The core and balancedness of TU games with
infinitely many players∗
David Bartl†and Miklós Pintér‡
January 4, 2017
Abstract
Transferable utility cooperative games with infinitely many players are considered. We generalize the notions of core and balancedness, and present a generalized Bondareva-Shapley Theorem for games
without and with restricted cooperation. Our generalized BondarevaShapley Theorem extends previous results by Bondareva (1963), Shapley (1967), Schmeidler (1967), Faigle (1989), and Kannai (1969, 1992)
among others.
Keywords: TU games with infinitely many players, BondarevaShapley Theorem, κ-core, κ-balancedness, κ-additive set function, duality theorem for infinite LPs, decision theory, ambiguity
1
Introduction
The core (Gillies, 1959) is perhaps the most important solution concept of
cooperative game theory. In the transferable utility setting (henceforth TU
games) the Bondareva-Shapley Theorem (Bondareva, 1963; Shapley, 1967)
provides a necessary and sufficient condition for the non-emptiness of the
core; it states that the core of a TU game is not empty if and only if the TU
∗
David Bartl acknowledges the support under the project Strengthening the International Extent of the Scientific Activities at the Faculty of Science, University of Ostrava,
reg. no. 0924/2016/ŠaS, provided by the Statutory City of Ostrava. Miklós Pintér acknowledges the support by National Research, Development and Innovation Office (NKFIH, K
101224, K 115538 and K 119930).
†
Department of Mathematics, University of Ostrava, [email protected].
‡
Corresponding author: Faculty of Business and Economics, University of Pécs and
MTA-BCE “Lendület” Strategic Interactions Research Group, [email protected].
1
game is balanced. The textbook proof of the Bondareva-Shapley Theorem
goes by the strong duality theorem of 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 TU games with finitely many players. It
is a question how we can generalize this result to the infinitely many players
case.
The finitely many players case is special in (at least) two counts: (1) it
can be handled by finite LPs, (2) since the power set of the players set is also
finite, it is natural to take the solution of a game from the set of additive set
functions (additive TU 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. σ-additivity). Naturally, a stronger notion does matter
only if there are infinitely many players. This latter approach is considered
in this paper.
Schmeidler (1967), Kannai (1969, 1992), and Pintér (2011) considered
TU 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 respectively that
the additive core of a non-negative TU game with infinitely and countably
infinitely many players is not empty if and only if the TU game is Schmeidler
balanced (6).
Kannai (1992) put the following two research questions: (1) When does
there exist a bounded σ-additive set function in the core? (2) When are all
elements in the core bounded σ-additive?
Kannai (1969) gave a necessary and sufficient condition for that the σadditive core of a non-negative TU game with infinitely many players is
not empty, that is, he answered Question (1). That result (the necessary
and sufficient condition) is, however, only slightly similar to the classical
balancedness condition (Propositions 29 and 30). Moreover it works only for
non-negative TU games without restricted cooperation. Schmeidler (1972)
and Einy et al. (1997) answered Question (2) respectively for exact and for
continuous convex TU games.
In this paper we generalize Kannai’s first research question; we put the
following question, where κ is an infinite cardinal number: When does there
exist a bounded κ-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 κ-additive set func2
tion (Definition 4), κ-core (Definition 8) and κ-balancedness (Definition 9).
Then we apply a strong duality theorem for infinite LPs by Anderson and
Nash (1987) (Proposition 3) and prove that the κ-core of a TU game is not
empty if and only if it is κ-balanced (Theorem 12).
The notion of κ-balancedness is a “double” extension of Schmeidler balancedness. First, we do not take each 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
κ-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
(Lemma 13).
We discuss in detail three special cases: the finite many player case (Subsection 4.1), the additive core case (ba-core case, Subsection 4.2) and the
σ-additive core case (ca-core case, Subsection 4.3). In each of the three cases
we consider games without and with restricted cooperation. In the finite
case Corollary 14 says that our κ-balancedness is an extension of the notion
of balancedness known for finite games without (Bondareva, 1963; Shapley,
1967) restricted cooperation as well as with restricted cooperation (Faigle,
1989). In other words, κ-balancedness and the “finite game balancedness”
are equivalent for finite games without and with restricted cooperation.
Regarding the additive case, when the core is the ba-core, that is, it
consists of bounded additive set functions, Table 1 and Table 2 summarizes
our results for games without and with restricted cooperation, respectively.
In words, Table 1 is as follows. Schmeidler (1967) covered only the nonnegative games case. For non-negative games our κ-balancedness condition and Schmeidler balancedness condition are equivalent (Corollary 21).
