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

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 (Problemy Kibernetiki 10:119–139, 1963), Shapley (Naval Res Logist Q 14:453–460, 1967), Schmeidler (On balanced games with infinitely many players, The Hebrew University, Jerusalem, 1967), Faigle (Zeitschrift für Oper Res 33(6):405–422, 1989), Kannai (J Math Anal Appl 27:227–240, 1969; The core and balancedness, handbook of game theory with economic applications, North-Holland, 1992), Pintér (Linear Algebra Appl 434(3):688–693, 2011) and Bartl and Pintér (Oper Res Lett 51(2):153–158, 2023).

1 Introduction

The core (Aumann, 1961; Gillies, 1959; Shapley, 1955) 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; Faigle, 1989; Shapley, 1967) 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 reformulate the notion of additive set function. The first, when we weaken (generalize) the notion of additivity; this leads to the notion of k-additive core (Grabisch & Miranda, 2008), where k is a finite cardinal (natural number). The second, when we use a notion stronger than additivity (e.g. $\sigma$-additivity). This latter approach is considered here.

Schmeidler (1967), Kannai (1969, 1992), Pintér (2011), and Bartl and Pintér (2023) 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 17). Bartl and Pintér (2023) 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 (1969, 1992) considered the following question: When does there exist a bounded $\sigma$-additive set function in the core? In this paper we generalize the above question as follows: When does there exist a bounded $\kappa$-additive set function in the core, where $\kappa$ is an infinite cardinal number? 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 15 and 21). 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 25).

The set-up of the paper is as follows. In the next section we recall the main mathematical notions and results, which are related to infinite LPs, and used in this paper. In Sect. 3 we define the notion of $\kappa$-additive set functions and discuss some related concepts and results. In Sect. 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 25). 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 is the space of all linear functionals on X; that is, all linear mappings $\varphi :X \rightarrow \mathbb {R}$, which are also known as linear forms on X. We denote the algebraic dual of X by $X'$. Similarly $Y'$ denotes the algebraic dual of Y. Moreover, $Y^* \subseteq Y'$ denotes a linear subspace of $Y'$ 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 $y(f) \ne 0$. For any linear mapping $A :X \rightarrow Y$ its adjoint mapping is $A' :Y' \rightarrow X'$ with $(A'(y))(x) = y\bigl (A(x)\bigr )$ for all $x \in X$ and $y \in Y'$. 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 $f,g :X \rightarrow \mathbb {R}$ we write $f \ge _P g$ if $f(x) \ge g(x)$ for all $x \in P$.

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

(1)

where $P \subseteq X$ is a convex cone and $Y^*$ is a subspace of $Y'$ 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 $\bigl (A' (y)\bigr )(x) \ge c(x)$ for all $x \in P$. The value of a consistent program $(\mathrm {D_{LP}})$ is $\inf \bigl \{\, y(b): A'(y) \ge _P c, \; y \in Y^* \,\bigr \}$.

In the next definition we assume the weak topology on the space Y with respect to $Y^*$. To define 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 $y_1, \ldots , y_n \in Y^*$ 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 $(I,\le )$ is right-directed if I is a preordered set and for any $i,j \in I$ there exists a $k \in I$ such that $i \le k$ and $j \le k$. A net (generalized sequence) of X is $(x_i)_{i\in I}$ where $(I,\le )$ 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 $A(x_i) {\mathop {\longrightarrow }\limits ^{w}} b$, 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 $z^*$ 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 $A(x_i) {\mathop {\longrightarrow }\limits ^{w}} b$ and $c(x_i) \longrightarrow z$.

Proposition 3

Consider the programs in (1). Program $(\mathrm {P_{LP}})$ is superconsistent and $z^*$ is its finite supervalue if and only if program $(\mathrm {D_{LP}})$ is consistent and $z^*$ 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 $S_1, \ldots , S_n \in \mathcal {A}$, 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 ${\textrm{ba}}(\mathcal {A})$ and ${\textrm{ca}}(\mathcal {A})$ denote, respectively, the set of bounded additive set functions and the set of bounded $\sigma$-additive (i.e. countably additive) set functions $\mu :\mathcal {A} \rightarrow \mathbb {R}$.

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 $\#I \le \kappa$, 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 $i \le j$ implies $S_i \supseteq S_j$ or $S_i \subseteq S_j$, respectively, for any $i, j \in I$.

Let $\mu :\mathcal {A} \rightarrow \mathbb {R}$ 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 Armstrong and Prikry (1980) or Schervish et al. (2017).

Definition 4

A set function $\mu :\mathcal {A} \rightarrow \mathbb {R}$ is $\kappa$-additive if it is additive and $\kappa$-continuous. Let ${\textrm{ba}}^\kappa (\mathcal {A})$ denote the set of $\kappa$-additive set functions over $\mathcal {A}$.

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

Proposition 5

If the set function $\mu :\mathcal {A} \rightarrow \mathbb {R}$ 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 $\kappa$-continuous 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 $\kappa \ge \mathfrak {c}$, where $\mathfrak {c}$ denotes the cardinality of the real numbers; but it is $\kappa$-additive for $\kappa = \aleph _0$.

