数理解析研究所講究録 1264 巻 2002 年 253-262

253

AMarket Game with Infinitely Many Players
Hidetoshi Komiya (小宮英敏)
Faculty of Business and Commerce
Keio University (慶応大学商学部)

1Intro function
Amarket game that derive ffom an exchange economy in which the finite
number of traders have continuous concave monetary utility functions wasstudied fully in [4] and amarket game with infinitely many traders describedwith anon-atomic measure space was extensively investigated in [1]. The
non-atomic measure space played acrucial role to remove the concavity ofutility functions from the assumption in [4]. In this paper, we shall study
amarket game with infinite traders described with ageneral measure spacepreserving the concavity assumption for utilities. It will be shown that suchamarket game has properties parallel to those of an exact game studied in[3] and each member of the core of amarket game has an outcome densitywith respect to the measure.
Let
be ameasurable space. Agame is anonnegative real valued
function, defined on the -field , which maps the empty set to zero. An
outcome of agame is afinitely additive real valued function on
scuh
that
. For an outcome of , an integrable function satisfying$\int_{S}fd\mu=\alpha(S)$ for all
is said to be an outcome density of withrespect to . An outcome indicates outcomes to each coalitions while anoutcome density designates outcomes to every players. The core of is theset of outcomes asatisfying $\alpha(S)\geq v(S)$ for all.
To every game we associate an extended real number
defined by$(\Omega, \ovalbox{\tt\small REJECT})$$v$$*\varphi$$\mathrm{c}\mathrm{r}$$v$$\ovalbox{\tt\small REJECT}$$\alpha$$\alpha(\Omega)=v(\Omega)$$\alpha$$f$$v$$S\in*\varphi$$\alpha$

$\mu$

$v$

$S\in\ovalbox{\tt\small REJECT}$

$|v|$

$v$

$|v|= \sup\{\sum_{i=1}^{n}\lambda_{i}v(S_{i})$:$\sum_{i=1}^{n}\lambda_{i}\chi s_{:}\leq\chi_{\Omega}\}$

,

(1)

,
where $n=1$ , 2,,
is areal number. The notation
denotes$|v|<\infty$
the characteristic function of asubset of . For agame with,$\ldots$$S_{i}\in*\varphi$$\lambda_{i}$$\chi_{A}$$A$$\Omega$$v$

254
we define two games

and$\overline{v}$$\overline{v}(S)=\sup\{\sum_{i=1}^{n}\lambda:v(S_{i})$$\mathrm{v}(\mathrm{S})=\min${$\mathrm{v}(\mathrm{S})$:$\alpha$by$\hat{v}$:$\sum_{\dot{l}=1}^{n}\lambda_{i}\chi_{S}\dot{.}\leq\chi s\}$is additive,$\alpha\geq v$,(2)

S $\in P$ ,, $\alpha(\Omega)=|v|$ },S$\in\ovalbox{\tt\small REJECT}$,(3)following [3]. Agame is said to be balanced if $v(\Omega)=|v|$ , totally balanced, respectively. It is proved in [3] that the core ofifand exact if
agame is nonempty if and only if it is balanced, every exact game is totallybalanced, and every totally balanced game is balanced.
Agameis said to be monotone if $S\subset T$ implies $v(S)\leq v(T)$ .
Agame is said to be inner continuous at $S\in*$?if it follows that$\lim_{narrow\infty}v(S)=v(S)$ for any nondecreasing sequenceof measurable. Similarly, agame is said to be outer continusets such that
if it follows that $\lim_{narrow\infty}v(S_{n})=v(S)$ for any nondecreasingous at. Agame issequenceof measurable sets such that
if it is both inner and outer continuous at .
continuous at$v$$v=\hat{v}$$v=\overline{v}$

$v$

$v$

$\{S_{n}\}$$\bigcup_{n=1}^{\infty}S_{n}=S$

$S$

$v$

$\in\ovalbox{\tt\small REJECT}$$\mathrm{g}$$\bigcap_{n=1}^{\infty}S_{n}=S$$\{S_{n}\}$

