. is Df Ž m Ž h.. s dŽ D¨ *Ž h..rdm g
L`Ž m. Žs L1Ž m.*., and it follows that Df Ž m Ž?.. is continuous on I . Hence,
Lemma 5 in the Appendix implies that Df is weakly continuous. Therefore, by Theorem 1, f ( m g pNA` . The formula for the value of ¨ Žs f ( m .
now follows from Ž2. and from the above expression for Df.
Conversely, suppose that ¨ * is the continuous extension of a game ¨ in
pNA` . There exists a sequence ¨ n4nG 1 of games in pNA` whose continuous extensions satisfy the condition of the theorem such that 5 ¨ n y ¨ 5 ` 4yn for every n. Indeed, we can take these games to be polynomials in NA
measures. For every u g pNA` and for every « ) 0 there is a nonatomic
probability measure m such that < m*Ž h q u g . y u*Ž h.< F < u <Ž5 u 5 ` q
« . mŽ g . for every g, h g I and u g R such that h q u g g I Žsee the
Appendix.. It follows that if u* is differentiable at h then
< Du* Ž h . Ž g . < F Ž 5 u 5 ` q « . m Ž g . F 5 u 5 ` q «
< D¨ nU Ž h.Ž g .
Ž ggI ..
U
Ž h.Ž g .< F 5 ¨ n y ¨ n9 5 ` for every n, n9, and
y D¨ n9
U
D¨ n Ž?.Ž?. converges uniformly to a real-valued
In particular,
g, h g I , and therefore
function g Ž?, ? . on I = I . For every h, g Ž h, ? . is Žthe continuous extension of. a nonatomic measure and, for every g, g Ž?, g . is continuous on I .
Since, for every g, h g I and u g R such that h q u g g I and for every
n, ¨ nU Ž h q u g . y ¨ nU Ž h. s H0u D¨ nU Ž h q tg .Ž g . dt, in the limit we get ¨ *Ž h
q u g . y ¨ *Ž h. s H0u g Ž h q tg, g . dt. This equation implies that ¨ * is differentiable at h wand D¨ *Ž h. s g Ž h, ? .x.
For every n, let m n be a nonatomic probability measure such that
< D¨ nU Ž h.Ž g . y D¨ *Ž h.Ž g .< F 4yn m nŽ g . Ž g, h g I .. The nonatomic probability measure m s Ý k G 1 2yk m k satisfies 4yn m nŽ g . F 2yn mŽ g .. Therefore, D¨ nU Ž h. y D¨ *Ž h. is absolutely continuous with respect to m and
d Ž D¨ nU Ž h . y D¨ * Ž h . .
dm
L`Ž m .
F 2yn .
Since ¨ n satisfies the condition of the theorem, we may assume without
loss of generality that, for every h, D¨ nU Ž h. is absolutely continuous with
respect to m n , its Radon]Nikodym derivative with respect to m n is
essentially bounded, and dŽ D¨ nU Ž?..rdm n is continuous at h as a function
into L`Ž m n .. This remains true when m n is replaced by m. It follows that,
for every h, D¨ *Ž h. too is absolutely continuous with respect to m and its
Radon]Nikodym derivative with respect to m is essentially bounded. And
since the function dŽ D¨ *Ž?..rdm: C ª L`Ž m. is the uniform limit of the
continuous functions dŽ D¨ nU Ž?..rdm n4n G 1 , this function is continuous, too.
B
VECTOR MEASURE GAMES
37
Proof of Theorem 1 Ž continued.. Suppose that dim X - `. If a vector
measure game f ( m is in pNA` then by Theorem 2 its continuous
extension ¨ * is differentiable. It is shown in the Appendix that ¨ *Ž h. s
f Ž m Ž h.. Ž h g I .. Hence, D¨ *Ž h.Ž g y h. s drdu < us0q f Ž m Ž h q u Ž g y h...
for every g and h. The value of this derivative clearly depends only on
m Ž h. and on m Ž g y h., and can therefore be written as g
ˆ Ž mŽ h..Ž mŽ g y h..,
where g
ˆ is a function from mŽ I . to X *. By Theorem 2, D¨ *Ž?.Ž g . is
continuous for every g g I . Therefore, by Lemma 5 in the Appendix,
gˆ Ž?.Ž m Ž g .. is continuous for every g. This proves that f is continuously
differentiable on m Ž I .. B
Proof of Lemma 2. If f is weakly continuously differentiable on the
extended range of a nonatomic vector measure of bounded variation m
then f ( m g pNA` by Theorem 1. Conversely, if ¨ is in pNA` then it is
shown in the proof of Theorem 2 that ¨ can be represented as a vector
measure game f ( m , with f weakly continuously differentiable.
