(137) $\displaystyle\mathcal{\tilde{H}}(\omega)$
$\displaystyle=\frac{1}{\epsilon+i\omega+\mathcal{R}_{1}(\omega)\mathcal{R}_{3}(\omega)}-\frac{1}{\epsilon+i\omega(1+r)}=\frac{i\omega
r-\mathcal{R}_{1}(\omega)\mathcal{R}_{3}(\omega)}{(\epsilon+i\omega+\mathcal{R}_{1}(\omega)\mathcal{R}_{3}(\omega))(\epsilon+i\omega(1+r))},$

where $r=\lim_{\omega\to
0}\frac{1}{i\omega}\mathcal{R}_{1}(\omega)\mathcal{R}_{3}(\omega)$. We see
that rather than diverging like $\mathcal{H}(\omega)$, this function
$\mathcal{\tilde{H}}(\omega)$ vanishes as $\omega\to 0$. We therefore
numerically perform Fourier integral against $\tilde{H}(\omega)$ and then add
the singular component which can be computed separately.

$\displaystyle
H(\tau)=\int\frac{d\omega}{2\pi}e^{i\omega\tau}\mathcal{\tilde{H}}(\omega)+\int\frac{d\omega}{2\pi}\frac{e^{i\omega\tau}}{\epsilon+i\omega(1+r)}=\int\frac{d\omega}{2\pi}e^{i\omega\tau}\mathcal{\tilde{H}}(\omega)+\frac{1}{1+r}$
(138)

where we used the fact that

$\displaystyle\frac{1}{\epsilon+i\omega(1+r)}=\frac{\pi}{1+r}\delta(\omega)-\frac{i}{1+r}\mathcal{P}(\omega^{-1})\
,\ \epsilon\to 0$ (139)

The Dirac mass is trivial to integrate over giving $\frac{1}{2(1+r)}$. Lastly,
we must perform an integral of the type

$\displaystyle-\frac{i}{1+r}\mathcal{P}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{e^{i\omega\tau}}{\omega}=\frac{1}{\pi(1+r)}\int_{0}^{\infty}d\omega\frac{\sin(\omega\tau)}{\omega}=\frac{1}{2(1+r)}$
(140)

Adding these two terms together, our transfer function has the form

$\displaystyle
H(\tau)=\frac{1}{1+r}+\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}e^{i\omega\tau}\mathcal{\tilde{H}}(\omega).$
(141)

The last integral can be performed numerically, giving a more stable result.

##  Appendix M Compute Optimal Scaling from Sum of Power-Laws

We suppose that the loss scales as (neglecting irrelevant prefactors)

$\displaystyle\mathcal{L}=t^{-r_{t}}+N^{-r_{N}}+P^{-r_{P}}+\mathcal{L}_{\infty}$
(142)

Our goal is to minimize the above expression subject to the constraint that
compute $C=Nt$ is fixed. Since $C$ is fixed we can reduce this to a one-
dimensional optimization problem

$\displaystyle\min_{N}\left[C^{-r_{t}}N^{r_{t}}+N^{-r_{N}}\right]$ (143)

The optimality condition $\partial_{N}L=0$ is

$\displaystyle r_{t}C^{-r_{t}}N^{r_{t}-1}-r_{N}N^{-r_{N}-1}=0$
$\displaystyle\implies N\propto C^{\frac{r_{t}}{r_{t}+r_{N}}}\implies t\propto
C^{\frac{r_{t}}{r_{t}+r_{N}}}$ (144)

From this last expression one can evaluate the loss at the optimum

$\displaystyle\mathcal{L}_{\star}(C)\propto
C^{-\frac{r_{t}r_{N}}{r_{t}+r_{N}}}.$ (145)


# The $\kappa$-core and the $\kappa$-balancedness of TU games††thanks: David
Bartl acknowledges the support of the Czech Science Foundation under grant
number GAČR 21-03085S. A part of this research was done while he was visiting
the Corvinus Institute for Advanced Studies; the support of the CIAS during
the stay is gratefully acknowledged. Miklós Pintér acknowledges the support by
the Hungarian Scientific Research Fund under projects K 133882 and K 119930.

David Bartl and Miklós Pintér Department of Informatics and Mathematics,
School of Business Administration in Karviná, Silesian University in Opava,
<EMAIL_ADDRESS>author: Corvinus Center for Operations
Research, Corvinus Institute of Advanced Studies, Corvinus University of
Budapest<EMAIL_ADDRESS>
###### Abstract

We consider transferable utility cooperative games with infinitely many
players. In particular, we generalize the notions of core and balancedness,
and also the Bondareva-Shapley Theorem for infinite TU-games with and without
restricted cooperation, to the cases where the core consists of
$\kappa$-additive set functions. Our generalized Bondareva-Shapley Theorem
extends previous results by Bondareva (1963), Shapley (1967), Schmeidler
(1967), Faigle (1989), Kannai (1969, 1992), Pintér (2011) and Bartl and Pintér
(2022).

Keywords: TU games with infinitely many players, Bondareva-Shapley Theorem,
$\kappa$-core, $\kappa$-balancedness, $\kappa$-additive set function, duality
theorem for infinite LPs

##  1 Introduction

The core (Shapley, 1955; Gillies, 1959) is definitely one of the most
important solution concepts of cooperative game theory. In the transferable
utility setting (henceforth games) the Bondareva-Shapley Theorem (Bondareva,
1963; Shapley, 1967; Faigle, 1989) provides a necessary and sufficient
condition for the non-emptiness of the core; it states that the core of a game
with or without restricted cooperation is not empty if and only if the game is
balanced. The textbook proof of the Bondareva-Shapley Theorem goes by the
strong duality theorem for linear programs (henceforth LPs), see e.g. Peleg
and Sudhölter (2007). The primal problem corresponds to the concept of
balancedness and so does the dual problem to the notion of core. However, this
result is formalized for games with finitely many players. It is a question
how one can generalize this result to the infinitely many player case.

The finitely many player case is special in (at least) two counts: (1) it can
be handled by finite linear programs, (2) since the power set of the player
set is also finite, it is natural to take the solution of a game from the set
of additive set functions (additive games).

There are two main directions to generalize the notion of additive set
function. The first, when we weaken the notion of additivity; this leads to
the notion of $k$-additive core (Grabisch and Miranda, 2008), where $k$ is a
finite cardinal (natural number). The second, when we use a notion stronger
than additivity (e.g. $\sigma$-additivity). Naturally, a stronger notion does
matter only if there are infinitely many players. This latter approach is
considered here.

Schmeidler (1967), Kannai (1969, 1992), Pintér (2011), and Bartl and Pintér
(2022) considered games with infinitely many players. All these papers studied
the additive core; that is, the case when the core consists of bounded
additive set functions. Schmeidler (1967) and Kannai (1969) showed that the
additive core of a non-negative game without restricted cooperation with
infinitely many players is not empty if and only if the game is Schmeidler
balanced (Definition 12). Bartl and Pintér (2022) extended these results and
showed that the additive core of a game bounded below with our without
restricted cooperation with infinitely many players is not empty if and only
if the game is (bounded-)Schmeidler balanced.