Moreover, we introduce a notion between Schmeidler balancedness (which is
weaker) and κ-balancedness (which is stronger): Schmeidler κ-balancedness
(Definition 16). Roughly speaking, Schmeidler κ-balancedness is an extension of Schmeidler balancedness where we take nets of balancing weight systems. This is clearly an extension of Schmeidler balancedness and weaker
than κ-balancedness. By the equivalence of Schmeidler balancedness and
κ-balancedness, however, we have that all three considered balancedness notions are equivalent for non-negative games.
Next, we consider the ba-core of bounded below games. At this point
it is worth mentioning that according to Kannai (1992) Schmeidler’s result
(Schmeidler, 1967) is a direct corollary of Fan (1956) (Theorem 13, p. 126).
However, as far as we can see, an extension of Schmeidler’s result to bounded
below TU games cannot be obtained by either Schmeidler (1967) or Fan
(1956). In other words, both Schmeidler (1967) and the application of Fan
(1956) heavily use that the considered TU game is non-negative. We extend
3
case
Balancedness Condition
κ = ℵ0
class of TU games
Schmeidler
Schmeidler
κ-balancedness
without restricted
balancedness
κ-balancedness
(Definition 9)
cooperation
(Schmeidler, 1967) (Definition 16)
√
√
√
non-negative
(v ≥ 0)
(Schmeidler, 1967) (Schmeidler (1967) (Theorem 12)
and Theorem 12)
√
√
√
bounded below
(Theorem 15)
(Theorems 12
(Theorem 12)
(v ≥ L)
and 15)
√
general
×
×
(Theorem 12)
(unbounded
(Example 18)
(Example 18)
below)
Table 1: The main results for κ = ℵ0 pertaining games without restricted
cooperation.
Schmeidler’s result to bounded below TU games, that is, we show that the
ba-core of a bounded below TU game without restricted cooperation is not
empty if and only if it is Schmeidler balanced (Theorem 15). Therefore, for
bounded below TU games without restricted cooperation all the considered
three balancedness conditions are equivalent.
Lastly, we consider the general case, when the TU game without restricted cooperation can be unbounded below. Example 18 shows that neither Schmeidler balancedness nor Schmeidler κ-balancedness can characterize
the non-emptiness of the ba-core in general; naturally, κ-balancedness can.
Therefore in general κ-balancedness is the only notion that works.
In the additive case of games with restricted cooperation Table 2 shows
that Schmeidler balancedness and Schmeidler κ-balancedness do not work
even for non-negative games, but the notion of κ-balancedness must be applied.
Table 3 summarizes our results for the ca-core case, that is, when the core
consists of bounded σ-additive set functions, for TU games without restricted
cooperation.
In words, Table 3 is as follows. Even for non-negative TU games without
restricted cooperation Example 26 says that Schmeidler κ-balancedness does
not reveal the non-emptiness of the ca-core. Put differently, as in the general
case of the ba-core, in the ca-core case only the κ-balancedness condition
4
case
Balancedness Condition
κ = ℵ0
class of TU games
Schmeidler
Schmeidler
κ-balancedness
with restricted
balancedness
κ-balancedness
(Definition 9)
cooperation
(Schmeidler, 1967) (Definition 16)
√
non-negative
×
×
(v ≥ 0)
(Example 23)
(Example 23)
(Theorem 12)
Table 2: The main results for κ = ℵ0 pertaining games with restricted cooperation.
case
κ = ℵ1
class of TU games
non-negative
(v ≥ 0)
Balancedness Condition
Schmeidler
κ-balancedness
(Definition 16)
×
(Example 26)
κ-balancedness
(Definition 9)
√
(Theorem 12)
Table 3: The main results for κ = ℵ1 pertaining games without restricted
cooperation.
works. Moreover, it is naturally tempting to generalize the balancing weight
systems so that those can have countable support (countable many non-zero
coefficients) as in (15). However, by Remark 28 these kinds of extensions
are covered by Schmeidler κ-balancedness, therefore, they cannot reveal the
non-emptiness of the ca-core either.
Back to Theorem 12, 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 results is the application of
a separating hyperplane theorem. In other words, both the strong duality
theorem and Shapley’s approach are based on the same separating hyperplane
theorem, practically both are direct corollaries of that theorem. Here we use
a strong duality theorem for infinite LPs (Anderson and Nash, 1987); this
theorem is also a direct corollary of the same separating hyperplane theorem.
Regarding applications, the non-emptiness of the core is also relevant from
the viewpoint of decision theory, where a non-additive belief is a monotone
5
TU game, and the so called priors (possible distributions, beliefs) are the
elements of the core of the corresponding TU game (Schmeidler, 1989; Gilboa
and Schmeidler, 1989; Ghirardato and Marinacci, 2002). Recently CerreiaVioglio et al. (2011) showed that under quite general assumptions a nonadditive belief expresses ambiguity aversion if and only if its core is not empty.