If f ( m is a vector measure game such that f is weakly continuous on
the extended range of m , which is a subset of some Banach space X, then
by Stone]Weierstrass theorem f can be uniformly approximated by polynomials in elements of the dual space X *. Specifically, since m Ž I . is
weakly compact, and since the continuous linear functionals on X separate points in this set, for every « ) 0 there are a finite sequence of
functionals xU1 , xU2 , . . . , xUn g X * and a polynomial p in n variables such
that < f Ž m Ž h.. y pŽ xU1 Ž m Ž h.., xU2 Ž m Ž h.., . . . , xUn Ž m Ž h...< - « for every h g
I . Since xUi ( m g NA for every i, it follows from Lemma 7.2 of Aumann
and Shapley Ž1974. that f ( m g pNA9.
Conversely, it is shown in the Appendix that if ¨ * is the continuous
extension of a game ¨ in pNA9 then there is a nonatomic probability
measure m such that, for every g, h g I which are equal m-almost
everywhere, ¨ *Ž g . s ¨ *Ž h.. It follows that there exists a real-valued function f, defined on the extended range of the nonatomic vector measure of
bounded variation m defined as in the Example, such that ¨ *Ž h. s f Ž m Ž h..
Ž h g I .. Since ¨ * is continuous, by Lemma 5 in the Appendix f is weakly
continuous. B
Proof of Proposition 2. It suffices to show that if ¨ * satisfies the
conditions of the proposition then there exist a vector measure m and a
function f, which satisfy the conditions of Proposition 1, such that ¨ *Ž h.
s f Ž m Ž h.. Ž h g I .. Once we establish that ¨ *Ž h. s 0 for every h which
is equal m-almost everywhere to 0 and that ¨ *Ž g . s ¨ *Ž1. for every g
which is equal m-almost everywhere to 1 we can proceed almost exactly as
in the proof of Theorem 2: define m and f as in that proof, and use the
same arguments to show that Df Ž m Ž?.. exists and is continuous in h g I N
0 - mŽ h. - 14 s h g I N m Ž h. / 0, m Ž I .4 , and that Df is therefore weakly
38
IGAL MILCHTAICH
continuous in m Ž I . _ 0, m Ž I .4 . The proof of Eq. Ž5. is also similar. Hence,
it only remains to show that ¨ *Ž h. ª 0 when m Ž h. ª 0 wor, equivalently,
when mŽ h. ª 0x and that ¨ *Ž g . ª ¨ *Ž1. when m Ž g . ª m Ž I . wor, equivalently, when mŽ g . ª 1x.
For every g, h g I such that h F g and mŽ h. - mŽ g ., ¨ *Ž h q t Ž g y h..
is nondecreasing as a function of t in the interval w0, 1x and has a
nonnegative continuous derivative in the interior of that interval. For
h s 0 Židentically. and g s 1 Židentically. this function is continuous also
at the end points. Therefore, for every sequence h n4 : I such that
m Ž h n . ª 0 and for every sequence g n4 : I such that h n F g n and m Ž g n .
ª m Ž I .,
lim inf n ¨ * Ž g n . y ¨ * Ž h n .
1
D¨ * Ž h n q t Ž g n y h n . . Ž g n y h n . dt
1
² m Ž g n y h n . , Df Ž m Ž tg n q Ž 1 y t . h n . . : dt
G lim inf n
H0
s lim inf n
H0
G
1
H0
² m Ž I . , Df Ž t m Ž I . . : dt s ¨ * Ž 1 .
by Fatou’s lemma and the continuity of Df in m Ž I . _ 0, m Ž I .4 . Taking
g n s 1 for every n shows that ¨ *Ž h n . ª 0. Taking h n s 0 for every n
shows that ¨ *Ž g n . ª ¨ *Ž1.. B
APPLICATION: MARKET GAMES
We give a new proof to the following result, due to Aumann and Shapley
Ž1974, Chapter VI..