Kannai (1992) raised the following two research questions: (1) When does there
exist a bounded $\sigma$-additive set function in the core? (2) When are all
elements in the core bounded $\sigma$-additive?

Kannai (1969) gave a necessary and sufficient condition for that the
$\sigma$-additive core of a non-negative game without restricted cooperation
and with infinitely many players is not empty; that is, he answered Question
(1). This result (the necessary and sufficient condition) is, however, only
slightly similar to the classical balancedness condition. Moreover, it works
only for non-negative games without restricted cooperation. Schmeidler (1972)
and Einy et al (1997) answered Question (2) respectively for exact and for
continuous convex games.

In this paper we raise the following question, where $\kappa$ is an infinite
cardinal number, and generalize Kannai’s first question thus: When does there
exist a bounded $\kappa$-additive set function in the core? Moreover, we
consider this question in the case of games with restricted cooperation too.

Addressing this question, we introduce the notions of $\kappa$-core and
$\kappa$-balancedness (Definitions 11 and 14). Then, we apply the strong
duality theorem for infinite LPs by Anderson and Nash (1987) (Proposition 3)
and prove that the $\kappa$-core of a game with or without restricted
cooperation and with arbitrarily many players is not empty if and only if the
game is $\kappa$-balanced (Theorem 17).

The set-up of the paper is as follows. In the next section we introduce the
main mathematical notions and results, which are related to infinite LPs, and
used in this paper. In Section 3 we introduce the notion of $\kappa$-additive
set functions and discuss some related concepts and results. In Section 4 we
present game theory notions and define various cores (such as $\kappa$-core)
and balancedness conditions (such as $\kappa$-balancedness) we consider in
this paper. Section 5 presents our main result. We give an answer to the
question we have raised: there is a bounded $\kappa$-additive set function in
the core if and only if the game is $\kappa$-balanced (Theorem 17). The last
section briefly concludes.

##  2 Duality theorem

In this section we discuss the duality theorem for infinite linear programs
that we will use later.

Let $X$ and $Y$ be real vector spaces; the algebraic dual of $X$, which is the
space of all linear functionals on $X$, is denoted by $X^{\prime}$; similarly
$Y^{\prime}$ denotes the algebraic dual of $Y$. Moreover, $Y^{*}\subseteq
Y^{\prime}$ denotes a linear subspace of $Y^{\prime}$ such that $(Y,Y^{*})$ is
a dual pair of spaces; that is, if $f\in Y$ is non-zero, then there exists a
$y\in Y^{*}$ such that $y(f)\neq 0$. For any linear mapping $A\colon X\to Y$
its adjoint mapping is $A^{\prime}\colon Y^{\prime}\to X^{\prime}$ with
$\bigl{(}A^{\prime}(y)\bigr{)}(x)=y\bigl{(}A(x)\bigr{)}$ for all $x\in X$ and
$y\in Y^{\prime}$. Moreover, a subset $P\subseteq X$ of the vector space $X$
is a convex cone if $\alpha x+\beta y\in P$ for all $x,y\in P$ and all non-
negative $\alpha,\beta\in\mathbb{R}$. For any two functionals $f,g\colon
X\to\mathbb{R}$ we write $f\geq_{P}g$ if $f(x)\geq g(x)$ for all $x\in P$.

Now, given a linear mapping $A\colon X\to Y$, a point $b\in Y$ and a linear
functional $c\colon X\to\mathbb{R}$, let us consider the following infinite
LP-pair (cf. Anderson and Nash, 1987, Section 3.3):

$(\mathrm{P_{LP}})\qquad\qquad c(x$ | ${})$ | ${}\to\sup$ |  $(\mathrm{D_{LP}})\qquad\qquad y(b$ | ${})$ | ${}\to\inf$
---|---|---|---|---|---
$\hbox{s.t.}\enskip A(x$ | ${})$ | ${}={}$ | $\hphantom{(}b$ |  $\hbox{s.t.}\enskip A^{\prime}(y$ | ${})$ | ${}\geq_{P}{}$ | $c$
$x$ |  | ${}\in{}$ | $\hphantom{(}P$ |  $y$ |  | ${}\in{}$ | $Y^{*}$
(1)

where $P\subseteq X$ is a convex cone and $Y^{*}$ is a subspace of
$Y^{\prime}$ such that $(Y,Y^{*})$ is a dual pair of spaces.

######  Definition 1.

The program $(\mathrm{D_{LP}})$ is consistent if there exists a linear
functional $y\in Y^{*}$ such that $\bigl{(}A^{\prime}(y)\bigr{)}(x)\geq c(x)$
for all $x\in P$. The value of a consistent program $(\mathrm{D_{LP}})$ is
$\inf\bigl{\\{}\,y(b):A^{\prime}(y)\geq_{P}c,\allowbreak\\!\;y\in
Y^{*}\,\bigr{\\}}$.

In the next definition we assume the weak topology on the space $Y$ with
respect to $Y^{*}$. To introduce that, we describe all the neighborhoods of a
point. A set $U\subseteq Y$ is a weak neighborhood of a point $f_{0}\in Y$ if
there exist a natural number $n$ and functionals $y_{1},\ldots,\allowbreak
y_{n}\in Y^{*}$ such that $\bigcap_{j=1}^{n}\bigl{\\{}\,f\in
Y:\bigl{|}y_{j}(f)-y_{j}(f_{0})\bigr{|}<1\,\bigr{\\}}\subseteq U$.

######  Definition 2.

Put $D=\bigl{\\{}\,\bigl{(}A(x),\,c(x)\bigr{)}:x\in P\,\bigr{\\}}$. The
program $(\mathrm{P_{LP}})$ is superconsistent if there exists a
$z\in\mathbb{R}$ such that $(b,z)\in\,\overline{\\!D}$, where
$\overline{\\!D}$ is the closure of $D$. The supervalue of a superconsistent
program $(\mathrm{P_{LP}})$ is
$\sup\bigl{\\{}\,z:(b,z)\in\,\overline{\\!D}\,\bigr{\\}}$.

We recall that a pair $(I,\leq)$ is right-directed if $I$ is a preordered set
and for any $i,j\in I$ there exists a $k\in I$ such that $i\leq k$ and $j\leq
k$. A net (generalized sequence) of $X$ is $(x_{i})_{i\in I}$ where $(I,\leq)$
is a right-directed pair and $x_{i}\in X$ for all $i\in I$.

Notice that the program $(\mathrm{P_{LP}})$ is superconsistent if there exists
a net $(x_{i})_{i\in I}$ from $P$ such that $A(x_{i})\buildrel
w\over{\longrightarrow}b$, which means that $A(x_{i})$ converges to $b$ in the
weak topology, and $(c(x_{i}))_{i\in I}$ is bounded. Furthermore, a number
$z^{\ast}$ is the supervalue of a superconsistent program $(\mathrm{P_{LP}})$
if it is the least upper bound of all numbers $z$ such that there exists a net
$(x_{i})_{i\in I}$ from $P$ such that $A(x_{i})\buildrel
w\over{\longrightarrow}b$ and $c(x_{i})\longrightarrow z$.