In other words, a necessary and sufficient condition for the non-emptiness of
the core is a necessary and sufficient condition of that a non-additive belief
expresses ambiguity aversion.
The set-up of the paper is as follows. In the next section we introduce
the main mathematical notions and results on LPs used in the paper. In
Section 3 we introduce the κ-additive set functions and discuss some related
notions and results. In Section 4 we give the game theory results. The last
section briefly concludes.
2
Duality theorem
In this section we present the duality theorem for infinite linear programs
(LPs) we will use later.
Let X and Y be real vector spaces, the algebraic dual of X, that is, the
space of all linear functionals on X, is denoted by X 0 ; similarly Y 0 denotes
the algebraic dual of Y . Moreover, Y ∗ ⊆ Y 0 is a linear subspace of Y 0 such
that (Y, Y ∗ ) is a dual pair of spaces, that is, if f ∈ Y is non-zero, then there
exists a y ∈ Y ∗ such that y(f ) 6= 0. For any linear mapping A : X → Y its
adjoint mapping is A0 : Y 0 → X 0 with A0 (y) (x) = y A(x) for all x ∈ X
and y ∈ Y 0 . Moreover, a subset P ⊆ X of the vector space X is a convex
cone if αx + βy ∈ P for all x, y ∈ P and all non-negative α, β ∈ R. For any
two functionals f, g : X → R we write f ≥P g if f (x) ≥ g(x) for all x ∈ P .
Now, given a linear mapping A : X → Y , a point b ∈ Y and a linear functional c : X → R, let us consider the following infinite LP-pair (cf. Anderson
and Nash (1987), Section 3.3):
(PLP )
c(x) → sup
(DLP )
y(b) → inf
s.t. A0 (y) ≥P c
y ∈ Y∗
s.t. A(x) = b
x ∈ P
(1)
where P ⊆ X is a convex cone and Y ∗ is a subspace of Y 0 such that (Y, Y ∗ )
is a dual pair of spaces.
Definition 1. We say that program
(DLP ) is consistent if there exists a
∗
0
functional y ∈ Y such that A (y) (x) ≥ c(x) for all x ∈ P . The value of a
consistent program (DLP ) is inf y(b) : A0 (y) ≥P c, y ∈ Y ∗ .
6
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 ⊆ Y is a weak neighborhood of a point f0 ∈ Y T
if there
exist a
n ∗
natural number n and functionals y1 , . . . , yn ∈ Y such that j=1 f ∈ Y :
yj (f ) − yj (f0 ) < 1 ⊆ U .
Definition 2. Put D =
A(x), c(x) : x ∈ P . We say that program
(PLP ) is superconsistent if there exists a z ∈ R such that (b, z) ∈ D, where
D is the closure of D in the product topology
of Y ×R. The supervalue of a
superconsistent program (PLP ) is sup z : (b, z) ∈ D .
We recall that a pair (I, ≤) is right-directed if I is a preordered set and
for any i, j ∈ I there exists a k ∈ I such that i ≤ k and j ≤ k. A net
(generalized sequence) of X is (xi )i∈I where (I, ≤) is a right-directed pair
and xi ∈ X for all i ∈ I.
Notice that program (PLP ) is superconsistent if there exists a net (xi )i∈I
w
of P such that A(xi ) −→ b, that is, A(xi ) converges to b in the weak topology,
and c(xi ) is bounded. Furthermore, a number z ∗ is the supervalue of a
superconsistent program (PLP ) if it is the least upper bound of all numbers
w
z such that there exists a net (xi )i∈I of P such that A(xi ) −→ b and c(xi ) −→
z.
Proposition 3. Consider the programs in (1). Program (PLP ) is superconsistent and z ∗ is its finite supervalue if and only if program (DLP ) 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 κ-structures
Throughout this section κ is an infinite cardinal number. Let N be a nonemptySset and let A ⊆ P(N ) be a field of sets, that is, if S1 , . . . , Sn ∈ A,
then nj=1 Sj ∈ A, and N ∈ A with N \ S ∈ A for any S ∈ A. The pair
(N, A) is a premeasurable space.
Let (Si )i∈I be a net of sets of A. Then (Si )i∈I is a κ-sequence if #I < κ,
where #I is the cardinality of the set I. In addition, we say that (Si )i∈I
7
is monotone decreasing or monotone increasing if i ≤ j implies Si ⊇ Sj or
Si ⊆ Sj , respectively, for any i, j ∈ I.