THEOREM 3. Suppose that Ž I, C . s Žw0, 1x, the Borel sets .. Let m Ž the
population measure. be a nonatomic probability measure, k Ž the number of
different goods. a positi¨ e integer, a Ž the endowment . an m-integrable funck
tion from I into the interior of the k-dimensional nonnegati¨ e orthant Rq
, and
k
u Ž the utility function. a real-¨ alued function that is defined on Rq= I and
satisfies the following assumptions:
k
for e¨ ery j g Rq
, u Ž j , ? . is a measurable function on I ;
Ž 11 .
k
for e¨ ery i g I, u Ž ?, i . is a continuous function on Rq
;
Ž 12 .
VECTOR MEASURE GAMES
39
for e¨ ery i g I, u Ž ?, i . is strictly increasing Ž in each component separately . ,
and u Ž 0, i . s 0;
Ž 13 .
k
for e¨ ery i g I and j, ­ u Ž j , i . r­j j exists and is continuous at each j g Rq
for which j j ) 0; and
uŽ j , i . s o
ž Ýj /
j
as
Ý j j ª `, integrably in i ,
j
Ž 14 .
Ž 15 .
j
that is, for e¨ ery « ) 0 there is an m-integrable function g : I ª R such that,
k
for e¨ ery j g Rq
and i g I, Ý j j j G g Ž i . implies uŽ j , i . F « Ý j j j . Then for
e¨ ery coalition S the maximum in Ž1. is attained, and the market game ¨
defined by this equation is in pNA. The ¨ alue of this game coincides with the
unique competiti¨ e payoff distribution of the Ž transferable utility . market.
Proof. It follows from Ž11. and Ž12. that u is Borel measurable ŽKlein
and Thompson, 1984, Lemma 13.2.3.. Define an ideal game ¨ *: I ª R by
¨ * Ž h . s max
H u Ž x . h dm,
Ž 16 .
k
where the maximum is taken over the set x: I ª Rq
N x is measurable and
H xh dm s H ah dm4 of all Žfeasible . allocations and uŽ x . denotes the function on I whose value at i is uŽ x Ž i ., i .. This maximum is attained Žwhich
means, in particular, that it is finite; Aumann and Shapley, 1974, Proposition 36.1.. Furthermore, by Proposition 36.4 and the discussion in pp.
189]190 of Aumann and Shapley Ž1974., for every h such that H h dm ) 0
k
there is a unique vector pŽ h. in the interior of Rq
, called the vector of
competiti¨ e prices corresponding to h, such that, for some allocation x,
u Ž x Ž i . , i . y p Ž h . ? x Ž i . s max u Ž j , i . y p Ž h . ? j
k
jgR q
Ž 17 .
for m-almost every i for which hŽ i . / 0 Žthe dot stands for scalar product..
It is not difficult to see that such an allocation x maximizes the integral in
Ž16.. If x is an allocation which satisfies Ž17. for e¨ ery i then the pair
Ž x, pŽ h.. is called a transferable utility competiti¨ e equilibrium corresponding
to h. Such an allocation always exists: Since u is Borel measurable on
k
Rq
= I and is continuous in the first argument, there exists a measurable
k
function x: I ª Rq
that satisfies Ž17. for every i for which the maximum
on the right hand side of that equation is attained ŽWagner, 1977, Theorem 9.2.. But, as we show next, this maximum is in fact attained for every i.
It follows that, given an allocation x that satisfies Ž17. for m-almost every i
such that hŽ i . / 0, we can change the values that x takes at those points
where Ž17. does not hold in such a way that the new function be an
allocation that satisfies Ž17. everywhere.
40
IGAL MILCHTAICH
For every positive integer s there exists, by Ž15., an m-integrable
k
function gs : I ª Ž0, `. such that, for every i and j g Rq
,
1
e ? j G gs Ž i . implies u Ž j , i . - e ? j ,
Ž 18 .
s
where e is the vector with all components 1. If s ) 1rmin j pj Ž h. then
1rs e ? j F p ? j . It then follows from Ž18. that, for every i, the maximum
k
on the right-hand side of Ž17. is attained in, and only in, the set j g Rq
N
e ? j - gs Ž i .4 . Consequentially, if Ž17. holds, x Ž i . / 0, and min j pj Ž h. ) 1rs
then gs Ž i .re ? x Ž i . ) 1, and hence
0 F uŽ x Ž i . , i . y pŽ h. ? x Ž i . - u
ž
gs Ž i .
e ? xŽ i.
x Ž i . , i y pŽ h. ? x Ž i .