$v$

$S$

$S\in\ovalbox{\tt\small REJECT}$2Market GamesLet
be afinite measure space throughout this paper. We denoteutilities of players by aCaratheodory type function defined on, wheretodenotes the nonnegative orthant of the -dimensional Euclidean space , andis the set of nonnegative real numbers. The nondesignates the density of the utility of aplayer
negative number
getting goods . We always use the ordinary coordinatewise order when
. We suppose that the function :having concern with an order insatisfies the conditions:$(\Omega, P, \mu)$$\Omega \mathrm{x}R_{+}^{l}$$u$$R_{+}$$l$$R_{+}^{l}$$R^{l}$$R_{+}$$u(\omega, x)$$\omega$$x$$u$$R_{+}^{l}$$\Omega\cross R_{+}^{l}arrow R_{+}$

1. The function$\omega$x) is measurable for all x$\mapsto u(\omega,$$\in R_{+}^{l}$;x) is continuous, concave, nondecreasing, and
2. The function x$u(\omega, 0)=0$ , for almost allin ;$\mapsto \mathrm{v}(\mathrm{S})$$\omega$3.

x) :$(\omega, x)\in\Omega \mathrm{x}B_{+}\}<\infty$ , where $B_{+}=$ {x$\sigma\equiv\sup\{u(\omega,$$||x||\leq 1\}$

$R_{+}^{l}$$\Omega$, and$||x||$

denotes the Euclidean norm of x$\in R_{+}^{l}$$\in R_{+}^{l}$:.

For any measurable set S $\in*r$ , the set of integrable furictions on S to
. We take an element e ofas theis denoted by$L_{1}(S, R_{+}^{l})$$L_{1}(S, R_{+}^{l})$255
density of initial endowments for the players. For any$v(S) \equiv\sup\{\int_{S}u(\omega, x(\omega))d\mu,(w)s$ $x\in L_{1}(S, R_{+}^{l})$,$S\in\ovalbox{\tt\small REJECT}$, defifine$\int_{S}xd\mu=\int_{S}ed\mu\}.(4)$The set function defifined above is called a market game derived from the.
marketWe shall confirm that the market game is actually a game in theisrest of this section. It is well known that the function
measurable for any $x\in L_{1}(S, R_{+}^{l})$ . Moreover we need to show that the
mapping
is integrable in order to defifine $v(S)$ as a real
number.
$v$$(\Omega, \ovalbox{\tt\small REJECT}, \mu, u, e)$$v$$\omega\mapsto u(\omega, x(\omega))$$\omega\mapsto u(\omega, x(\omega))$Lemma 1 If $x\in L_{1}(S, R_{+}^{l})$ , then $u(\cdot, x(\cdot))\in L_{1}(S, R+)$ for any $S\in*$? andis continuous with respect to the norm topologies ofthe map
and $L_{1}(S, R+)$ .
$x\mapsto u(\cdot, x(\cdot))$$L_{1}(S, R_{+}^{l})$Proof Let $x\in L_{1}(S, R_{+}^{l})$ . Since$||x||>1$ , we have the inequality$u(\omega$,$\cdot$$)$is concave, for any$\frac{u(\omega,x)-u(\omega,x/||x||)}{||x-x/||x||||}\leq\frac{u(\omega,x/||x||)-u(\omega,0)}{||x/||x||||}$,$x\in R_{+}^{l}$

with

(5)

and hence we have $u(\omega, x)\leq||x||\sigma$ . It is obvious ffom the definition ofthat
for all with $||x||\leq 1$ . Thus we have $u(\omega, x)\leq\sigma(1+||x||)$for any
and this leads to the inequalities$\sigma$$u(\omega, x)\leq\sigma$$x$$x\in R_{+}^{l}$$\int_{S}u(\omega, x(\omega))d\mu\leq\int_{S}\sigma(1+||x(\omega)||)d\mu=\sigma$$( \mu(S)+\int_{S}||x(\omega)||d\mu)<\infty$.

(6)

Thus it follows that $u(\cdot, x(\cdot))\in L_{1}(S, R_{+})$ . The second part of the assertionis verifified in Theorem 2.1 of [2]. Although Theorem 2.1 of [2] is proved under
and the second argument
the hypotheses that is ameasurable set in$R$of the function runs over , the proof of Theorem 2.1 of [2] is valid evenin our setting. Thus the map
is norm continuous. Q.E.D.
$R^{l}$$\Omega$$x$$u$$x\mapsto u(\cdot, x(\cdot))$

Remark 1The assumption of the finiteness of a is necessary to prove
Lemma 1. The following example violates the assumption and shows that
does not necessarily convey an integrable function to an integrable function
$u$

256
Example 1 Let l $=1$ and $\Omega=(0,$ 1). Defifine u : (0, 1). Then, for the function $x(\omega)=1$ for all $\omega\in(0,$ 1), it
and obviously it is not integrable.
follows$\cross R_{+}arrow R+\mathrm{b}\mathrm{y}$$\mathrm{u}(\mathrm{u}, x)=\sqrt{x}/\omega$$u(\omega, \mathrm{x}(\mathrm{u})=1/\omega,$

Lemma 2 A market game$v$

is actually a game and is monotone.