Let µ : A → R be a set function. We say that µ is upper κ-continuous or
lower κ-continuous at S ∈ A T
if for any monotone
S decreasing or increasing
κ-sequence (Si )i∈I of A with i∈I Si = S or i∈I Si = S, respectively, it
holds that limi∈I µ(Si ) = µ(S). The set function µ is κ-continuous if it is
both upper and lower κ-continuous at every set S ∈ A.
Definition 4. Given a set function µ : A → R, we say that µ is κ-additive
if it is additive and κ-continuous.
The following proposition is easy to see.
Proposition 5. If µ : A → R is additive, then µ is
• upper κ-continuous if and only if it is lower κ-continuous;
• lower κ-continuous if andTonly if for any monotone decreasing κ-sequence (Si )i∈I of A with i∈I Si = ∅ it holds that limi∈I µ(Si ) = 0;
• ℵ0 -continuous or ℵ0 -additive if and only if it is additive;
• ℵ1 -continuous or ℵ1 -additive if and only if it is σ-additive.
Example 6. The Lebesgue measure on B([0, 1]), the Borel σ-field of [0, 1],
is not κ-additive for any κ > c, where c denotes the cardinality of the real
numbers, but it is κ-additive for κ ≤ ℵ1 .
Given the premeasurable space (N, A), the space R(A) consists of all functions λ : A → R with a finite support, that is,
R(A) = λ ∈ RA : #{ S ∈ A : λS 6= 0 } < ∞ .
Denoting λ(S) and the characteristic function of a set S ∈ A by λS and χS ,
respectively, let Λ(A) = { λS1 χS1 + · · · + λSn χSn : n ∈ N, λS1 , . . . , λSn ∈ R,
S1 , . . . , Sn ∈ A } be the space of all simple measurable functions on (N, A),
that is,


X

Λ(A) =
λS χS : λ ∈ R(A) .


S∈A
λS 6=0
Henceforth, in the summations like
explicitly to simplify the notation.
P
8
S∈A
λS χS we do not indicate λS 6= 0
We introduce a norm on Λ(A) as follows. For a simple function f =
λS1 χS1 + · · · + λSn χSn ∈ Λ(A) let
kf k = supf (x) .
x∈N
Then the topological dual (Λ(A))? of the vector space Λ(A), that is, the
space of all continuous linear functionals on Λ(A), is isometrically isomorphic
to ba(A), the space of all bounded additive set functions on A. For simplicity,
we shall identify the space (Λ(A))? with ba(A). Indeed, a set function µ ∈
ba(A) induces a continuous linear functional µ0 ∈ (Λ(A))? on Λ(A) as follows:
µ0 (f ) = λS1 µ(S1 ) + · · · + λSn µ(Sn )
(2)
for any f = λS1 χS1+ · · · + λSn χSn ∈ Λ(A).
Let baκ (A) = µ ∈ ba(A) : µ is κ-additive , the space of all bounded
κ-additive set functions on A. Note that baκ (A) is a linear subspace of
ba(A).
Lemma 7. It holds that Λ(A), baκ (A) is a dual pair of spaces.
Proof. Let f ∈ Λ(A) be non-zero, whence there is an x ∈ N such that
f (x) 6= 0. Then δx , the Dirac measure concentrated at point x on A, is a
κ-additive set function, and δx0 (f ) = f (x) 6= 0.
4
The main results
Let κ be an arbitrary infinite cardinal number as in the previous section.
First, we introduce the TU games. Let N be a non-empty set of players and
let A0 ⊆ P(N ) be a collection of sets such that ∅, N ∈ A0 , and let A denote
the field hull of A0 , that is, the smallest field of sets that contains A0 . Then
a TU game (henceforth a game) on coalitions A0 is a set function v : A0 → R
0
such that v(∅) = 0. We denote the class of games on coalitions A0 by G A . If
0
A0 = A, that is, A0 is a field of sets, then v ∈ G A is a game without restricted
0
cooperation. Otherwise, if A0 is not a field, then v ∈ G A is a game with
restricted cooperation.
Recalling baκ (A) is the space of all bounded κ-additive set functions on
the field A, we introduce the notion of κ-core of a game.
0
Definition 8. For a game v ∈ G A its κ-core is defined as follows:
κ-core(v) = µ ∈ baκ (A) : µ(N ) = v(N ) and µ(S) ≥ v(S) for all S ∈ A0 .
9
In other words, the κ-core consists of κ-additive set functions defined on
the field hull A of the feasible coalitions A0 that meet the efficiency (µ(N ) =
v(N )) and the coalitional rationality (µ(S) ≥ v(S) for all S ∈ A0 ) conditions.