/
- gs Ž i . y p Ž h . ? x Ž i .
Ž 19 .
by Ž13. and Ž18.. It follows that if x is an allocation that satisfies Ž17. for
every i then H uŽ x . dm - H gs dm - `.
LEMMA 3. The function pŽ?. that sends each element of the set Im s h
g I N mŽ h. ) 04 to the corresponding ¨ ector of competiti¨ e prices is continuous Ž with respect to the NA-topology..
Proof. It suffices to show that if h n4n G 0 : Im is such that H gh n dm ª
H gh 0 dm for every g g L1Ž m. then pŽ h n . ª pŽ h 0 .. The idea Žborrowed
from Aumann and Shapley, 1974, p. 188. is to identify the competitive
prices corresponding to h g Im with equilibrium prices of a suitable
exchange economy Eh Žsee Hildenbrand, 1974, Chapter 2.. There are
k q 1 kinds of goods in Eh : the k original ones plus ‘‘money’’ Žthe 0-th
k
kq1 Ž
good.. The consumption set of each player i is Rq= Rq
s Rq
thus, no
.
player is allowed to hold a negative amount of money , his utility function
is given by Ž j 0 , j . ¬ uŽ j , i . q j 0 , and his endowment is Žgs Ž i ., aŽ i ..,
where s is some positive integer, that does not depend on i, such that
min j pj Ž h. ) 1rs. The population measure is h dm. As is readily verified
wsee Ž19.x, an allocation x satisfies Ž17. if and only if the bundle Žgs Ž i . y
pŽ h. ? x Ž i . q pŽ h. ? aŽ i ., x Ž i .. maximizes player i’s utility in the set Ž j 0 , j .
k
g Rq= Rq
N j 0 q pŽ h. ? j s gs Ž i . q pŽ h. ? aŽ i .4 . It follows that Ž1, pŽ h.. are
equilibrium prices for Eh . Moreover, by the uniqueness of the competitive
prices corresponding to h, these are the only equilibrium prices for Eh
which are of the form Ž1, p ., with min j pj ) 1rs.
If there is some positive integer s such that inf j, n pj Ž h n . ) 1rs then, for
every i and n, we can choose the monetary endowment of player i in Eh n
to be equal to gs Ž i .. The exchange economies then differ only in their
population measures. The assumption concerning h n4 implies, in this case,
that the Žpreference]endowment. distribution of Eh n tends to that of Eh 0 ,
and similarly for the aggregate endowments. It follows, by Proposition 4 in
VECTOR MEASURE GAMES
41
Section 2.2 of Hildenbrand Ž1974., that every cluster point of the corresponding sequence of normalized equilibrium prices 1r Ž1 q
Ý j pj Ž h n .. Ž1, pŽ h n ..4 is a Ž k q 1.-tuple of normalized equilibrium prices
for Eh 0 . The first component of such a cluster point cannot be zero: the
equilibrium price of money must be positive. Therefore, the sequence
pŽ h n .4 must be bounded. If p is a cluster point of this sequence then
Ž1, p . are equilibrium prices for Eh . And since min j pj ) 1rs, p s pŽ h 0 ..
0
This proves that pŽ h n .4 converges to pŽ h 0 ..
It remains to show that inf j, n pj Ž h n . s 0 is impossible. We will prove
this by assuming that pj Ž h n . ª 0 for some j and showing that this leads to
a contradiction. If, for every n, Ž x n , pŽ h n .. is a transferable utility competitive equilibrium corresponding to h n then in particular uŽ x n . y pŽ h n . ? x n
G uŽ x n q e j . y pŽ h n . ? Ž x n q e j ., where e j is the jth unit vector in R k ,
and therefore 0 - uŽ x n q e j . y uŽ x n . F pj Ž h n . ª 0. This is consistent with
Ž12. and Ž13. only if x nŽ i . ª ` for every i, and hence only if H max e ? Ž2 a
y x n ., 04 h n dm ª 0. But this contradicts the fact that H max e ? Ž2 a y
x n ., 04 h n dm G e ? H Ž2 a y x n . h n dm s e ? H ah n dm ª e ? H ah 0 dm ) 0.