Proof It is obvious $v(\emptyset)=0$ . The fifinitenaes of $v(S)$ follows since theinequalities$\int_{S}u(\omega, x(\omega))d\mu(\omega)\leq\sigma\int_{S}(1+||x||)d\mu$(7)$\leq\sigma(\mu(S)+\sum_{i=1}^{l}\int_{S}x^{i}d\mu)=\sigma(\mu(S)+\sum_{i=1}^{l}\int_{S}e^{i}d\mu)$hold if$\int_{S}xd\mu=\int_{S}ed\mu$(8),

are the -th coordinate functions of and , respectively.
where
Moreover is monotone because the function $x\mapsto u(\omega,x)$ is nondecreasing. Q.E.D.
for almost all$x^{i}\mathrm{m}\mathrm{d}$$e^{i}$

$x$

$i$

$e$

$v$

$\omega\in\Omega$

Remark 2 The supremum in the defifinition of a market game cannot be
replaced by maximum in general as the following example shows.

Let $l=1$ , $\Omega=[0,1]$ and be the Lebesgue
Example 2
measure. Defifine : $[0, 1]\cross R+arrow R+\mathrm{b}\mathrm{y}u(\omega, x)=\omega x$ and let $e(\omega)=1$. Then $v([0,1])=1$ but, for any $x\in L_{1}([0,1], R+)$ withfor all$\int_{0}^{1}xd\mu=1$ ,never reaches 1.
$[[1], \mathrm{p}\mathrm{p}.

204]$

$\mu$

$u$

$\omega\in\Omega$$\int_{0}^{1}\omega x(\omega)d\mu(\omega)$

3Cores of Market GamesWe shall investigate properties of the cores of the market games in thissection. We start with a lemma on concave functions.
Lemma 3 If :and , ,$f$$R_{+}^{l}$$\lambda_{1}$$\ldots$

is concave and $f(0)=0$ , then for any, it follows thatwith

$R_{+}^{l}arrow R$$\lambda_{n}\geq 0$$x_{1}$,$\ldots$,$x_{n}\in$$\sum_{\dot{l}=1}^{n}\lambda_{i}\leq 1$$\sum_{i=1}^{n}\lambda_{i}f(x_{i})\leq f(\sum_{=1}^{n}\lambda_{i}x_{i})$

.

(9)

257
Proof We can assume that
follows that$\lambda=\sum_{i=1}^{n}\lambda_{i}>0$

without loss of generality. It(10)$\sum_{i=1}^{n}\lambda_{i}f(x_{i})=\lambda$ $\sum_{i=1}^{n}\frac{\lambda_{i}}{\lambda}f(x_{i})$(11)$\leq\lambda f(\sum_{i=1}^{n}\frac{\lambda_{i}}{\lambda}x_{i})$(12)$=(1- \lambda)f(0)+\lambda f(\frac{1}{\lambda}\sum_{i=1}^{n}\lambda_{i}x_{i})$$\leq f(\sum_{i=1}^{n}\lambda_{i}x_{i})$

.

(13)

Q.E.D.
Let $S’$ and be measurable sets with $S’\subset S$ . For any $x\in L_{1}(S’, R_{+}^{l})$ ,defifine an extensionof to by$S$$\overline{x}\in L_{1}(S, R_{+}^{1})$$x(\omega)$$\overline{x}(\omega)=\{$

0,

Proposition 1 A market game

$v$

$S$

$x$

,

if
if

;

$\omega$$\in S’$$\omega$$\in S\backslash S’$

(14)

.

is totally balanced.