######  Proposition 3.

Consider the programs in (1). Program $(\mathrm{P_{LP}})$ is superconsistent
and $z^{\ast}$ is its finite supervalue if and only if program
$(\mathrm{D_{LP}})$ is consistent and $z^{\ast}$ is its finite value.

Proposition 3 is a restatement of Theorem 3.3, p. 41, in Anderson and Nash
(1987). Notice that we differ from Anderson and Nash (1987) in the point that
Anderson and Nash use slightly different notions of superconsistency and
supervalue. However, they also remark that their notions and the ones we use
here are equivalent (p. 41 above Theorem 3.3). This is why we omit the proof
of Proposition 3 here.

##  3 The $\kappa$-structures

Throughout this section $\kappa$ is an infinite cardinal number. Let $N$ be a
non-empty set and let $\mathcal{A}\subseteq\mathcal{P}(N)$ be a field of sets;
that is, if $S_{1},\ldots,\allowbreak S_{n}\in\mathcal{A}$, then
$\bigcup_{j=1}^{n}S_{j}\in\mathcal{A}$, and $N\in\mathcal{A}$ with $N\setminus
S\in\mathcal{A}$ for any $S\in\mathcal{A}$. The pair $(N,\mathcal{A})$ is
called chargeable space.

Given a chargeable space $(N,\mathcal{A})$, let $\mathop{\rm
ba}\nolimits(\mathcal{A})$ and $\mathop{\rm ca}\nolimits(\mathcal{A})$ denote,
respectively, the set of bounded additive set functions and the set of bounded
$\sigma$-additive set functions $\mu\colon\mathcal{A}\to\mathbb{R}$.

Let $(S_{i})_{i\in I}$ be a net of sets of $\mathcal{A}$; a net $(S_{i})_{i\in
I}$ is a $\kappa$-net if $\\#I\leq\kappa$, where $\\#I$ is the cardinality of
the set $I$. In addition, the net $(S_{i})_{i\in I}$ is monotone decreasing or
monotone increasing if $i\leq j$ implies $S_{i}\supseteq S_{j}$ or
$S_{i}\subseteq S_{j}$, respectively, for any $i,j\in I$.

Let $\mu\colon\mathcal{A}\to\mathbb{R}$ be a set function. We say that $\mu$
is upper $\kappa$-continuous or lower $\kappa$-continuous at $S\in\mathcal{A}$
if for any monotone decreasing or increasing $\kappa$-net $(S_{i})_{i\in I}$
from $\mathcal{A}$ with $\bigcap_{i\in I}S_{i}=S$ or $\bigcup_{i\in
I}S_{i}=S$, respectively, it holds that $\lim_{i\in I}\mu(S_{i})=\mu(S)$. The
set function $\mu$ is $\kappa$-continuous if it is both upper and lower
$\kappa$-continuous at every set $S\in\mathcal{A}$.

Next we define the notion of $\kappa$-additivity. Our definition is similar to
the one by Schervish et al (2017).

######  Definition 4.

A set function $\mu\colon\mathcal{A}\to\mathbb{R}$ is $\kappa$-additive if it
is additive and $\kappa$-continuous. Let $\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A})$ denote the set of $\kappa$-additive set
functions over $\mathcal{A}$.

Note that $\mathop{\rm ba}\nolimits^{\kappa}(\mathcal{A})$ is a linear
subspace of $\mathop{\rm ba}\nolimits(\mathcal{A})$. Futhermore, the following
proposition is easy to see.

######  Proposition 5.

If the set function $\mu\colon\mathcal{A}\to\mathbb{R}$ is additive, then it
is

  * •

upper $\kappa$-continuous if and only if it is lower $\kappa$-continuous;

  * •

$\kappa$-continuous if and only if it is lower $\kappa$-continuos at
$\emptyset$;

  * •

$\aleph_{0}$-continuous if and only if it is $\sigma$-additive.

######  Example 6.

The Lebesgue measure on $B([0,1])$, the Borel $\sigma$-field of $[0,1]$, is
not $\kappa$-additive for any $\kappa\geq\mathfrak{c}$, where $\mathfrak{c}$
denotes the cardinality of the real numbers; but it is $\kappa$-additive for
$\kappa=\aleph_{0}$.

If $\kappa$ is not countable and the field of sets on which the
$\kappa$-additive set function is defined is rich enough, then one may ask
whether there are enough or just few $\kappa$-additive set functions. Without
going into the details we remark that this problem is related to the notion of
measurable cardinal (Ulam, 1930).

The next example shows that there are many $\kappa$-additive set functions in
the space $\mathop{\rm ba}\nolimits^{\kappa}(\mathcal{A})$ even in the case
when the field $\mathcal{A}$ is large; that is, the theory is not trivial nor
vacuous.

######  Example 7.

Let $X$ be an arbitrary set such that $\\#X=\kappa\geq\aleph_{0}$. Consider
$\mathcal{P}(X)$, the power set of $X$. It is clear that the Dirac measures on
$\mathcal{P}(X)$ are $\kappa$-additive. Let

$\displaystyle{\it\Delta}=\biggl{\\{}\,\sum_{n=1}^{\infty}\alpha_{n}\delta_{n}:{}$ | $(\alpha_{n})_{n=1}^{\infty}\in\ell^{1},$
---|---
|  $\hphantom{(}\delta_{n}$ being Dirac measures on $\mathcal{P}(X)$ for
$n=1$, $2$, $3$, …$\,\biggr{\\}}\,.$

It is clear that each $\mu\in{\it\Delta}$ is a $\kappa$-additive set function
on $\mathcal{P}(X)$. Notice that $\\#{\it\Delta}\geq\\#\mathop{\rm
ca}\nolimits\bigl{(}\mathbb{N},\mathcal{P}(\mathbb{N})\bigr{)}$; that is, even
in the “worst” case, when there does not exist a non-trivial
$\\{0,1\\}$-valued $\kappa$-additive set function on $\mathcal{P}(X)$, which
means the cardinal $\kappa$ is not measurable (Ulam, 1930), the collection
${\it\Delta}$ of the trivial $\kappa$-additive set functions on
$\mathcal{P}(X)$ is at least as large as the collection of the
$\sigma$-additive ones on $\mathcal{P}(\mathbb{N})$. In other words, even in
the “worst” case, the problem of the non-emptiness of the
$\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ is at least as complex as
the non-emptiness of the $\sigma$-core with player set $\mathbb{N}$ and all
coalitions feasible, the case considered by Kannai (1969, 1992).