We noted in the previous section that Y ∗ = baκ (A) is a linear subspace of
Y ? = ba(A), which can be identified with the topological dual of the normed
linear space Y = Λ(A).
We consider the weak topology on Λ(A) by the dual
pair Λ(A), baκ (A) (see Lemma 7).
0
(A0 )
Take a game v ∈ G A and define
a
mapping
A
:
R
→
P
PΛ(A) and a
(A0 )
functional c : R
→ R by A(λ) = S∈A0 λS χ
c(λ) = S∈A0 λS v(S),
S and A
0
(A0 )
(A0 )
respectively,
= λ ∈ R : #{ S ∈ A0 : λS 6=
for all λ ∈ R , where R
0 } < ∞ . Considering the convex cone
n
o
(A0 )
Kv =
A(λ) ,
c(λ)
: λ ∈ R∗
)
(
(3)
X
X
(A0 )
,
λS v(S) : λ ∈ R∗
=
λS χS ,
S∈A0
S∈A0
0
(A0 )
where R∗ = λ ∈ R(A ) : λS ≥ 0 for all S ∈ A0 \ {N } , we introduce the
notion of κ-balancedness of the game v.
0
Definition 9. A game v ∈ G A is κ-balanced if
z ≤ v(N )
for all z ∈ R such that (χN , z) ∈ Kv , where the closure is understood in
the product of the weak topology with respect to baκ (A) and the Euclidean
topology on the space Λ(A) × R.
Remark 10. Notice that the notion of κ-balancedness is very closely related to
the notion of supervalue introduced in Definition 2. The cone Kv is precisely
the set D of Definition 2. Then the game is κ-balanced if and only if the
supervalue of the related primal problem (PLP ) is not greater than v(N ).
The following proposition presents an equivalent formulation of Definition 9.
0
Proposition 11. A game v ∈ G A is κ-balanced if and only if for any ε > 0
there exist a δ > 0, a natural number n and µ1 , . . . , µn ∈ baκ (A) such that
!
X
0)
∀λ ∈ R(A
:
λS µi (S) − µi (N ) < δ for i = 1, . . . , n
∗
S∈A0
(4)
X
=⇒
λS v(S) < v(N ) + ε .
S∈A0
10
Proof. Recalling that a point (χN , z) ∈ Kv if and only if each neighborhood
of (χN , z) intersects the cone Kv , we prove both parts indirectly.
To prove the “if” part, assume the game v is not κ-balanced. Then there
exists a z > v(N ) such that (χN , z) ∈ Kv . Consider ε = 12 z − v(N ) .
κ
Choosing any δ > 0, any natural
n and
any µ1 , . . . , µn ∈ ba
(A), the
0 number
0
neighborhood
f ∈ Λ(A) : µi (f ) − µi (χN ) < δ for i = 1, . . . , n × z − ε,
z + ε intersects the cone Kv . Hence (4) does not hold either.
To prove the “only if” part, assume that there exists an ε > 0 such that
for any δ > 0, any natural number n and any µ1 , . . . , µn ∈ baκ (A) formula
P
(A0 )
(4) does
not
hold,
that
is,
there
exists
a
λ
∈
R
such
that
∗
S∈A0 λS µi (S)−
P
µi (N ) < δ for i = 1, . . . , n and S∈A0 λS v(S) ≥ v(N ) + ε. Letting z =
v(N ) + ε, we show that (χN , z) ∈ Kv . To that end, choose a δ > 0 and
a natural number n with µ1 , . . . , µn ∈ baκ (A). By hypothesis there exists
P
(A0 )
aPλ ∈ R∗ such that S∈A0 λS µi (S) − µi (N ) < δ for i = 1, . . . , n and
v(N ) + ε.
S∈A0 λS v(S)≥P
Let α = ε
S∈A0 λS v(S) − v(N ) . It is easy to see 0 < α ≤ 1. Let
λ̄N = αλN + (1 − α) and let λ̄S = αλS for all other S ∈ A0 \ {N }. Then
X
X
= α
λ
µ
(S)
+
(1
−
α)µ
(N
)
−
µ
(N
)
λ̄
µ
(S)
−
µ
(N
)
S
i
i
i
S
i
i
0
S∈A0
S∈A
X
≤
λS µi (S) − µi (N ) < δ
S∈A0
for i = 1, . . . , n and
X
X
λS v(S) + (1 − α)v(N ) = v(N ) + ε = z .
λ̄S v(S) = α
S∈A0
S∈A0
It follows that (χN , z) ∈ Kv , and the game is not κ-balanced.