B
Proof of Theorem 3 Ž continued.. Let g and h be two elements of Im ,
and let Ž y, pŽ g .. and Ž x, pŽ h.. be two corresponding transferable utility
competitive equilibria. Since Ž17. holds for every i,
¨ *Ž g . y ¨ *Ž h. s
H u Ž y . g dm y H u Ž x . h dm
s
H u Ž x . Ž g y h . dm y H
u Ž x . y u Ž y . g dm
F
H u Ž x . Ž g y h . dm y H p Ž h . ? Ž x y y . g dm
s
H u Ž x . Ž g y h . dm y p Ž h . ? H Ž x y a. g dm
s
H
Ž 20 .
u Ž x . y p Ž h . ? Ž x y a . Ž g y h . dm.
Similarly,
¨ *Ž h. y ¨ *Ž g . F
H
u Ž y . y p Ž g . ? Ž y y a . Ž h y g . dm.
Ž 21 .
If h F g then the right-hand side of Ž21. is nonpositive. Hence, ¨ * is
monotonic.
42
IGAL MILCHTAICH
Let 0 - c 0 - c1 - ??? be such that the series g s 1rc0 e ? a q Ý s G 1 1rc s
gs converges m-almost everywhere and H g dm s 1. Let mg be the
nonatomic probability measure defined by dmg s g dm. It follows from
Ž21. and Ž22. that, for some 0 F u F 1,
¨ *Ž g . y ¨ *Ž h. s
1
Hg
u Ž x . y p Ž h . ? Ž x y a . q u« g , h Ž g y h . dmg ,
where « g, h : I ª R is defined by « g, hŽ i . s max j g Rqk w uŽ j , i . y pŽ g . ? Ž j y
aŽ i ..x y max j g Rqk w uŽ j , i . y pŽ h. ? Ž j y aŽ i ..x. If s is a positive integer
such that pj Ž g ., pj Ž h. ) 1rs for all j then, as shown above, both maxima
k
are attained in the set j g Rq
N e ? j - gs Ž i .4 . Therefore, m-almost every<
<
where, 1rg « g, h F 1rg max e?j F g s < Ž pŽ g . y pŽ h.. ? Ž j y a.< F 1rg Žgs q
e ? a. max j < pj Ž g . y pj Ž h.< F c s max j < pj Ž g . y pj Ž h.<. Since, by Lemma 3,
pŽ?. is continuous on Im , max j < pj Ž g . y pj Ž h.< ª 0 when g ª h. This
proves that ¨ * is differentiable at h, that its derivative there is absolutely
continuous with respect to mg , and that
d Ž D¨ * Ž h . .
dmg
s
1
g
u Ž x . y p Ž h . ? Ž x y a . g L`Ž mg . .
ŽThe essential boundness of 1rg w uŽ x . y pŽ h. ? Ž x y a.x follows from Ž19.
and from the definition of g .. Since, for every g and h as above,
5 d Ž D ¨ * Ž g .. rdm g y d Ž D ¨ * Ž h .. rdm g 5 L ` Ž m g . s 5 1r g « g , h 5 L ` Ž m g . F
c s max j < pj Ž g . y pj Ž h.<, the function dŽ D¨ *Ž?..rdmg is continuous at h.
A transferable utility competitive equilibrium Ž x, pŽ h.. corresponding to
an ideal coalition h g Im is easily seen to correspond also to th, for every
0 - t - 1. Therefore, ¨ *Ž th. s t¨ *Ž h. for every 0 F t F 1. ŽIncidentally,
this shows that ¨ * is differentiable at 0 if and only if ¨ g NA.. In
particular, lim t ª 0q ¨ *Ž t . s 0 and lim t ª 1y ¨ *Ž t . s ¨ *Ž1., and if Ž x, pŽ1.. is
a transferable utility competitive equilibrium corresponding to the ideal
coalition h s 1 then dŽ D¨ *Ž t ..rdmg s 1rg w uŽ x . y pŽ1. ? Ž x y a.x for
every 0 - t F 1. It follows, by Proposition 2, that the market game ¨ is in
pNA and its value is given by
Ž w ¨ . Ž S . s H u Ž x . y p Ž 1 . ? Ž x y a. dm
S
ŽS g C..