, with, $i=1$ ,. We can assume that $\mu(S)>0$ without loss of generality.
such thatbe an arbitrary positive number. Take

Proof Take any$s\in\ovalbox{\tt\small REJECT}$and$S_{\mathrm{i}}\in\ovalbox{\tt\small REJECT}$

and$\lambda_{i}>0$$\ldots$$n$$\sum_{i=1}^{n}\lambda_{i}\chi s_{i}\leq\chi s$Let$\epsilon$$x_{i}\in L_{1}(S_{i}, R_{+}^{l})$$\int_{S}.\cdot x_{i}d\mu=\int_{S_{i}}ed\mu$

and defifine y$\in L_{1}(S, R_{+}^{l})$and$v(S_{i})- \frac{\epsilon}{n}<\int_{S}.\cdot u(\omega, x_{i}(\omega))d\mu(\omega)$

,

(15)

by

y

$= \sum_{i=1}^{n}\lambda_{i}\overline{x}_{i}$

.

(16)

258
Then we have the following:(17)$\int_{S}yd\mu=\sum_{i=1}^{n}\lambda:\int_{S}\overline{x}_{\dot{l}}d\mu$(18)$= \sum_{=\dot{l}1}^{n}\lambda_{i}\int_{S}.\cdot ed\mu$(19)$= \int_{S}e.\cdot\sum_{=1}^{n}\lambda:\chi s\dot{.}d\mu$$\leq\int_{S}ed\mu$

.

(20)

Defifine $y’\in L_{1}(S, R_{+}^{l})$ by$y’=y+ \frac{1}{\mu(S)}(\int_{S}ed\mu-\int_{S}yd\mu)$

.

(21)

Then it is easily seen that $\int_{S}y’d\mu=\int_{S}ed\mu$ .
On the other hand, let $A$ be the family of all nonempty subsets of$\{1, \ldots, n\}$ such that
. Then it is easily
seen that $S_{i}=\cup A\ni iTA$ for $i=1$ , , and {TA : $A\in A$ } is a partition of, andfor all $A\in A$ . For any and with $i\in A\in A$ ,. Then we have
, the restrictiontodefifine$A$$T_{A} \equiv\bigcap_{:\in A}S_{\dot{l}}\cap\bigcap_{j\in A^{\mathrm{c}}}(S\backslash S_{j})\neq\emptyset$$\ldots$$\bigcup_{\dot{|}=1}^{n}S_{i}$$n$$x_{i}^{A}=x:|\tau_{A}$$\mathrm{o}\mathrm{f}_{X:}$$\overline{x}_{\dot{l}}=\sum_{A\in i}\overline{x}_{\dot{l}}^{A}$

$A$

$i$

$\sum_{i\in A}\lambda_{i}\leq 1$

and y$T_{A}$$= \sum_{A\in A}\sum_{\dot{l}\in A}\lambda_{i}\overline{x}_{\dot{l}}^{A}$

.

(22)

259
Thus we have(23)$\sum_{i=1}^{n}\lambda_{i}v(S_{i})-\epsilon<\sum_{i=1}^{n}\lambda_{i}\int_{S}\dot{.}u(\omega, x_{i}(\omega))d\mu(\omega)$$= \sum_{i=1}^{n}\sum_{A\ni i}\lambda_{i}\int_{T_{A}}u(\omega, x_{i}^{A}(\omega))d\mu(\omega)$(24)$= \sum_{A\in A}\sum_{i\in A}\lambda_{i}\int_{T_{A}}u(\omega, x_{i}^{A}(\omega))d\mu(\omega)$(25)$= \sum_{A\in A}\int_{T_{A}}\sum_{i\in A}\lambda_{i}u(\omega, x_{i}^{A}(\omega))d\mu(\omega)$(26)$\leq\sum_{A\in A}\int_{T_{A}}u(\omega,\sum_{i\in A}\lambda_{i}x_{i}^{A}(\omega))d\mu(\omega)$$= \int_{S}u(\omega,\sum_{A\in A}\sum_{i\in A}\lambda_{i}\overline{x}_{i}^{A}(\omega))d\mu(\omega)$

by Lemma 3(27)

by $u(\omega, 0)=0$

(28)

(29)

$= \int_{S}u(\omega, y(\omega))d\mu(\omega)$$\leq\int_{S}u(\omega, y’(\omega))d\mu(\omega)$$\leq v(S)$by monotonicity of$u(\omega$,$\cdot$

(30)

$)$

.