The next result is our generalized Bondareva-Shapley Theorem.
0
Theorem 12. For any game v ∈ G A it holds κ-core(v) 6= ∅ if and only if
the game is κ-balanced.
0
(A0 )
Proof. Put X = R(A ) , P = R∗ , Y = Λ(A), andPY ∗ = baκ (A), moreover
0
define the mapping A : R(A ) → Λ(A) by A(λ) = S∈A0 λS χS , let b = χN ,
P
0
and define the functional c : R(A ) → R by c(λ) =
S∈A0 λS v(S). Now,
consider the programs (PLP ) and (DLP ) of (1).
Notice that program (PLP ) is superconsistent
and its supervalue is at
0
least v(N ). (Consider that A(λ), c(λ) ∈ Kv ⊆ Kv for λ ∈ R(A ) with
λN = 1 and λS = 0 for S 6= N .) Then the game is κ-balanced (Definition 9)
11
if and only if the supervalue of (PLP ) is finite and not greater than v(N )
(Remark 10).
Moreover, observe that a set function µ ∈ baκ (A) is feasible for (DLP ) if
and only if µ(S) ≥ v(S) for all S ∈ A0 and µ(N ) = v(N ). Thus program
(DLP ) is equivalent to finding an element of κ-core(v), and its value is v(N )
if it is consistent, and its value is +∞ otherwise.
Therefore by Proposition 3 the game has a non-empty κ-core (program
(DLP ) is consistent) if and only if it is κ-balanced (program (PLP )’s supervalue
is not greater than v(N )).
We will use the following lemma in the next subsections.
Recall that
P
P
(A0 )
(A0 )
A(λ) = S∈A0 λS χS , c(λ) = S∈A0 λS v(S), R∗ = λ ∈ R
: λS ≥ 0
0)
0
(A
for all S ∈ A0 \ {N } , and put R+ = λ ∈ R(A ) : λS ≥ 0 for all S ∈ A0 .
0
Lemma 13. For a game v ∈ G A it holds
z ≤ v(N )
sup
if and only if
(A0 )
z ≤ v(N ) .
sup
(A0 )
(λj )j∈J ⊆R∗
w
A(λj )−→χN
j
c(λ )−→z
lim inf λjN >−∞
(λi )i∈I ⊆R+
w
A(λi )−→χN
c(λi )−→z
(A0 )
Proof. The “if” part is obvious. Given a net (λi )i∈I ⊆ R+ , consider the
(A0 )
same net (λj )j∈J = (λi )i∈I ⊆ R∗ . Notice that lim inf λjN ≥ 0.
We prove the “only if” part indirectly. Suppose the right-hand side does
(A0 )
not hold. Then there exists a net (λj )j∈J ⊆ R∗ such that lim inf λjN = L >
w
−∞ and A(λj ) −→ χN with c(λj ) −→ z > v(N ).
If L > 0, then there exists a j0 ∈ J such that j ≥ j0 implies λjN ≥ 0.
(A0 )
Consider the index set I = { j ∈ J : j ≥ j0 } and the net (λi )i∈I ⊆ R+ ,
w
which satisfies A(λi ) −→ χN and c(λj ) −→ z > v(N ).
Assume L ≤ 0. There exists a subnet (λji )i∈I of (λj )j∈J such that λjNi −→
L. Define the net (λ̄i )i∈I as follows: for any i ∈ I and for any S ∈ A0 let
(
0
if S = N ,
λ̄iS =
ji
λS /(1 − L) otherwise.
Then
A(λ̄i ) =
X λji χS
A(λji ) − λjNi χN
S
=
1−L
1−L
0
S∈A
S6=N
w
−→
χN − LχN
= χN ,
1−L
12
and
X λji v(S)
c(λji ) − λjNi v(N )
S
c(λ̄ ) =
=
1−L
1−L
0
i
S∈A
S6=N
z − Lv(N )
v(N ) − Lv(N )
>
= v(N ) .
1−L
1−L
It follows that the left-hand side does not hold in either case, which
concludes the proof.
−→
In the following subsections we consider some special cases already discussed in the literature.
4.1
The finite case
If the player set N is finite, then so is A0 ⊆ P(N ), whence the cone Kv is
closed. Consequently by Lemma 13 the definition of κ-balancedness (Defini0
tion 9) can be written as: a game v ∈ G A is κ-balanced if
X
X
A0
max
λS v(S) :
λS χS = χN , λ ∈ R+ ≤ v(N ) .
(5)
S∈A0
S∈A0
Notice that the κ-core is the classical core and (5) is the classical definition
of balancedness (Bondareva, 1963; Shapley, 1967; Faigle, 1989), that is, the
definition of κ-balancedness is a natural generalization of the balancedness
of finite games (without and with restricted cooperation). Hence we have
obtained the following corollary of Theorem 12.