However, if $\mu$ is a measure such that it is a linear combination of Dirac measures, then it is $\kappa$-additive for every cardinal number $\kappa$.

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 ${\textrm{ba}}^\kappa (\mathcal {A})$ 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 $\#X = \kappa \ge \aleph _0$. Consider $\mathcal {P}(X)$, the power set of X. It is clear that the Dirac measures on $\mathcal {P}(X)$ are $\kappa$-additive. Let

$$\begin{aligned} \varDelta = \left\{ \sum _{n=1}^\infty \alpha _n \delta _n: (\alpha _n)_{n=1}^\infty \in \ell ^1, \delta _n\,\text {being Dirac measures on}\,\mathcal {P}(X)\, \text {for}\,n = 1,2,3,\ldots \right\} . \end{aligned}$$

It is clear that each $\mu \in \varDelta$ is a $\kappa$-additive set function on $\mathcal {P}(X)$. Notice that $\#\varDelta \ge \#{\textrm{ca}}\bigl (\mathbb {N},\mathcal {P}(\mathbb {N})\bigr )$; 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 $\varDelta$ 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 ${\kappa \text {-core}}$ 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 $\lambda :\mathcal {A} \rightarrow \mathbb {R}$ with a finite support; that is,

$$\begin{aligned} \mathbb {R}^{(\mathcal {A})} = \left\{ \lambda \in \mathbb {R}^\mathcal {A} : \#\{\, S \in \mathcal {A} : \lambda _S \ne 0 \,\} < \infty \right\} . \end{aligned}$$

Denoting $\lambda (S)$ and the characteristic function of a set $S \in \mathcal {A}$ by $\lambda _S$ and $\chi _S$, respectively, let $\Lambda (\mathcal {A}) = \{\lambda _{S_1} \chi _{S_1} + \cdots + \lambda _{S_n} \chi _{S_n}: n \in \mathbb {N}, \!\; \lambda _{S_1}, \ldots , \lambda _{S_n} \!\in \mathbb {R}, \!\; S_1, \ldots , S_n \in \mathcal {A}\}$ be the space of all simple functions on $(N,\mathcal {A})$; that is,

$$\begin{aligned} \Lambda (\mathcal {A}) = \left\{ \sum _{S\in \mathcal {A}} \lambda _S \chi _S : \lambda \in \mathbb {R}^{(\mathcal {A})} \right\} . \end{aligned}$$

We define 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

$$\begin{aligned} \Vert f\Vert = \sup _{x \in N} |f(x)|. \end{aligned}$$

The defined norm induces a topology on $\Lambda (\mathcal {A})$, and the topological dual $(\Lambda (\mathcal {A}))^\star$ of the vector space $\Lambda (\mathcal {A})$ consists of all linear functionals $\mu ':\Lambda (\mathcal {A}) \rightarrow \mathbb {R}$ continuous with respect to the norm. It is well-known that the dual $(\Lambda (\mathcal {A}))^\star$ is isometrically isomorphic to ${\textrm{ba}}(\mathcal {A})$, the space of all bounded additive set functions on $\mathcal {A}$ (see e.g. Dunford and Schwartz (1958), Theorem IV.5.1, p. 258); that is, we can identify the space $(\Lambda (\mathcal {A}))^\star$ with ${\textrm{ba}}(\mathcal {A})$ for simplicity. Indeed, a set function $\mu \in {\textrm{ba}}(\mathcal {A})$ induces a continuous linear functional $\mu ' \in (\Lambda (\mathcal {A}))^\star$ on $\Lambda (\mathcal {A})$ as follows:

$$\begin{aligned} \mu '(f) = \lambda _{S_1} \mu (S_1) + \cdots + \lambda _{S_n} \mu (S_n) \end{aligned}$$

(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 $\bigl (\Lambda (\mathcal {A}), {\textrm{ba}}^\kappa (\mathcal {A})\bigr )$ 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 $f(x) \ne 0$. Then $\delta _x$, the Dirac measure concentrated at point x on $\mathcal {A}$, is a $\kappa$-additive set function, and $\delta '_x(f) = f(x) \ne 0$. $\square$

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 recall the notion of TU games. Let N be a non-empty set of players and let $\mathcal {A}' \subseteq \mathcal {P}(N)$ be a collection of sets such that $\emptyset ,N \in \mathcal {A}'$. Then a TU game (henceforth a game) on $\mathcal {A}'$ is a set function $v:\mathcal {A}' \rightarrow \mathbb {R}$ such that $v(\emptyset ) = 0$. Every $A \in \mathcal {A}'$ is called a (feasible) coalition, the set $A = N$ is the grand coalition, and v(A) is the payoff of A if the coalition A is formed. We denote the class of games on $\mathcal {A}'$ by $\mathcal {G}^{\mathcal {A}'}$. Let $\mathcal {A}$ denote the field hull of $\mathcal {A}'$; that is, the smallest field of sets that contains $\mathcal {A}'$. We say that $v \in \mathcal {G}^{\mathcal {A}'}$ is a game without restricted cooperation if $\mathcal {A}' = \mathcal {A}$; that is, $\mathcal {A}'$ is a field. Otherwise, if $\mathcal {A}'$ is not a field, we say $v \in \mathcal {G}^{\mathcal {A}'}$ is a game with restricted cooperation.