(31)

Therefore, we have$\sum_{i=1}^{n}\lambda_{i}v(S_{i})\leq v(S)$Thus$\overline{v}(S)\leq v(S)$

.

(32)

and the reverse inequality is obvious. Hence we have

. Q.E.D.
Amarket game has acontinuity property by nature.

$\overline{v}=v$

Proposition 2 A market game$v$

is inner continuous at any

$S$

in

$\ovalbox{\tt\small REJECT}$.

andbe asequence of measurable sets with
Proof Let$x\in
L_{1}(S,R_{+}^{l})$such thatan arbitrary positive number. Then, there is$\bigcup_{n=1}^{\infty}S_{n}=S$$\{S_{n}\}$$v(S)- \epsilon<\int_{S}u(\omega, x(\omega))d\mu(\omega)$

and$\int_{S}xd\mu=\int_{S}ed\mu$.

$\epsilon$

(33)

260
Let

$x_{n}$

be the restrictionby$x|s_{\mathfrak{n}}$and defifine a sequence$\{y_{n}\}$of functions in$L_{1}(S_{n}, R_{+}^{l})$$y_{n}^{i}=\{\frac{\int_{S_{\mathrm{J}1}}e}{xJs_{\mathfrak{n}}^{x_{\dot{\hslash}}},ni+}...\frac{d\mu d\mu^{X_{n}}1}{\mu(S_{n})}..,(\int_{S_{f*}}e^{i}d\mu-\int_{S_{n}}x_{n}^{i}d\mu)$$\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{f}\int_{\int_{S_{\mathfrak{n}}}}s_{\mathfrak{n}}x_{n}^{\dot{1}}x_{n}^{i}$(34)$d \mu>\int_{\leq d\mu\int_{S_{\mathfrak{n}}}}s_{\mathfrak{n}}e^{i}d\mu e^{}d\mu’.$,for $i=1$ ,, . It is obvious that$l$$\ldots$$\int_{S_{\mathfrak{n}}}y_{n}d\mu=\int_{S_{\tau*}}ed\mu$

.

(35)

On the other hand, since$\lim_{narrow\infty}\int_{S_{\mathfrak{n}}}|y_{n}^{\dot{l}}-x_{n}.\cdot|d\mu=\lim_{narrow\infty}|\int_{S_{\mathfrak{n}}}e^{:}d\mu-\int_{S_{\mathrm{B}}}x_{n}^{\dot{l}}d\mu|=0$

for $i=1$ ,

,

(36)

, , we have$l$$\ldots$, (37)$\lim_{narrow\infty}\int_{S}||\overline{y}_{n}-x||d\mu=\lim_{narrow\infty}\int_{S_{\mathfrak{n}}}||y_{n}-x||d\mu+\lim_{narrow\infty}\int_{S\backslash S_{\mathfrak{n}}}||x||d\mu=0$

converges to with respect to the norm topoloy of
Therefore, by Lemma 1, it follows that$\mathrm{m}\mathrm{d}$$x$$\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\overline{y}_{n}$$L_{1}(S, R^{l})+\cdot$$\lim_{narrow\infty}\int_{S_{\mathfrak{n}}}u(\omega, y_{n}(\omega))d\mu(\omega)=\lim_{narrow\infty}\int_{S}u(\omega,\overline{y}_{n}(\omega))d\mu(\omega)=\int_{S}u(\omega, x(\omega))d\mu(\omega)$(38)

and hence, for sufficiently large ,$n$$v(S)- \epsilon<\int_{S_{*}}.u(\omega, y_{n}(\omega))d\mu(\omega)\leq v(S_{n})$

.

(39)

Thus we have $\lim_{narrow\infty}v(S_{n})=v(S)$ . Q.E.D.

Remark 3 Every exact game which is continuous at , equivalently inner
according to [3]. A market
continuous at , is continuous at every
game, however, is not necessarily continuous at each $S\in*\Psi$ . Consider again
the market game in Example 2. The game is not outer continuous at eachwith $0<\mu(S)<\mu(\Omega)$ according to [1].
$\Omega$$\Omega$$S\in\ovalbox{\tt\small REJECT}$$S\in f\ovalbox{\tt\small REJECT}$

we have reached our main theorem combining Proposition 1 and
Proposition 2.
$\mathrm{N}\mathrm{o}\iota \mathrm{v}$261
Theorem 1 A market game v has anonempty core, and every elementof the core is countably additive and has a unique outcome density f$L_{1}(\Omega, R+)$ , and hence it follows that$\alpha(S)=\int_{S}fd\mu$,$S\in\ovalbox{\tt\small REJECT}$.