Corollary 14. If N is finite, then the core is non-empty if and only if the
game is balanced.
4.2
The additive case
Allow the player set N to be infinite and let κ = ℵ0 , that is, κ is the smallest
κ
infinite cardinal.
Then ba (A) = ba(A) and the κ-core is the additive core
ba-core(v) = µ ∈ ba(A) : µ(N ) = v(N ) and µ(S) ≥ v(S) for all S ∈ A0
(Schmeidler, 1967; Kannai, 1969; Pintér, 2011). Schmeidler (1967) gave the
following balancedness condition for infinite games with additive core:
X
X
(A0 )
λS χS = χN , λ ∈ R+
≤ v(N ) .
(6)
sup
λS v(S) :
S∈A0
S∈A0
13
Henceforth we call (6) Schmeidler balancedness. Moreover, unless we mention
otherwise, we assume throughout this subsection that A0 = A ⊆ P(N ) is a
field of sets, that is, the game is without restricted cooperation.
Schmeidler (1967) proved that a non-negative game without restricted
cooperation is Schmeidler balanced if and only if its ba-core is not empty.
Next, we present another main result of this paper, which extends Schmeidler’s (1967) result to bounded below games without restricted cooperation.
Theorem 15. Let N be a player set and v ∈ G A be a game, where A ⊆ P(N )
is a field of sets. If the game is bounded below, that is,
∃L ∈ R ∀S ∈ A : v(S) ≥ L ,
then its additive core is non-empty if and only if it is Schmeidler balanced.
Before we present a proof of Theorem 15, we find it appropriate to recall
several notions and concepts.
We mentioned in Section 3 that the topological dual (Λ(A))? of Λ(A), the
space of simple measurable functions on (N, A), is isometrically isomorphic
to the space ba(A). The norm of a µ ∈ ba(A) is defined as follows:
kµk =
sup µ(S1 ) + · · · + µ(Sn ) .
(7)
n∈N
S1 ,...,Sn ∈A
S1 ∪···∪Sn =N
Si ∩Sj =∅, i6=j
Assume that a µ ∈ ba-core(v), where the game v is bounded below by L. Let
S1 , . . . , Sn ∈ A be pairwise disjoint and such that N = S1 ∪ · · · ∪ Sn . Then
n
n
n
X
X
X
µ(Sj ) =
µ(Sj ) −
µ(Sj )
j=1
j=1
µ(Sj )≥0
=µ
j=1
µ(Sj )<0
[
n
−µ
Sj
j=1
µ(Sj )≥0
=µ N\
[
n
Sj
j=1
µ(Sj )<0
n
\
[
n
(N \ Sj ) − µ
Sj
j=1
µ(Sj )≥0
= µ(N ) − µ
\
n
j=1
µ(Sj )<0
[
n
(N \ Sj ) − µ
Sj
j=1
µ(Sj )≥0
≤ µ(N ) − 2L = v(N ) − 2L .
14
j=1
µ(Sj )<0
It follows the additive
core
is
contained
in
the
closed
ball
B
=
µ ∈ ba(A) :
R
kµk ≤ v(N ) − 2L of radius R = v(N ) − 2L.
We endow the space ba(A) with the weak* topology with respect to Λ(A).
A set U ⊆ ba(A) is a weak* neighborhood of a µ0 ∈ ba(A) if there
a
T exist
natural number n and functions f1 , . . . , fn ∈ Λ(A) such that nj=1 µ ∈
ba(A) : µ0 (fj ) − µ00 (fj ) < 1 ⊆ U , where µ0 and µ00 is the continuous linear
functional induced by µ and µ0 , respectively, see (2). By Alaoglu’s Theorem
(see e.g. Aliprantis and Border (2006), Theorem 6.21, p. 235) the closed ball
BR is compact in the weak* topology. That is, if Gi ⊆Sba(A) are weakly*
open
Sn sets for i ∈ I, where I is an index set, such that i∈I Gi ⊇ BR , then
j=1 Gij ⊇ BR for some natural number n and some i1 , . . . , in ∈ I.
Let Fi ⊆ BR be weakly* closed sets for i ∈ I, where ITis an index set. We
say the collection {Fi }i∈I is a centered system of sets if nj=1 Fij 6= ∅ for any
natural number n and any iT
1 , . . . , in ∈ I. By considering the complements
(Gi = ba(A) \ Fi ) it follows i∈I Fi 6= ∅.