Thus, the value of ¨ is equal to the competitive payoff distribution
ŽAumann and Shapley, 1974, p. 184. of the market. B
VECTOR MEASURE GAMES
43
APPENDIX
Approximation Lemma
The method of approximation employed in the proofs of Theorem 1 and
Proposition 1 can be used more generally for approximating games in
pNA` , pNA, or pNA9. All three spaces of games are generated by powers
of nonatomic probability measures, but the norm is different in each case.
The norm on pNA9 is the supremum norm 5 ? 59, the norm on pNA is the
variation norm 5 ? 5 BV , and pNA` is endowed with the norm 5 ? 5 ` . These
norms satisfy 5 ? 59 F 5 ? 5 BV F 5 ? 5 ` , and the spaces themselves satisfy
pNA` : pNA : pNA9. Each of the three norms can be ‘‘extended’’ in a
natural way to a norm on the linear space of extensions of games in the
respective space. Specifically, we define 5 ¨ * 59 s sup h g I < ¨ *Ž h.<, 5 ¨ * 5 BV s
sup 0 F h 0 F ? ? ?F h n F 1 Ý nis1 5 ¨ *Ž h i . y ¨ *Ž h iy1 .5, or 5 ¨ * 5 ` s inf mŽ I . N m g
NA, and < ¨ *Ž g . y ¨ *Ž h.< F mŽ g y h. for every g, h g I , h F g 4 when the
ideal game ¨ * is the continuous extension of a game ¨ in pNA9, in pNA,
or in pNA` , respectively. The extension operator ¨ ¬ ¨ * is linear and
norm-preserving on each of the three spaces ŽAumann and Shapley, 1974,
p. 151; Hart and Monderer, 1997.. If f ( m is a vector measure game in
pNA9 such that the range of m is finite dimensional then the continuous
extension ¨ * of f ( m is given by ¨ *Ž h. s f Ž m Ž h... This can be shown as
follows. First, since the range of a finite dimensional nonatomic vector
measure coincides with its extended range, for every h g I we can find a
coalition S such that m Ž h. s m Ž S .. Second, for the same reason, for every
given neighborhood of h in I we can choose S in such a way that xS is in
that neighborhood. Therefore, by the continuity of ¨ *, for every given
« ) 0 we may assume that S satisfies < ¨ *Ž xS . y ¨ *Ž h.< - « . But ¨ *Ž xS . s
f Ž m Ž S .. s f Ž m Ž h...
For every game ¨ in pNA9 there exists a nonatomic probability measure
m such that ¨ is in pNA9Ž m., the closed linear subspace of pNA9 that is
generated by powers of nonatomic probability measures which are absolutely continuous with respect to m. wIf mŽ k .4k G 1 is a sequence of vectors
Žk.
of nonatomic probability measures, mŽ k . s Ž mŽ1k ., mŽ2k ., . . . , m nŽ k . ., and
p Ž k .4k G 1 is a sequence of polynomials such that 5 ¨ y p Ž k . ( mŽ k . 59 ª 0,
Žk.
then m can be chosen as Ý k G 1 1rŽ2 k nŽ k . .Ž mŽ1k . q mŽ2k . q ??? qm nŽ k . ..x For
every g, h g I such that g s h m-almost everywhere, ¨ *Ž g . s ¨ *Ž h..
Similarly, if ¨ is in pNA or in pNA` then there exists a nonatomic
probability measure m such that ¨ is in pNAŽ m. or in pNA`Ž m., respectively. These subspaces are defined in a similar way to pNA9Ž m.. For every
finite subfield F of C and for every g g L1Ž m., the conditional expectation EŽ g N F . is defined as that function on I which is constant on
44
IGAL MILCHTAICH
each atom S of the field F and is equal there to 1rmŽ S . HS g dm ws 0,
by convention, if mŽ S . s 0x. The ideal game ¨ FU defined by ¨ FU Ž h. s
¨ *Ž EŽ h N F .. is easily seen to be continuous. Its ‘‘restriction’’ to C is the
game ¨ F defined by ¨ F Ž S . s ¨ FU Ž xS ..
LEMMA 4. Let X be one of the three spaces, pNAŽ m., pNA`Ž m., or
pNA9Ž m., and let 5 ? 5 be the norm on that space. Then a game ¨ is in X if and
only if ¨ F is in X for e¨ ery finite field F : C and lim F 5 ¨ F y ¨ 5 s 0 as F
¨ aries o¨ er the finite subfields of C , directed by inclusion.