Given a set system $\mathcal{A}$, the space $\mathbb{R}^{(\mathcal{A})}$
consists of all functions $\lambda\colon\mathcal{A}\to\mathbb{R}$ with a
finite support; that is,

$\mathbb{R}^{(\mathcal{A})}=\bigl{\\{}\,\lambda\in\mathbb{R}^{\mathcal{A}}:\\#\\{\,S\in\mathcal{A}:\lambda_{S}\neq
0\,\\}<\infty\,\bigr{\\}}\,.$

Denoting $\lambda(S)$ and the characteristic function of a set
$S\in\mathcal{A}$ by $\lambda_{S}$ and $\chi_{S}$, respectively, let
$\Lambda(\mathcal{A})=\\{\,\lambda_{S_{1}}\chi_{S_{1}}+\cdots+\lambda_{S_{n}}\chi_{S_{n}}:n\in\mathbb{N},\allowbreak\\!\;\lambda_{S_{1}},\ldots,\allowbreak\lambda_{S_{n}}\\!\in\mathbb{R},\allowbreak\\!\;S_{1},\ldots,\allowbreak
S_{n}\in\mathcal{A}\,\\}$ be the space of all simple functions on
$(N,\mathcal{A})$; that is,

$\Lambda(\mathcal{A})=\biggl{\\{}\,\sum_{S\in\mathcal{A}}\lambda_{S}\chi_{S}:\lambda\in\mathbb{R}^{(\mathcal{A})}\,\biggr{\\}}\,.$

We introduce a norm on $\Lambda(\mathcal{A})$ as follows. For a simple
function
$f=\lambda_{S_{1}}\chi_{S_{1}}+\cdots+\lambda_{S_{n}}\chi_{S_{n}}\\!\in\Lambda(\mathcal{A})$
let

$\mathopen{\|}f\mathclose{\|}=\sup_{x\in N}\bigl{|}f(x)\bigr{|}\,.$

Then the topological dual $(\Lambda(\mathcal{A}))^{\star}$ of the vector space
$\Lambda(\mathcal{A})$, which is the space of all continuous linear
functionals on $\Lambda(\mathcal{A})$, is isometrically isomorphic to
$\mathop{\rm ba}\nolimits(\mathcal{A})$, the space of all bounded additive set
functions on $\mathcal{A}$ (see e.g. Dunford and Schwartz (1958), Theorem
IV.5.1, p. 258). For simplicity, we shall identify the space
$(\Lambda(\mathcal{A}))^{\star}$ with $\mathop{\rm ba}\nolimits(\mathcal{A})$.
Indeed, a set function $\mu\in\mathop{\rm ba}\nolimits(\mathcal{A})$ induces a
continuous linear functional $\mu^{\prime}\in(\Lambda(\mathcal{A}))^{\star}$
on $\Lambda(\mathcal{A})$ as follows:

$\mu^{\prime}(f)=\lambda_{S_{1}}\mu(S_{1})+\cdots+\lambda_{S_{n}}\mu(S_{n})$
(2)

for any
$f=\lambda_{S_{1}}\chi_{S_{1}}+\cdots+\lambda_{S_{n}}\chi_{S_{n}}\\!\in\Lambda(\mathcal{A})$.

######  Lemma 8.

It holds that $\bigl{(}\Lambda(\mathcal{A}),\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A})\bigr{)}$ is a dual pair of spaces.

###### Proof.

Let $f\in\Lambda(\mathcal{A})$ be non-zero, whence there is an $x\in N$ such
that $f(x)\neq 0$. Then $\delta_{x}$, the Dirac measure concentrated at point
$x$ on $\mathcal{A}$, is a $\kappa$-additive set function, and
$\delta^{\prime}_{x}(f)=f(x)\neq 0$. ∎

##  4 The $\kappa$-core and the $\kappa$-balancedness of TU games

Let $\kappa$ be an arbitrary infinite cardinal number as in the previous
section. First, we introduce the notion of TU games. Let $N$ be a non-empty
set of players, let $\mathcal{A}^{\prime}\subseteq\mathcal{P}(N)$ be a
collection of sets such that $\emptyset,N\in\mathcal{A}^{\prime}$, and let
$\mathcal{A}$ denote the field hull of $\mathcal{A}^{\prime}$; that is, the
smallest field of sets that contains $\mathcal{A}^{\prime}$. Then a TU game
(henceforth a game) on $\mathcal{A}^{\prime}$ is a set function
$v\colon\mathcal{A}^{\prime}\to\mathbb{R}$ such that $v(\emptyset)=0$. We
denote the class of games on $\mathcal{A}^{\prime}$ by
$\mathcal{G}^{\mathcal{A}^{\prime}}$. If $\mathcal{A}^{\prime}=\mathcal{A}$,
then $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ is a game without restricted
cooperation. Otherwise, if $\mathcal{A}^{\prime}$ is not a field,
$v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ is a game with restricted
cooperation.

In the following subsections we introduce the three notions of core and the
three notions of balancedness that we consider in this paper.

###  4.1 The core of a TU game

First, we introduce the notion of additive core of a game, which was
considered by Schmeidler (1967), Kannai (1969, 1992), Pintér (2011), and Bartl
and Pintér (2022).

######  Definition 9.

For a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ its additive core
(henceforth $\mathop{\rm ba\mathchar 45\relax core}\nolimits$) is defined as
follows:

$\mathop{\rm ba\mathchar 45\relax
core}\nolimits(v)=\bigl{\\{}\,\mu\in\mathop{\rm
ba}\nolimits(\mathcal{A}):\mu(N)=v(N)\text{ and }\mu(S)\geq v(S)\text{ for all
}S\in\mathcal{A}^{\prime}\,\bigr{\\}}\,.$

We shall also need the notion of $\sigma$-additive core of a game.

######  Definition 10.

For a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ its $\sigma$-additive core
(henceforth $\mathop{\rm ca\mathchar 45\relax core}\nolimits$) is defined as
follows:

$\mathop{\rm ca\mathchar 45\relax
core}\nolimits(v)=\bigl{\\{}\,\mu\in\mathop{\rm
ca}\nolimits(\mathcal{A}):\mu(N)=v(N)\text{ and }\mu(S)\geq v(S)\text{ for all
}S\in\mathcal{A}^{\prime}\,\bigr{\\}}\,.$

In general, for an infinite cardinal number $\kappa$ we introduce the notion
of $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ of a game.

######  Definition 11.

For a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ its $\kappa$-core is
defined as follows:

$\mathop{\rm\kappa\mathchar 45\relax
core}\nolimits(v)=\bigl{\\{}\,\mu\in\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A}):\mu(N)=v(N)\text{ and }\mu(S)\geq
v(S)\text{ for all }S\in\mathcal{A}^{\prime}\,\bigr{\\}}\,.$

In words, the $\mathop{\rm ba\mathchar 45\relax core}\nolimits$, the
$\mathop{\rm ca\mathchar 45\relax core}\nolimits$, and the
$\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ consists of bounded
additive, bounded $\sigma$-additive, and bounded $\kappa$-additive,
respectively, set functions defined on the field hull $\mathcal{A}$ of the
feasible coalitions $\mathcal{A}^{\prime}$ that meet the conditions of
efficiency ($\mu(N)=v(N)$) and coalitional rationality ($\mu(S)\geq v(S)$ for
all $S\in\mathcal{A}^{\prime}$). Observe that the $\mathop{\rm ca\mathchar
45\relax core}\nolimits$ is a special case of the $\mathop{\rm\kappa\mathchar
45\relax core}\nolimits$ when $\kappa=\aleph_{0}$.