Example 9

Consider the following game: Let the player set N be the set of natural numbers; that is, let $N = \mathbb {N}$. Moreover, let $\mathcal {A}'$, the class of the feasible coalitions, be the class of the finite and co-finite subsets of $\mathbb {N}$; that is, $\mathcal {A}' = \{A \subseteq \mathbb {N}$: either $\# A < \infty$ or $\# (N {\setminus } A) < \infty \}$. Furthermore, let game v be defined as follows:

$$\begin{aligned} v(A) = {\left\{ \begin{array}{ll} 0 &{} \text {if}\;\#\,A <\infty ,\\ 1 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Since $\mathcal {A}'$ is a field, we have that v is a game without restricted cooperation.

Let us modify the game v above as follows: Let the set of the feasible coalitions be $\mathcal {A}'' = \mathcal {A}' \cup \{2k: k \in \mathbb {N}\}$; that is, the coalitions from $\mathcal {A}'$ and the set of the even numbers. Moreover, let the modified game $v'$ be the following:

$$\begin{aligned} v'(A) = {\left\{ \begin{array}{ll} v(A) &{} \text {if}\;A \in \mathcal {A}',\\ 10 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Since $\mathcal {A}'' = \mathcal {A}' \cup \{2k: k \in \mathbb {N} \}$ is not a field, we conclude that $v'$ 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 recall the notion of additive core of a game, which was considered by Schmeidler (1967), Kannai (1969, 1992), Pintér (2011), and Bartl and Pintér (2023).

Definition 10

For a game $v \in \mathcal {G}^{\mathcal {A}'}$ its additive core (henceforth ${{\mathrm{ba`-core}}}$) is defined as follows:

$$\begin{aligned} {{\mathrm{ba`-core}}}(v) = \left\{ \mu \in {\textrm{ba}}(\mathcal {A}) : \mu (N) = v(N)\; \text {and}\; \mu (S) \ge v(S)\;\text {for all}\; S \in \mathcal {A}' \right\} . \end{aligned}$$

Example 11

Consider the games v and $v'$ from Example 9. Since v itself is an additive set function, we have that

$$\begin{aligned} {{\mathrm{ba`-core}}}(v) = \{v\} . \end{aligned}$$

Since the domain $\mathcal {A}'' = \mathcal {A}' \cup \{2k: k \in \mathbb {N} \}$ of $v'$ is not a field, it is meaningless to use the notion of additivity to characterize $v'$. Moreover, since the domain of v is a proper subset of the domain of $v'$, and $v'$ itself is an extension of v from $\mathcal {A}'$ onto $\mathcal {A}''$, we have the following:

$$\begin{aligned} {{\mathrm{ba`-core}}}(v') = \left\{ \mu \in {\textrm{ba}}(\overline{\mathcal {A}''}) : \mu \bigl (\{2k : k \in \mathbb {N} \}\bigr ) \ge 10 , \; \mu (A) = v(A)\;\text {for all}\; A \in \mathcal {A}' \right\} , \end{aligned}$$

where $\overline{\mathcal {A}''}$ is the field hull of $\mathcal {A}''$.

In other words, each element of ${{\mathrm{ba`-core}}}(v')$ is such an extension of v, the only element of ${{\mathrm{ba`-core}}}(v)$, that its value at the set of the even numbers is at least 10. It is clear that there exists such an extension of v; that is, ${{\mathrm{ba`-core}}}(v') \ne \emptyset$. Even more, it is easy to see that there are continuum many such extensions of v because the value of any extension at the set of even numbers can be any real number not smaller than 10, hence the cardinality of ${{\mathrm{ba`-core}}}(v')$ is continuum.

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

Definition 12

For a game $v \in \mathcal {G}^{\mathcal {A}'}$ its $\sigma$-additive core (henceforth ${{\mathrm{ca`-core}}}$) is defined as follows:

$$\begin{aligned} {{\mathrm{ca`-core}}}(v) = \bigl \{\mu \in {\textrm{ca}}(\mathcal {A}) : \mu (N) = v(N) \;\text {and}\; \mu (S) \ge v(S)\;\text {for all}\; S \in \mathcal {A}' \bigr \}. \end{aligned}$$