Proof. One direction in trivial: if ¨ is the limit of a net in X then
¨ g X. Conversely, if ¨ is in X and F is a finite subfield of C then it is
not too difficult to see that 5 ¨ F 5 F 5 ¨ * 5 Žs 5 ¨ 5.. Hence, for every game u
that is a linear combination of powers of nonatomic probability measures
which are absolutely continuous with respect to m, 5 ¨ F y u F 5 F 5 ¨ y u 5.
Therefore, it suffices to prove that u F g X and lim F 5 u F y u 5 s 0 for
every such u. Consider, then, a game of the form h k , where h is a
nonatomic probability measure that is absolutely continuous with respect
to m and k is a positive integer.
For every F , hF is a nonatomic probability measure that is absolutely
continuous with respect to m. In fact, dhFrdm s EŽ dhrdm N F .. Therefore, Žh k .F s ŽhF . k g X. Since 5 uw 5 ` F 4 5 u 5 ` 5 w 5 ` for every u, w g pNA`
and 5 l 5 ` F 2 5 l 5 BV for every l g NA ŽMonderer and Neyman, 1988.,
5 h Fk y h k 5 F 5 h Fk y h k 5 ` s 5 Ý klsy01 h Fk y ly 1h l Ž h F y h . 5 ` F
1 k5
5 ky ly1 5h 5 `l 5hF y h 5 BV s k4 k 5 EŽ dhrdm N F . y dhrdm 5 L 1 Ž m. .
Ý ky
ls0 4 hF `
By Theorem IV.8.18 of Dunford and Schwartz Ž1958., the limit of the last
expression as F varies over the finite subfields of C , directed by inclusion,
is zero. B
Topological Lemma
The relative weak topology on the extended range of a nonatomic vector
measure of bounded variation m is the strongest topology on that set with
respect to which m Ž?. is continuous on I . This result constitutes the first
part of the following lemma.
LEMMA 5. If m is a nonatomic ¨ ector measure of bounded ¨ ariation with
¨ alues in a Banach space X, then a set A : m Ž I . is open with respect to the
relati¨ e weak topology on m Ž I . if and only if h g I N m Ž h. g A4 is open
Ž with respect to the NA-topology on I .. In this case, a function f from A to
some topological space Y is weakly continuous if and only if f Ž m Ž?..: h g
I N m Ž h. g A4 ª Y is continuous.
VECTOR MEASURE GAMES
45
Proof. We have to show that if ha ª h is a converging net in I then
m Ž ha . ª m Ž h. weakly, that is, x*Ž m Ž ha .. ª x*Ž m Ž h.. for every continuous
linear functional x* g X *. But this follows immediately from the fact that
x*( m g NA. Conversely, we have to show that if xa ª x is a weakly
converging net in m Ž I . then there exists h g I such that m Ž h. s x and in
every neighborhood of h there is some h9 such that m Ž h9. g xa 4 . Let
ha 4 : I be such that m Ž ha . s xa for every a . It follows from Alaoglu’s
theorem that, by passing to a subnet if necessary, we may assume that
there is some h g I such that mŽ ha . ª mŽ h. for every m g NA which is
absolutely continuous with respect to < m <. In particular, x*Ž xa . s
x*Ž m Ž ha .. ª x*Ž m Ž h.., and therefore x*Ž m Ž h.. s x*Ž x ., for every x* g
X *. This proves that m Ž h. s x. Every neighborhood of h contains an open
neighborhood of the form g g I N max 1 F iF l < m i Ž g y h.< - « 4 , where « )
0, m1 , m 2 , . . . , m k g NA are absolutely continuous with respect to < m <, and
m kq 1 , . . . , m l g NA are singular with respect to < m <. This follows from the
fact that every nonatomic measure can be written as the sum of two
nonatomic measures, one absolutely continuous with respect to < m < and
the other singular with respect to < m <. Let a be such that < m i Ž ha y h.< - «
for every i F k. Let h9 g I be equal to h in some subset of I of
< m <-measure zero in which m kq 1 , . . . , m l are supported and equal to ha
elsewhere. Then m Ž h9. s m Ž ha . s xa , and < m i Ž h9 y h.< - « for every k i F l as well as for every i F k.
The second part of the lemma follows from that fact that, for every set
B in Y, fy1 Ž B . is weakly open in m Ž I . if and only if h g I N m Ž h. g
fy1 Ž B .4 is open. B
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