Notice that in the finite case all the three notions of $\mathop{\rm
ba\mathchar 45\relax core}\nolimits$, $\mathop{\rm ca\mathchar 45\relax
core}\nolimits$, and $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ are
equivalent with the notion of (ordinary) core.

###  4.2 Balancedness of a TU game

In the case of infinite games without restricted cooperation with additive
core Schmeidler (1967) defined the notion of balancedness. Here, we generalize
his notion to the restricted cooperation case, and call it Schmeidler
balancedness.

######  Definition 12.

We say that a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ is Schmeidler
balanced if

$\sup\biggl{\\{}\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S):\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}=\chi_{N},\,\lambda\in\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}\biggr{\\}}\leq
v(N)\,.$ (3)

Notice that for finite games the notions of Schmeidler balancedness and
(ordinary) balancedness (Bondareva, 1963; Shapley, 1967; Faigle, 1989)
coincide, hence Schmeidler balancedness is an extension of (ordinary)
balancedness.

Recall that $Y^{*}=\mathop{\rm ba}\nolimits^{\kappa}(\mathcal{A})$ is a linear
subspace of $Y^{\star}=\mathop{\rm ba}\nolimits(\mathcal{A})$, which can be
identified with the topological dual of the normed linear space
$Y=\Lambda(\mathcal{A})$. In the next two definitions, where we introduce two
new notions of balancedness, we consider the weak topology on
$Y=\Lambda(\mathcal{A})$ with respect to $Y^{*}=\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A})$ (see Lemma 8).

First, for a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ consider the convex
cone

$K_{v}^{+}=\Biggl{\\{}\biggl{(}\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S},\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)\biggr{)}:\lambda\in\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}\Biggr{\\}}\,.$
(4)

######  Definition 13.

We say that a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ is Schmeidler
$\kappa$-balanced if

$z\leq v(N)$

for all $z\in\mathbb{R}$ such that $(\chi_{N},z)\in\,\overline{\\!K_{v}^{+}}$,
where $\overline{\\!K_{v}^{+}}$ is the closure of $K_{v}^{+}$.

Observe that Schmeidler $\kappa$-balancedness implies Schmeidler balancedness,
which implies (ordinary) balancedness.

Lastly, for a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ let

$\mathbb{R}_{\ast}^{(\mathcal{A}^{\prime})}=\bigl{\\{}\,\lambda\in\mathbb{R}^{(\mathcal{A}^{\prime})}:\lambda_{S}\geq
0\text{ for all }S\in\mathcal{A}^{\prime}\setminus\\{N\\}\,\bigr{\\}}$

and consider the convex cone

$K_{v}=\Biggl{\\{}\biggl{(}\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S},\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)\biggr{)}:\lambda\in\mathbb{R}_{\ast}^{(\mathcal{A}^{\prime})}\Biggr{\\}}\,.$
(5)

######  Definition 14.

A game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ is $\kappa$-balanced if

$z\leq v(N)$

for all $z\in\mathbb{R}$ such that $(\chi_{N},z)\in\,\overline{\\!K_{v}}$,
where $\overline{\\!K_{v}}$ is the closure of $K_{v}$.

######  Remark 15.

The notion of $\kappa$-balancedness and Schmeidler $\kappa$-balancedness is
very closely related to the notion of supervalue introduced in Definition 2.
The cone $K_{v}$ or $K_{v}^{+}$ is precisely the set $D$ if
$A(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}$ and
$c(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)$ with
$P=\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}$ or
$P=\mathbb{R}_{*}^{(\mathcal{A}^{\prime})}$, respectively, in Definition 2.
Then the game is $\kappa$-balanced or Schmeidler $\kappa$-balanced,
respectively, if and only if the supervalue of the related primal problem
$(\mathrm{P_{LP}})$ is not greater than $v(N)$.

Notice that the notion of $\kappa$-balancedness is a “double” extension of
Schmeidler balancedness. First, we do not take the balancing weight system
alone, but we take nets of balancing weight systems. Second, we let the weight
of the grand coalition be sign unrestricted. It is worth noticing that the
notion of $\kappa$-balancedness applies its full strength when in a net of
balancing weight systems the net of the weights of the grand coalition is not
bounded below (see Lemma 16 below).

The insight why we need the “double” extension is the following: As we shall
see, the proof of our generalized Bondareva-Shapley theorem is based on the
strong duality theorem for infinite LPs (Proposition 3), which is based on
separation of a closed convex set from a point (not in the set). Therefore, we
need to take the weak closure of a convex set in our proof. This is why we use
the nets of balancing set systems.

Regarding that the weight of the grand coalition is sign unrestricted, notice
that the linear combinations of Dirac measures are $\kappa$-additve for any
$\kappa$, moreover, it is easy to see that the linear space spanned by the
Dirac measures is weak* dense in the set of bounded additive set functions.
Hence, by the results of Schmeidler (1967), Kannai (1969, 1992), and Bartl and
Pintér (2022), we have a necessary and sufficient condition for the non-
emptiness of the “approximate” $\mathop{\rm\kappa\mathchar 45\relax
core}\nolimits$ for any $\kappa$ for free: Schmeidler balancedness. However,
we analyze the non-emptiness of the (exact) $\mathop{\rm\kappa\mathchar
45\relax core}\nolimits$ for any $\kappa$. Therefore, we set the appropriate
variable (the weight of the grand coalition) in the primal problem be sign
unrestricted, by which we get equality in the related constraint in the dual
problem (the total mass of an allocation must exactly be the value of the
grand coalition), hence we will have a necessary and sufficient condition for
the non-emptiness of the $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$
for any $\kappa$: $\kappa$-balancedness.

Between Schmeidler balancedness and $\kappa$-balancedness, there lies the
“double” extension of the former one, Schmeidler $\kappa$-balancedness, where
only the first step is taken: we take nets of balancing weight systems. Even
though we shall see later that Schmeidler $\kappa$-balancedness does not lead
to new characterization results, it provides deeper understanding of the
problem.

Since Schmeidler $\kappa$-balancedness is the same as $\kappa$-balancedness
except that $K_{v}$ in Definition 13 is replaced by $K_{v}^{+}$ in Definition
14, by $K_{v}^{+}\subseteq K_{v}$, it is clear that $\kappa$-balancedness
implies Schmeidler $\kappa$-balancedness. Furthermore, Schmeidler
$\kappa$-balancedness and $\kappa$-balancedness are related by the following
lemma.