Example 13

Consider the games v and $v'$ from Example 9 again. Since the ${{\mathrm{ca`-core}}}$ is a subset of the ${{\mathrm{ba`-core}}}$ for any game, and v is not $\sigma$-additive ($\sum _{n \in \mathbb {N}} v\bigl (\{n\}\bigr ) = 0 \ne 1 = v(N)$), we conclude

$$\begin{aligned} {{\mathrm{ca`-core}}}(v) = \emptyset . \end{aligned}$$

Since the restriction of any element of ${{\mathrm{ca`-core}}}(v')$ onto $\mathcal {A}'$ is an element of ${{\mathrm{ca`-core}}}(v)$, but ${{\mathrm{ca`-core}}}(v) = \emptyset$, we conclude that

$$\begin{aligned} {{\mathrm{ca`-core}}}(v') = \emptyset . \end{aligned}$$

Example 14

Consider the following game: Let the player set N be the set of real numbers, that is, let $N = \mathbb {R}$. Moreover, let $\mathcal {A}'$, the class of the feasible coalitions, be the class of the finite and co-finite subsets of $\mathbb {R}$; that is, $\mathcal {A}' = \{A \subseteq \mathbb {R}:$ either $\# A < \infty$ or $\# (N {\setminus } A) < \infty \}$. Furthermore, let game w be defined as follows:

$$\begin{aligned} w(A) = {\left\{ \begin{array}{ll} 0 &{} \text {if}\;\#\,A < \infty ,\\ 1 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Then it is easy to see that w itself is a measure (non-negative and $\sigma$-additive), hence

$$\begin{aligned} {{\mathrm{ca`-core}}}(w) = \{w\}. \end{aligned}$$

In general, for an infinite cardinal number $\kappa$ we introduce the notion of ${\kappa \text {-core}}$ of a game.

Definition 15

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

$$\begin{aligned} {\kappa \text {-core}}(v) = \bigl \{\mu \in {\textrm{ba}}^\kappa (\mathcal {A}) : \mu (N) = v(N)\;\text {and}\; \mu (S) \ge v(S)\;\text {for all}\; S \in \mathcal {A}' \bigr \}. \end{aligned}$$

Example 16

Consider the game w from Example 14. Then it is clear that w is not $\kappa$-additive for any $\kappa \ge \mathfrak {c}$, hence

$$\begin{aligned} {\kappa \text {-core}}(w) = \emptyset \end{aligned}$$

for every $\kappa \ge \mathfrak {c}$.

In words, the ${{\mathrm{ba`-core}}}$, the ${{\mathrm{ca`-core}}}$, and the ${\kappa \text {-core}}$ 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 $\mathcal {A}'$ that meet the conditions of efficiency ($\mu (N) = v(N)$) and coalitional rationality ($\mu (S) \ge v(S)$ for all $S \in \mathcal {A}'$). Observe that the ${{\mathrm{ca`-core}}}$ is a special case of the ${\kappa \text {-core}}$ when $\kappa = \aleph _0$.

Notice that in the finite case all the three notions of ${{\mathrm{ba`-core}}}$, ${{\mathrm{ca`-core}}}$, and ${\kappa \text {-core}}$ 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, letting

$$\begin{aligned} \mathbb {R}_+^{(\mathcal {A}')} = \bigl \{\lambda \in \mathbb {R}^{(\mathcal {A}')}: \lambda _S \ge 0\;\text {for all}\; S \in \mathcal {A}' \bigr \} , \end{aligned}$$

we generalize his notion to the restricted cooperation case, and call it Schmeidler balancedness.

Definition 17

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

$$\begin{aligned} \sup \left\{ \sum _{S\in \mathcal {A}'} \lambda _S v(S) : \sum _{S\in \mathcal {A}'} \lambda _S \chi _S = \chi _N, \, \lambda \in \mathbb {R}_+^{(\mathcal {A}')} \right\} \le v(N). \end{aligned}$$

(3)

Example 18

Consider the game v from Example 9. Take any balancing weights ($\lambda \in \mathbb {R}_+^{(\mathcal {A}')}$ such that $\sum _{S \in \mathcal {A}'} \lambda _S \chi _S = \chi _N$). Then for any ${{\mathrm{ba`-core}}}$ element $\mu$ we have that

$$\begin{aligned} \sum _{S\in \mathcal {A}'} \lambda _S v(S) \le \sum _{S\in \mathcal {A}'} \lambda _S \mu (S) = \mu (N) = v(N) , \end{aligned}$$

therefore the game v is Schmeidler balanced.

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

Recall that $Y^* = {\textrm{ba}}^\kappa (\mathcal {A})$ is a linear subspace of $Y^\star = {\textrm{ba}}(\mathcal {A})$, 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 $Y^* = {\textrm{ba}}^\kappa (\mathcal {A})$ (see Lemma 8).

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

$$\begin{aligned} K_v^+ = \left\{ \left( \sum _{S\in \mathcal {A}'} \lambda _S \chi _S, \sum _{S\in \mathcal {A}'} \lambda _S v(S) \right) : \lambda \in \mathbb {R}_+^{(\mathcal {A}')} \right\} . \end{aligned}$$

(4)

Definition 19

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

$$\begin{aligned} z \le v(N) \end{aligned}$$

for all $z \in \mathbb {R}$ such that $(\chi _N,z) \in \,\overline{\!K_v^+}$, where $\overline{\!K_v^+}$ is the closure of $K_v^+$.