$\alpha$

$\in$

(40)

Proof The core is nonempty by Proposition 1. Each element of the core
is continuous at by Proposition 2, and hence is countably additive. To
prove existence of an outcome density for , it is sufficient to show thatis absolutely continuous with respect to by virtue of the Radon-Nikodym
theorem. If $\mu(S)=0$ , then $v(S^{c})=v(\Omega)$ by the definition of the game ,and hence we have $\alpha(S^{c})\geq v(S^{c})=v(\Omega)=\alpha(\Omega)$ , that is, $\alpha(S)=0$ . Q.E.D.
$\alpha$$\Omega$$\alpha$$\alpha$$\alpha$

$\mu$

$v$

Remark 4 Similar to the assertion of Theorem 1, an exact game whichis continuous at has anonempty core and every member of the core iscountably additive. Moreover, there is ameasuresuch that every
member of the core is absolutely continuous with respect to according to[3]. The following example shows that there is a market game which is notexact, and hence Theorem 1 is independent of the results of [3].
$\Omega$$\lambda \mathrm{o}\mathrm{n}*?$$\lambda$

Example 3$[[1], \mathrm{p}\mathrm{p}.

measure. Defifine

$u$

:

192]$

Let $l=1$ , $\Omega=[0,1]$ and$\mu$

be the Lebesgue$[0, 1]\cross R+arrow R+\mathrm{b}\mathrm{y}$$u(\omega, x)=\sqrt{x+\omega}-\sqrt{\omega}$and$e( \omega)=\frac{1}{32}$for all $\omega\in[0,1]$ .

According to [1], the core of the market game has only one member
the outcome density of is given by$f$$\alpha$and$\alpha$$f(\omega)=\{$

Thus it follows
we have(41),$\frac{(\frac{1}{12}}{32},-\sqrt{\omega})^{2}+\frac{1}{32}$$\alpha([\frac{1}{2},1])=\frac{1}{64}$, and hence

if
if

;
$\omega\in[0, \frac{1}{4}]$$\omega$$\in[\frac{1}{4},1]$$\hat{v}([\frac{1}{2},1])=\frac{1}{64}$

.

(42)

. On the other hand,$\sqrt{x+\omega}-\sqrt{\omega}\leq\sqrt{x+\frac{1}{2}}-\sqrt{\underline{\frac{1}{9}}}\leq\sqrt{\frac{1}{2}}x$(43)

for $1/2\leq\omega\leq 1$ and $x\geq 0$ . Thus, if $x\in L_{1}([0,1], R_{+})$ satisfies$\int_{\frac{1}{2}}^{1}xd\mu=\int_{\frac{1}{2}}^{1}ed\mu=\frac{1}{64}$

,

(44)

262

$\int_{\frac{1}{2}}^{1}u(\omega, x(\omega))d\mu(\omega)\leq\int_{\frac{1}{2}}^{1}\sqrt{\frac{1}{2}}xd\mu=\frac{1}{64\sqrt{2}}<\frac{1}{64}$Therefore we have4$v([ \frac{1}{2},1])<\hat{v}([\frac{1}{2},1])$

and

$v$

.

(45)

is not exact.

Concluding Remark

We have shown that every member of the core of a market game is countably
additive and hence has an outcome density, and an exact game which iscontinuous at has these properties as written in Remark 4. If we provedthat every totally balanced game that is continuous at is a game derived
from a market in our sense, then we could deduce from Theorem 1 thatevery totally balanced game that is continuous at has a nonempty core
whose members are all countably additive and have outcome densities.
problem is the infinite version of the problem solved in [4], but it is still
open.
$\Omega$$\Omega$$\Omega$$\mathrm{T}\acute{\mathrm{h}}\mathrm{i}\mathrm{s}$References[1] Aumann RJ and Shapley LS(1974) Values of Non-Atomic Games.
Princeton University Press, Princeton[2] Krasnosel’skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations, Pergmon Press, Oxford[3] Schmeidler D(1972) Cores of Exact Games, I. J Math Anal Appl 40:

214-225[4] Shapley LS and Shubik M(1969) On Market Games. J Econ Theory 1:
9-25.