######  Lemma 16.

For a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ it holds

$\sup_{\begin{subarray}{c}(\lambda^{i})_{i\in
I}\subseteq\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}\\\ A(\lambda^{i})\buildrel
w\over{\longrightarrow}\chi_{N}\\\ c(\lambda^{i})\longrightarrow
z\end{subarray}}z\,\,\leq\,v(N)\qquad\quad\text{if and only
if}\quad\qquad\sup_{\begin{subarray}{c}(\lambda^{j})_{j\in
J}\subseteq\mathbb{R}_{*}^{(\mathcal{A}^{\prime})}\\\ A(\lambda^{j})\buildrel
w\over{\longrightarrow}\chi_{N}\\\ c(\lambda^{j})\longrightarrow z\\\
\liminf\lambda^{j}_{N}>-\infty\end{subarray}}z\,\,\leq\,v(N)\,.$

where $A(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}$ and
$c(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)$ for any
$\lambda\in\mathbb{R}_{+}^{\mathcal{A}^{\prime}}$.

###### Proof.

The “if” part is obvious. Given a net $(\lambda^{i})_{i\in
I}\subseteq\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}$, consider the same net
$(\lambda^{j})_{j\in J}=(\lambda^{i})_{i\in
I}\subseteq\mathbb{R}_{*}^{(\mathcal{A}^{\prime})}$. Notice that
$\liminf\lambda^{j}_{N}\geq 0$.

We prove the “only if” part indirectly. Suppose the right-hand side does not
hold. Then there exists a net $(\lambda^{j})_{j\in
J}\subseteq\mathbb{R}_{*}^{(\mathcal{A}^{\prime})}$ such that
$\liminf\lambda^{j}_{N}=L>-\infty$ and $A(\lambda^{j})\buildrel
w\over{\longrightarrow}\chi_{N}$ with $c(\lambda^{j})\longrightarrow z>v(N)$.

If $L>0$, then there exists a $j_{0}\in J$ such that $j\geq j_{0}$ implies
$\lambda^{j}_{N}\geq 0$. Consider the index set $I=\\{\,j\in J:j\geq
j_{0}\,\\}$ and the net $(\lambda^{i})_{i\in
I}\subseteq\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}$, which satisfies
$A(\lambda^{i})\buildrel w\over{\longrightarrow}\chi_{N}$ and
$c(\lambda^{j})\longrightarrow z>v(N)$.

Assume $L\leq 0$. There exists a subnet $(\lambda^{j_{i}})_{i\in I}$ of
$(\lambda^{j})_{j\in J}$ such that $\lambda^{j_{i}}_{N}\longrightarrow L$.
Define the net $(\bar{\lambda}^{i})_{i\in I}$ as follows: for any $i\in I$ and
for any $S\in\mathcal{A}^{\prime}$ let

$\bar{\lambda}^{i}_{S}=\begin{cases}\hphantom{-}0&\text{if $S=N$,}\\\
\hphantom{-}\lambda^{j_{i}}_{S}/(1-L)&\text{otherwise.}\\\ \end{cases}$

Then

$\displaystyle A(\bar{\lambda}^{i})$ | $\displaystyle{}=\sum_{\begin{subarray}{c}S\in\mathcal{A}^{\prime}\\\ S\neq N\end{subarray}}\frac{\lambda^{j_{i}}_{S}\chi_{S}}{1-L}=\frac{A(\lambda^{j_{i}})-\lambda^{j_{i}}_{N}\chi_{N}}{1-L}$
---|---
| $\displaystyle{}\buildrel
w\over{\longrightarrow}\frac{\chi_{N}-L\chi_{N}}{1-L}=\chi_{N}\,,$

and

$\displaystyle c(\bar{\lambda}^{i})$ | $\displaystyle{}=\sum_{\begin{subarray}{c}S\in\mathcal{A}^{\prime}\\\ S\neq N\end{subarray}}\frac{\lambda^{j_{i}}_{S}v(S)}{1-L}=\frac{c(\lambda^{j_{i}})-\lambda^{j_{i}}_{N}v(N)}{1-L}$
---|---
|
$\displaystyle{}\longrightarrow\frac{z-Lv(N)}{1-L}>\frac{v(N)-Lv(N)}{1-L}=v(N)\,.$

It follows that the left-hand side does not hold in either case, which
concludes the proof. ∎

##  5 The main result

The next result is our generalized Bondareva-Shapley Theorem.

######  Theorem 17.

For any game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$ it holds that
$\mathop{\rm\kappa\mathchar 45\relax core}\nolimits(v)\neq\emptyset$ if and
only if the game is $\kappa$-balanced.

###### Proof.

Put $X=\mathbb{R}^{(\mathcal{A}^{\prime})}$,
$P=\mathbb{R}_{\ast}^{(\mathcal{A}^{\prime})}$, $Y=\Lambda(\mathcal{A})$, and
$Y^{*}=\mathop{\rm ba}\nolimits^{\kappa}(\mathcal{A})$, moreover define the
mapping $A\colon\mathbb{R}^{(\mathcal{A}^{\prime})}\to\Lambda(\mathcal{A})$ by
$A(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}$, let
$b=\chi_{N}$, and define the functional
$c\colon\mathbb{R}^{(\mathcal{A}^{\prime})}\to\mathbb{R}$ by
$c(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)$. Now, consider the
programs $(\mathrm{P_{LP}})$ and $(\mathrm{D_{LP}})$ of (1).

Notice that program $(\mathrm{P_{LP}})$ is superconsistent and its supervalue
is at least $v(N)$. (Consider that $\bigl{(}A(\lambda),c(\lambda)\bigr{)}\in
K_{v}\subseteq\,\overline{\\!K_{v}}$ for
$\lambda\in\mathbb{R}^{(\mathcal{A}^{\prime})}$ with $\lambda_{N}=1$ and
$\lambda_{S}=0$ for $S\neq N$.) Then the game is $\kappa$-balanced (Definition
14) if and only if the supervalue of $(\mathrm{P_{LP}})$ is finite and not
greater than $v(N)$ (Remark 15).

Moreover, observe that a set function $\mu\in\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A})$ is feasible for $(\mathrm{D_{LP}})$ if and
only if $\mu(S)\geq v(S)$ for all $S\in\mathcal{A}^{\prime}$ and
$\mu(N)=v(N)$. Thus program $(\mathrm{D_{LP}})$ is equivalent to finding an
element of $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits(v)$, and its
value is $v(N)$ if it is consistent, and its value is $+\infty$ otherwise.

Therefore by Proposition 3 the game has a non-empty
$\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ (program
$(\mathrm{D_{LP}})$ is consistent) if and only if it is $\kappa$-balanced (the
supervalue of program $(\mathrm{P_{LP}})$ is not greater than $v(N)$). ∎

If the player set $N$ is finite, then so is
$\mathcal{A}^{\prime}\subseteq\mathcal{P}(N)$, whence the cone $K_{v}$ is
closed. Then by Lemma 16 $\kappa$-balancedness reduces to Schmeidler
balancedness, which is (ordinary) balancedness (Bondareva, 1963; Shapley,
1967; Faigle, 1989), and the $\mathop{\rm\kappa\mathchar 45\relax
core}\nolimits$ is the (ordinary) core in the finite case. We thus obtain the
classical Bondareva-Shapley Theorem as a corollary of Theorem 17:

######  Corollary 18 (Bondareva-Shapley Theorem).

If $N$ is finite, then the core of a game with or without restricted
cooperation is non-empty if and only if the game is balanced.