Example 20

Consider the game w from Example 14. Knowing that ${{\mathrm{ca`-core}}}(w) \ne \emptyset$, it is easy to see that the game w is Schmeidler balanced. We show that it is Schmeidler $\aleph _0$-balanced too. Take a net of weights $(\lambda ^i)_{i\in I} \subseteq \mathbb {R}_+^{(\mathcal {A}')}$ such that $\sum _{S\in \mathcal {A}'} \lambda ^i_S \chi _S {\mathop {\longrightarrow }\limits ^{w}} \chi _N$; that is, for any $\mu \in {\textrm{ca}}(\mathcal {A}')$ we have $\sum _{S\in \mathcal {A}'} \lambda ^i_S \mu (S) \longrightarrow \mu (N)$; and such that $\sum _{S\in \mathcal {A}'} \lambda ^i_S w(S) \longrightarrow z$. Then for any ${{\mathrm{ca`-core}}}$ element $\mu$ and for any index $i \in I$ we have that

$$\begin{aligned} \sum _{S\in \mathcal {A}'} \lambda ^i_S v(S) \le \sum _{S\in \mathcal {A}'} \lambda ^i_S \mu (S) . \end{aligned}$$

Moreover, since both sides converge, it follows that

$$\begin{aligned} z = \lim _{i\in I} \sum _{S\in \mathcal {A}'} \lambda ^i_S v(S) \le \lim _{i\in I} \sum _{S\in \mathcal {A}'} \lambda ^i_S \mu (S) = \mu (N) = w(N) , \end{aligned}$$

therefore the game w is Schmeidler $\aleph _0$-balanced.

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

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

$$\begin{aligned} \mathbb {R}_*^{(\mathcal {A}')} = \bigl \{\lambda \in \mathbb {R}^{(\mathcal {A}')}: \lambda _S \ge 0\;\text {for all}\; S \in \mathcal {A}' {\setminus } \{N\} \bigr \} \end{aligned}$$

and consider the convex cone

$$\begin{aligned} K_v = \left\{ \left( \sum _{S\in \mathcal {A}'} \lambda _S \chi _S, \sum _{S\in \mathcal {A}'} \lambda _S v(S) \right) : \lambda \in \mathbb {R}_*^{(\mathcal {A}')} \right\} . \end{aligned}$$

(5)

Definition 21

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

$$\begin{aligned} z \le v(N) \end{aligned}$$

for all $z \in \mathbb {R}$ such that $(\chi _N,z) \in \,\overline{\!K_v}$, where $\overline{\!K_v}$ is the closure of $K_v$.

Example 22

Consider the game w from Example 14. We know by Example 20 that the game is Schmeidler $\aleph _0$-balanced, hence Schmeidler balanced. We now show that the game w is not Schmeidler $\kappa$-balanced if $\kappa \ge \mathfrak {c}$. Take the index set $I = \bigl \{i \subseteq N: \#(N {\setminus } i ) < \infty \bigr \}$, and define its ordering by $i \le j$ if $i \supseteq j$. Observe that $\# I = \mathfrak {c}$. Consider the net of weights $(\lambda ^i)_{i\in I}$, with $\lambda ^i \in \mathbb {R}_+^{(\mathcal {A})}$, defined as follows: for any $i \in I$ and for any $S \in \mathcal {A}$ let

$$\begin{aligned} \lambda ^i_S = {\left\{ \begin{array}{ll} 1 &{} \text {if}\;S = i\;or\;S = N,\\ 0 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Then $\sum _{S\in \mathcal {A}'} \lambda ^i_S \chi _S = \chi _i + \chi _N {\mathop {\longrightarrow }\limits ^{w}} \chi _N$; that is, for any $\mu \in {\textrm{ba}}^\kappa (\mathcal {A}')$, with $\kappa \ge \mathfrak {c}$, we have $\sum _{S\in \mathcal {A}'} \lambda ^i_S \mu (S) = \mu (i) + \mu (N) \longrightarrow \mu (N)$. However, for every $i \in I$ we have that $\sum _{S\in \mathcal {A}'} \lambda ^i_S w (S) = w(i) + w(N) = 2$, therefore $\lim _{i \in I} \sum _{S\in \mathcal {A}'} \lambda ^i_S w (S) = 2 > 1 = w(N)$. In other words, the game w is not Schmeidler $\kappa$-balanced, hence it is not $\kappa$-balanced, if $\kappa \ge \mathfrak {c}$.

Remark 23

The notion of $\kappa$-balancedness and Schmeidler $\kappa$-balancedness is very closely related to the notion of supervalue given in Definition 2. The cone $K_v$ or $K_v^+$ is precisely the set D if $A(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S \chi _S$ and $c(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S v(S)$ with $P = \mathbb {R}_+^{(\mathcal {A}')}$ or $P = \mathbb {R}_*^{(\mathcal {A}')}$, 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 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 weight systems the net of the weights of the grand coalition is not bounded below (see Lemma 24 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 and to approach a point in the closure. This is why we need to use the nets of weight systems.

Regarding that the weight of the grand coalition is sign unrestricted, notice that the linear combinations of Dirac measures are $\kappa$-additive 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 (2023), we have a necessary and sufficient condition for the non-emptiness of the “approximate” ${\kappa \text {-core}}$ for any $\kappa$ for free: Schmeidler balancedness. However, we analyze the non-emptiness of the (exact) ${\kappa \text {-core}}$ 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 ${\kappa \text {-core}}$ for any $\kappa$: $\kappa$-balancedness.