Regarding Theorem 17, it is worth mentioning that while Bondareva (1963)
applied the strong duality theorem to prove the Bondareva-Shapley Theorem,
Shapley (1967) used a different approach. We do not go into the details, but
we remark that the common point in both approaches is the application of a
separating hyperplane theorem. In other words, both Bondareva’s and Shapley’s
approaches are based on the same separating hyperplane theorem, practically
their result is a direct corollary of that. Here we use the strong duality
theorem for infinite LPs (Proposition 3, Anderson and Nash, 1987), which is
also a direct corollary of the same separating hyperplane theorem.

###  5.1 The $\sigma$-additive case

In this subsection let $\kappa=\aleph_{0}$. Then $\mathop{\rm
ba}\nolimits^{\kappa}(\mathcal{A})=\mathop{\rm ca}\nolimits(\mathcal{A})$, the
space of all bounded countably additive set functions on $\mathcal{A}$. Given
a game $v\in\mathcal{G}^{\mathcal{A}^{\prime}}$, its
$\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ is the $\sigma$-additive
core $\mathop{\rm ca\mathchar 45\relax core}\nolimits(v)$ introduced by
Definition 10.

In the next example we demonstrate that there exists a Schmeidler
$\kappa$-balanced ($\aleph_{0}$-balanced) non-negative game without restricted
cooperation having its $\mathop{\rm ca\mathchar 45\relax core}\nolimits$
empty.

######  Example 19.

Let the player set $N=\mathbb{N}$, the system of coalitions
$\mathcal{A}=\mathcal{P}(\mathbb{N})$, and the game $v$ be defined as follows:
for any $S\in\mathcal{A}$ let

$v(S)=\begin{cases}\hphantom{-}1&\text{if $\\#(N\setminus S)\leq 1$,}\\\
\hphantom{-}0&\text{otherwise.}\\\ \end{cases}$

We show that $\mathop{\rm ca\mathchar 45\relax core}\nolimits(v)=\emptyset$.
If $\mu\in\mathop{\rm ca\mathchar 45\relax core}\nolimits(v)$, then
$\mu\bigl{(}N\setminus\\{n\\}\bigr{)}\geq
v\bigl{(}N\setminus\\{n\\}\bigr{)}=1$ and $v(N)=1$, whence
$\mu\bigl{(}\\{n\\}\bigr{)}\leq 0$. So
$0\geq\sum_{n=1}^{\infty}\mu\bigl{(}\\{n\\}\bigr{)}=\mu(N)=v(N)=1$, a
contradiction.

We now show that, if $(\chi_{N},z)\in\,\overline{\\!K_{v}^{+}}$, see (4), then
$z\leq 1=v(N)$. We have $(\chi_{N},z)\in\,\overline{\\!K_{v}^{+}}$ if and only
if each neighborhood of the point $(\chi_{N},z)$ intersects the cone
$K_{v}^{+}$. In particular, if $(\chi_{N},z)\in\,\overline{\\!K_{v}^{+}}$,
then for any natural number $m$ and for any $\varepsilon>0$ there exists a
point $(f,t)\in K_{v}^{+}$ such that $f$ belongs to the weak neighborhood

$\bigl{\\{}\,f\in\Lambda(\mathcal{A}):\bigl{|}\delta^{\prime}_{i}(f)-1\bigr{|}<\varepsilon\enskip\text{for}\enskip
i=1,\,\ldots,\,m\,\bigr{\\}}\,,$

where $\delta^{\prime}_{i}$ is the continuous linear functional induced by the
Dirac measure $\delta_{i}$ concentrated at $i$, see (2), and $t$ belongs to
the neighborhood
$\bigl{\\{}\,t\in\mathbb{R}:\mathopen{|}t-z\mathclose{|}<\varepsilon\,\bigr{\\}}$.
Hence, we have a natural number $n$, some distinct sets
$S_{0},S_{1},\ldots,\allowbreak S_{n}\in\mathcal{A}$, and some non-negative
$\lambda_{S_{0}}$, $\lambda_{S_{1}}$, …, $\lambda_{S_{n}}$ such that
$f=\lambda_{S_{0}}\chi_{S_{0}}+\lambda_{S_{1}}\chi_{S_{1}}+\cdots+\lambda_{S_{n}}\chi_{S_{n}}$
and

$\biggl{|}\sum_{\begin{subarray}{c}j=0\\\ S_{j}\ni
i\,\end{subarray}}^{n}\lambda_{S_{j}}-1\biggr{|}<\varepsilon\qquad\text{for}\quad
i=1,\,\ldots,\,m$ (6)

with

$\biggl{|}\sum_{j=0}^{n}\lambda_{S_{j}}v(S_{j})-z\biggr{|}=\biggl{|}\sum_{\begin{subarray}{c}j=0\\\
\\#(N\setminus S_{j})\leq
1\,\end{subarray}}^{n}\lambda_{S_{j}}-z\biggr{|}<\varepsilon\,.$ (7)

We can assume w.l.o.g. that $S_{0}=N$, as well as $\\#(N\setminus S_{j})=1$
for $j=1$, …, $n_{1}$ and $\\#(N\setminus S_{j})>1$ for $j=n_{1}+1$, …, $n$,
where $n_{1}\leq n$.

Everything is clear if there exists an $i\in\\{1,\ldots,m\\}$ such that
$i\in\bigcap_{j=1}^{n_{1}}S_{j}$. Then by (6)

$\sum_{\begin{subarray}{c}j=0\\\ \\#(N\setminus S_{j})\leq
1\,\end{subarray}}^{n}\lambda_{S_{j}}=\sum_{j=0}^{n_{1}}\lambda_{S_{j}}\leq\sum_{\begin{subarray}{c}j=0\\\
S_{j}\ni i\,\end{subarray}}^{n}\lambda_{S_{j}}<1+\varepsilon\,,$

whence $z<1+2\varepsilon$ by (7).

In the other case we have $m\leq n_{1}$ and, because the sets $S_{0}$,
$S_{1}$, …, $S_{n}$ are pairwise distinct, for $i=1$, …, $m$ we can assume
w.l.o.g. that $S_{i}=N\setminus\\{i\\}$. By (6)

$\sum_{j=0}^{n_{1}}\lambda_{S_{j}}-\lambda_{S_{i}}=\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{n_{1}}\lambda_{S_{j}}\leq\sum_{\begin{subarray}{c}j=0\\\
S_{j}\ni
i\,\end{subarray}}^{n}\lambda_{S_{j}}<1+\varepsilon\qquad\text{for}\quad
i=1,\,\ldots,\,m\,.$

Summing up, we get
$m\sum_{j=0}^{n_{1}}\lambda_{S_{j}}-\sum_{i=1}^{m}\lambda_{S_{i}}<m+m\varepsilon$,
whence
$m\sum_{j=0}^{n_{1}}\lambda_{S_{j}}-\sum_{j=0}^{n_{1}}\lambda_{S_{i}}<m+m\varepsilon$.
It then follows

$\sum_{\begin{subarray}{c}j=0\\\ \\#(N\setminus S_{j})\leq
1\,\end{subarray}}^{n}\lambda_{S_{j}}=\sum_{j=0}^{n_{1}}\lambda_{S_{j}}<\frac{m}{m-1}(1+\varepsilon)\,.$

Taking (7) into account, we obtain

$z<\frac{m}{m-1}(1+\varepsilon)+\varepsilon\,.$ (8)

Since $1+2\varepsilon<(1+\varepsilon)m/(m-1)+\varepsilon$, inequality (8)
holds in both cases. By that $m\geq 2$ and $\varepsilon>0$ can be arbitrary,
we conclude that $z\leq 1$.