Between Schmeidler balancedness and $\kappa$-balancedness, there lies Schmeidler $\kappa$-balancedness, where only the first step is taken: we take nets of 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 19 is replaced by $K_v^+$ in Definition 21, 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 24

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

$$\begin{aligned} \sup _{\begin{array}{c} (\lambda ^i)_{i\in I} \subseteq \mathbb {R}_+^{(\mathcal {A}')} \\ A(\lambda ^i) {\mathop {\longrightarrow }\limits ^{w}} \chi _N \\ c(\lambda ^i) \longrightarrow z \end{array}} z \le v(N) \qquad \quad \text {if and only if} \quad \qquad \sup _{\begin{array}{c} (\lambda ^j)_{j\in J} \subseteq \mathbb {R}_*^{(\mathcal {A}')} \\ A(\lambda ^j) {\mathop {\longrightarrow }\limits ^{w}} \chi _N \\ c(\lambda ^j) \longrightarrow z \\ \liminf \lambda ^j_N > -\infty \end{array}} z \le v(N) . \end{aligned}$$

where $A (\lambda ) = \sum _{S \in \mathcal {A}'} \lambda _S \chi _S$ and $c (\lambda ) = \sum _{S \in \mathcal {A}'} \lambda _S v(S)$ for any $\lambda \in \mathbb {R}_+^{\mathcal {A}'}$.

Proof

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

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

If $L > 0$, then there exists a $j_0 \in J$ such that $j \ge j_0$ implies $\lambda ^j_N \ge 0$. Consider the index set $I = \{j \in J: j \ge j_0 \}$ and the net $(\lambda ^i)_{i\in I} \subseteq \mathbb {R}_+^{(\mathcal {A}')}$, which satisfies $A(\lambda ^i) {\mathop {\longrightarrow }\limits ^{w}} \chi _N$ and $c(\lambda ^j) \longrightarrow z > v(N)$.

Assume $L \le 0$. 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 $S \in \mathcal {A}'$ let

$$\begin{aligned} {\bar{\lambda }}^i_S = {\left\{ \begin{array}{ll} 0 &{}\text {if}\;S = N,\\ \lambda ^{j_i}_S / (1 - L) &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Then

$$\begin{aligned} A({\bar{\lambda }}^i)&= \sum _{\begin{array}{c} S\in \mathcal {A}' \\ S\ne N \end{array}} \frac{\lambda ^{j_i}_S \chi _S}{1 - L} = \frac{A(\lambda ^{j_i}) - \lambda ^{j_i}_N \chi _N}{1 - L} \\&{\mathop {\longrightarrow }\limits ^{w}} \frac{\chi _N - L \chi _N}{1 - L} = \chi _N , \end{aligned}$$

and

$$\begin{aligned} c({\bar{\lambda }}^i)&= \sum _{\begin{array}{c} S\in \mathcal {A}' \\ S\ne N \end{array}} \frac{\lambda ^{j_i}_S v(S)}{1 - L} = \frac{c(\lambda ^{j_i}) - \lambda ^{j_i}_N v(N)}{1 - L} \\&\longrightarrow \frac{z - L v(N)}{1 - L} > \frac{v(N) - L v(N)}{1 - L} = v(N) . \end{aligned}$$

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

5 The main result

The next result is our generalized Bondareva–Shapley Theorem.

Theorem 25

For any game $v \in \mathcal {G}^{\mathcal {A}'}$ it holds that ${\kappa \text {-core}}(v) \ne \emptyset$ if and only if the game is $\kappa$-balanced.

Proof

Put $X = \mathbb {R}^{(\mathcal {A}')}$, $P = \mathbb {R}_*^{(\mathcal {A}')}$, $Y = \Lambda (\mathcal {A})$, and $Y^* = {\textrm{ba}}^\kappa (\mathcal {A})$, moreover define the mapping $A:\mathbb {R}^{(\mathcal {A}')} \rightarrow \Lambda (\mathcal {A})$ by $A(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S \chi _S$, let $b = \chi _N$, and define the functional $c:\mathbb {R}^{(\mathcal {A}')} \rightarrow \mathbb {R}$ by $c(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S v(S)$. 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 $\lambda \in \mathbb {R}^{(\mathcal {A}')}$ with $\lambda _N = 1$ and $\lambda _S = 0$ for $S \ne N$.) Then the game is $\kappa$-balanced (Definition 21) if and only if the supervalue of $(\mathrm {P_{LP}})$ is finite and not greater than v(N) (Remark 23).

Moreover, observe that a set function $\mu \in {\textrm{ba}}^\kappa (\mathcal {A})$ is feasible for $(\mathrm {D_{LP}})$ if and only if $\mu (S) \ge v(S)$ for all $S \in \mathcal {A}'$ and $\mu (N) = v(N)$. Thus program $(\mathrm {D_{LP}})$ is equivalent to finding an element of ${\kappa \text {-core}}(v)$, 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 ${\kappa \text {-core}}$ (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)). $\square$

If the player set N is finite, then so is $\mathcal {A}' \subseteq \mathcal {P}(N)$, whence the cone $K_v$ is closed. Then by Lemma 24$\kappa$-balancedness reduces to Schmeidler balancedness, which is (ordinary) balancedness (Bondareva, 1963; Faigle, 1989; Shapley, 1967), and the ${\kappa \text {-core}}$ is the (ordinary) core in the finite case. We thus obtain the classical Bondareva–Shapley Theorem as a corollary of Theorem 25:

Corollary 26

(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 25, 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 & 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 ${\textrm{ba}}^\kappa (\mathcal {A}) = {\textrm{ca}}(\mathcal {A})$, the space of all bounded countably additive set functions on $\mathcal {A}$. Given a game $v \in \mathcal {G}^{\mathcal {A}'}$, its ${\kappa \text {-core}}$ is the $\sigma$-additive core ${{\mathrm{ca`-core}}}(v)$ given in Definition 12.

In the next example we demonstrate that there exists a Schmeidler $\aleph _0$-balanced non-negative game without restricted cooperation having its ${{\mathrm{ca`-core}}}$ empty.

Example 27

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

$$\begin{aligned} v(S) = {\left\{ \begin{array}{ll} 1 &{} \text {if}\;\#(N\,{\setminus }\,S) \le 1,\\ 0 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

We show that ${{\mathrm{ca`-core}}}(v) = \emptyset$. If $\mu \in {{\mathrm{ca`-core}}}(v)$, then $\mu \bigl (N {\setminus } \{n\}\bigr ) \ge v\bigl (N {\setminus } \{n\}\bigr ) = 1$ and $v(N) = 1$, whence $\mu \bigl (\{n\}\bigr ) \le 0$. So $0 \ge \sum _{n=1}^\infty \mu \bigl (\{n\}\bigr ) = \mu (N) = v(N) = 1$, a contradiction.

We now show that, if $(\chi _N,z) \in \,\overline{\!K_v^+}$, see (4), then $z \le 1 = v(N)$. 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

$$\begin{aligned} \bigl \{f \in \Lambda (\mathcal {A}) : \bigl |\delta '_i(f) - 1\bigr | < \varepsilon \text {for} i = 1,\, \ldots ,\, m \bigr \} , \end{aligned}$$

where $\delta '_i$ 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 $S_0, S_1, \ldots , S_n \in \mathcal {A}$, and some non-negative $\lambda _{S_0},\lambda _{S_1},\ldots ,\lambda _{S_n}$ such that $f = \lambda _{S_0} \chi _{S_0} + \lambda _{S_1} \chi _{S_1} + \cdots + \lambda _{S_n} \chi _{S_n}$ and

$$\begin{aligned} \left| \sum _{\begin{array}{c} j=0 \\ S_j\ni i \, \end{array}}^n \lambda _{S_j} - 1 \right| < \varepsilon \quad \text {for} \quad i = 1,\ldots , m \end{aligned}$$

(6)

with

$$\begin{aligned} \left| \sum _{j=0}^n \lambda _{S_j} v(S_j) - z \right| = \left| \sum _{\begin{array}{c} j=0 \\ \#(N{\setminus } S_j)\le 1\, \end{array}}^n \lambda _{S_j} - z \right| < \varepsilon . \end{aligned}$$

(7)

We can assume w.l.o.g. that $S_0 = N$, as well as $\#(N {\setminus } S_j) = 1$ for $j = 1,\ldots ,n_1$ and $\#(N {\setminus } S_j) > 1$ for $j = n_1 + 1,\ldots ,n$, where $n_1 \le n$.

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)

$$\begin{aligned} \sum _{\begin{array}{c} j = 0 \\ \#(N{\setminus } S_j) \le 1 \, \end{array}}^n \lambda _{S_j} = \sum _{j=0}^{n_1} \lambda _{S_j} \le \sum _{\begin{array}{c} j = 0 \\ S_j \ni i \, \end{array}}^n \lambda _{S_j} < 1 + \varepsilon , \end{aligned}$$

whence $z < 1 + 2\varepsilon$ by (7).

In the other case we have $m \le n_1$ and, because the sets $S_0,S_1,\ldots ,S_n$ are pairwise distinct, for $i = 1,\ldots ,m$ we can assume w.l.o.g. that $S_i = N {\setminus } \{i\}$. By (6)

$$\begin{aligned} \sum _{j=0}^{n_1} \lambda _{S_j} - \lambda _{S_i} = \sum _{\begin{array}{c} j=0 \\ j\ne i \end{array}}^{n_1} \lambda _{S_j} \le \sum _{\begin{array}{c} j=0 \\ S_j\ni i \, \end{array}}^n \lambda _{S_j} < 1 + \varepsilon \quad \text {for} \quad i = 1,\ldots , m . \end{aligned}$$

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

$$\begin{aligned} \sum _{\begin{array}{c} j=0 \\ \#(N\setminus S_j)\le 1 \, \end{array}}^n \lambda _{S_j} = \sum _{j=0}^{n_1} \lambda _{S_j} < \frac{ m }{ m - 1} (1 + \varepsilon ) . \end{aligned}$$