######  Remark 20.

Consider the game $v$ from Example 19. Since $\mathop{\rm ca\mathchar 45\relax
core}\nolimits(v)=\emptyset$, the game is not $\kappa$-balanced
($\aleph_{0}$-balanced). To see this, consider the sequence
$(\lambda^{i})_{i=1}^{\infty}$, with
$\lambda^{i}\in\mathbb{R}_{*}^{(\mathcal{A})}$, defined as follows: for any
$i\in\mathbb{N}$ and for any $S\in\mathcal{A}$ let

$\lambda^{i}_{S}=\begin{cases}-(i-2)&\text{if $S=N$,}\\\
\hphantom{-}1&\text{if $S=N\setminus\\{n\\}$ for $n=1$,~{}\ldots,~{}$i$,}\\\
\hphantom{-}0&\text{otherwise.}\\\ \end{cases}$

Then
$\sum_{S\in\mathcal{A}}\lambda^{i}_{S}\chi_{S}=2\chi_{N}-\chi_{\\{1,\ldots,i\\}}\buildrel
w\over{\longrightarrow}\chi_{N}$, where the weak convergence in the space
$\Lambda(\mathcal{A})$ is with respect to $\mathop{\rm
ca}\nolimits(\mathcal{A})$, and
$\sum_{S\in\mathcal{A}}\lambda^{i}_{S}v(S)=2>1=v(N)$.

Notice again that the sequence
$(\lambda^{i}_{N})_{i=1}^{\infty}=(2-i)_{i=1}^{\infty}$ is unbounded below. If
$(\lambda^{i}_{N})_{i=1}^{\infty}$ were bounded below, then by Lemma 16 we
would get a contradiction with Example 19.

Example 19 demonstrates that it is not sufficient to use
$\mathbb{R}_{+}^{(\mathcal{A})}$ and $K_{v}^{+}$ in the definition of
$\kappa$-balancedness; that is, Schmeidler $\kappa$-balancedness is unable to
reveal that the $\mathop{\rm ca\mathchar 45\relax core}\nolimits$ is empty
even for non-negative games without restricted cooperation.

######  Remark 21.

Reconsidering Schmeidler balancedness for the additive case, it is somehow
tempting to ask whether the following “$\sigma$-extension” of condition (3)
could lead to a similar result in the $\sigma$-additive case too:

$\sup\biggl{\\{}\,\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S):\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}=\chi_{N},\,\lambda\in\mathbb{R}_{+}^{[\mathcal{A}^{\prime}]}\biggr{\\}}\leq
v(N)\,,$ (9)

where
$\mathbb{R}^{[\mathcal{A}^{\prime}]}=\bigl{\\{}\,\lambda\in\mathbb{R}^{\mathcal{A}^{\prime}}:\\#\\{\,S\in\mathcal{A}^{\prime}:\lambda_{S}\neq
0\,\\}\leq\aleph_{0}\,\bigr{\\}}$ and
$\mathbb{R}_{+}^{[\mathcal{A}^{\prime}]}=\bigl{\\{}\,\lambda\in\mathbb{R}^{[\mathcal{A}^{\prime}]}:\lambda_{S}\geq
0$ for all $S\in\mathcal{A}^{\prime}\,\bigr{\\}}$. Moreover, the convergence
of the sum $\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}$ is understood
pointwise. In this case it is equivalent to say that the convergence is weak
in the space $\Lambda(\mathcal{A})$ with respect to $\mathop{\rm
ca}\nolimits(\mathcal{A})$. If the sum
$\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)$ is convergent, but not
absolutely convergent, then we put
$\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S):=+\infty$.

Denoting $A(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}\chi_{S}$ and
$c(\lambda)=\sum_{S\in\mathcal{A}^{\prime}}\lambda_{S}v(S)$, we can also
consider the following generalization of (9). Let $z\leq v(N)$ whenever there
exists a net $(\lambda_{i})_{i\in
I}\subseteq\mathbb{R}_{+}^{[\mathcal{A}^{\prime}]}$ such that
$A(\lambda^{i})\buildrel w\over{\longrightarrow}\chi_{N}$ and
$c(\lambda^{i})\longrightarrow z$ where $z$ is finite. Then for each $i\in I$
there exists a sequence
$(\lambda^{in})_{n=1}^{\infty}\subseteq\mathbb{R}^{(\mathcal{A}^{\prime})}_{+}$
such that $A(\lambda^{in})\buildrel w\over{\longrightarrow}A(\lambda^{i})$ and
$c(\lambda^{in})\longrightarrow c(\lambda^{i})$. Consequently, there exists a
net $(\lambda_{j})_{j\in J}\subseteq\mathbb{R}_{+}^{(\mathcal{A}^{\prime})}$
such that $A(\lambda^{j})\buildrel w\over{\longrightarrow}\chi_{N}$ and
$c(\lambda^{j})\longrightarrow z$. In other words, Schmeidler
$\kappa$-balancedness covers such extensions of Schmeidler balancedness
(Definition 12) like (9).

Moreover, in Example 19 we presented a non-negative Schmeidler
$\kappa$-balanced game. Therefore, the presented game is balanced according to
(9) too, but the $\mathop{\rm ca\mathchar 45\relax core}\nolimits$ of the game
is empty.

##  6 Conclusion

We have generalized the Bondareva-Shapley Theorem to TU games with and without
restricted cooperation, with infinitely many players, and with at least
$\sigma$-additive cores: we have proved for an arbitrary infinite cardinal
$\kappa$ that the $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ of a TU
game with or without restricted cooperation is not empty if and only if the TU
game is $\kappa$-balanced. The main conceptual messages of our results might
be that in the proper notion of balancing weight system the weight of the
grand coalition is sign unrestricted.

While $\kappa$-balancedness is universally a necessary and sufficient
condition for that the $\mathop{\rm\kappa\mathchar 45\relax core}\nolimits$ of
a game with or without restricted cooperation is not empty (Theorem 17), we
have shown that Schmeidler $\kappa$-balancedness (which implies Schmeidler
balancedness as well as its $\sigma$-extension, see Remark 21) is not suitable
for this purpose even in the case of $\sigma$-additive core of a non-negative
game without restricted cooperation (Example 19).

Notice that Kannai (1969, 1992) gave another necessary and sufficient
condition for that the $\sigma$-additive core of a non-negative game without
restricted cooperation is not empty. Kannai’s result is based on a very
different approach and not related directly to our $\aleph_{0}$-balancedness
condition.

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