Taking (7) into account, we obtain

$$\begin{aligned} z < \frac{ m }{ m - 1} (1 + \varepsilon ) + \varepsilon . \end{aligned}$$

(8)

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

Remark 28

Consider the game v from Example 27. Since ${{\mathrm{ca`-core}}}(v) = \emptyset$, the game is not $\aleph _0$-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

$$\begin{aligned} \lambda ^i_S = {\left\{ \begin{array}{ll} -(i - 2)&{} \text {if}\;S = N,\\ 1 &{} \text {if}\;S = N{\setminus } \{n\}\,\,{\textit{for}}\,\,n = 1,\ldots ,i,\\ 0 &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Then $\sum _{S\in \mathcal {A}} \lambda ^i_S \chi _S = 2\chi _N - \chi _{\{1,\ldots ,i\}} {\mathop {\longrightarrow }\limits ^{w}} \chi _N$, where the weak convergence in the space $\Lambda (\mathcal {A})$ is with respect to ${\textrm{ca}}(\mathcal {A})$, 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 24 we would get a contradiction with Example 27.

Example 27 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 ${{\mathrm{ca`-core}}}$ is empty even for non-negative games without restricted cooperation.

Remark 29

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:

$$\begin{aligned} \sup \left\{ \sum _{S\in \mathcal {A}'} \lambda _S v(S) : \sum _{S\in \mathcal {A}'} \lambda _S \chi _S = \chi _N, \, \lambda \in \mathbb {R}_+^{[\mathcal {A}']} \right\} \le v(N) , \end{aligned}$$

(9)

where $\mathbb {R}^{[\mathcal {A}']} = \bigl \{\lambda \in \mathbb {R}^{\mathcal {A}'}: \#\{ S \in \mathcal {A}': \lambda _S \ne 0 \} \le \aleph _0 \bigr \}$ and $\mathbb {R}_+^{[\mathcal {A}']} = \bigl \{\lambda \in \mathbb {R}^{[\mathcal {A}']}: \lambda _S \ge 0$ for all $S \in \mathcal {A}' \bigr \}$. Moreover, the convergence of the sum $\sum _{S\in \mathcal {A}'} \lambda _S \chi _S$ 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 ${\textrm{ca}}(\mathcal {A})$. If the sum $\sum _{S\in \mathcal {A}'} \lambda _S v(S)$ is convergent, but not absolutely convergent, then we put $\sum _{S\in \mathcal {A}'} \lambda _S v(S):=+\infty$.

Denoting $A(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S \chi _S$ and $c(\lambda ) = \sum _{S\in \mathcal {A}'} \lambda _S v(S)$, we can also consider the following generalization of (9). Let $z \le v(N)$ whenever there exists a net $(\lambda ^i)_{i\in I} \subseteq \mathbb {R}_+^{[\mathcal {A}']}$ such that $A (\lambda ^i) {\mathop {\longrightarrow }\limits ^{w}} \chi _N$ and $c (\lambda ^i) \longrightarrow z$ where z is finite. Then for each $i \in I$ there exists a sequence $(\lambda ^{in})_{n=1}^\infty \subseteq \mathbb {R}^{(\mathcal {A}')}_+$ such that $A(\lambda ^{in}) {\mathop {\longrightarrow }\limits ^{w}} A (\lambda ^i)$ and $c(\lambda ^{in}) \longrightarrow c(\lambda ^i)$. Consequently, there exists a net $(\lambda ^j)_{j\in J} \subseteq \mathbb {R}_+^{(\mathcal {A}')}$ such that $A(\lambda ^j) {\mathop {\longrightarrow }\limits ^{w}} \chi _N$ and $c(\lambda ^j) \longrightarrow z$. In other words, Schmeidler $\kappa$-balancedness covers such extensions of Schmeidler balancedness (Definition 17) like (9).

Moreover, in Example 27 we presented a non-negative Schmeidler $\kappa$-balanced game. Therefore, the presented game is balanced according to (9) too, but the ${{\mathrm{ca`-core}}}$ 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 ${\kappa \text {-core}}$ of a TU game with or without restricted cooperation is not empty if and only if the TU game is $\kappa$-balanced.

Perhaps the most interesting result of this paper is that in the proper notion of balancing weight system the weight of the grand coalition is sign unrestricted. More precisely, it is not surprising at all that in the definition of $\kappa$-balancedness we need nets (generalized sequences) since we have to take a weak closure of a set and to approach a point in the closure. However, the fact that in the elements of the net the weight of the grand coalition must be sign unrestricted, even more unbounded below, is interesting.

The reason why we need sign unrestricted weight of the grand coalition is simple. In the proof of Theorem 25 in the dual problem (which describes of the ${\kappa \text {-core}}$), when we set $\mu (N) = v (N)$ (equality), then it means the corresponding variable $\lambda _N$ in the primal problem must be sign unrestricted. If we do not set $\mu (N) = v (N)$, but only $\mu (N) \ge v (N)$, in the dual problem, then we get Schmeidler $\kappa$-balancedness. However, Example 27 shows that Schmeidler $\kappa$-balancedness does not imply the non-emptiness of the ${\kappa \text {-core}}$.

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.