In the proof of Theorem 15 we consider the weakly* closed sets
FS = µ ∈ ba(A) : µ(N ) = v(N ) and µ(S) ≥ v(S) and kµk ≤ R
for S ∈ A. The main idea is to show that, if the game v is Schmeidler
balanced,
then the system {FS }S∈A is centered. Noticing that ba-core(v) =
T
S∈A FS 6= ∅, the proof will be done.
Proof of Theorem 15. Assume that the given game v ∈ G A is bounded below by L. We are to show that ba-core(v) 6= ∅ if and only if the game
v is Schmeidler balanced. The “only if” part is P
obvious. Assume that a
(A)
µ
∈
ba-core(v)
and
that
a
λ
∈
R
is
such
that
+
S∈A λS χS = χN . Then
P
P
S∈A λS v(S) ≤
S∈A λS µ(S) = µ(N ) = v(N ), so (6) holds. It remains to
prove the “if” part.
Tn Pick up any sets S0 , S1 , . . . , Sn ∈ A. Our purpose is to show that
j=0 FSj 6= ∅. We can assume w.l.o.g. that the sets S0 , . . . , Sn are distinct with S0 = ∅ and Sn = N , and that the collection {S0 , . . . , Sn } ⊆ A
is a field of sets.T(Roughly speaking, the more sets we pick up, the smaller
the intersection nj=0 FSj is. Having to show the intersection is non-empty
anyway, we can include the empty and the grand coalition among the sets.
Moreover, we can add further sets from A so that the collection {S0 , . . . , Sn }
becomes a finite field of sets.) We can also assume w.l.o.g. that S1 , . . . , Sn0
are all the atoms of the field, that is, they are all the minimal elements in
the collection {S1 , . . . , Sn }. Obviously, the atoms S1 , . . . , Sn0 are pairwise
0
disjoint, and it holds n = 2n − 1.
15
Now, the sets ∅ = S0 , S1 , . . . , Sn being fixed, we apply Schmeidler balancedness to the sets S1 , . . . , Sn :
∀λ1 , . . . , λn ≥ 0 : λ1 χS1 + · · · + λn χSn = χN
=⇒ λ1 v(S1 ) + · · · + λn v(Sn ) ≤ v(N ) .
(8)
It shall follow hence that the system of relations
µ(N ) = v(N ) ,
µ(S1 ) ≥ v(S1 ) ,
...............
µ(Sn ) ≥ v(Sn )
(9)
has a solution µ ∈ ba(A) such that kµk ≤ R. To see that, we apply the
Bondareva-Shapley Theorem for finite games (Corollary 14).
Consider a new finite game v 0 : P(N 0 ) → R with the set of the players
0
N = {1, . . . , n0 }. Define the game as follows. Recall first that the collection
{S0 , . . . , Sn } is a field of sets and that S1 , . . . , Sn0 areSall its atoms, which
are pairwise disjoint. Now, for an S 0 ⊆ N 0 , let S = i0 ∈S 0 Si0 , notice S ∈
{S0 , . . . , Sn }, and put v 0 (S 0 ) = v(S). The new finite game v 0 has been defined
thus.
Now, condition (8) equivalently says that the new game v 0 is balanced. By
the Bondareva-Shapley Theorem
14), its core
there
Pis non-empty:
P(Corollary
n0
0
0
0
0
0
0
0
a
≥
v
(S
) for
a
=
v
(N
)
and
exist a1 , . . . , an ∈ R such that i0 =1 i
i0 ∈S 0 i
0
0
any S ⊆ N .
The atoms S1 , . . . , Sn0 being non-empty sets, there exist elements xi0 ∈ Si0
for i0 = 1, . . . , n0 . Consider the measure
µ = a1 δx1 + · · · + an0 δxn0 ,
where δxi0 is the Dirac measure concentrated at xi0 . We have µ(N ) = µ(S1 ∪
· · · ∪ Sn0 ) = a1 + · · · + aS
1, . . . , n let Sj0 = { i0 ∈ N 0 :
n0 = v(N ). For any j = P
Si0 ⊆ Sj }. Then Sj = i0 ∈S 0 Si0 , and µ(Sj ) = i0 ∈S 0 ai0 ≥ v 0 (Sj0 ) = v(Sj ).
j
j
We have shown thus that the µ is a solution to the system of inequalities (9).
Finally, let us calculate the norm kµk of the solution, see (7). For a
T ∈ A, we observe that
n0
X
µ(T ) =
ai0 .
i0 =1
xi0 ∈T
Given pairwise disjoint sets T1 , . . . , Ts ∈ A such that N = T1 ∪ · · · ∪ Ts , and
16
recalling
Pn0
ai0 = v(N ), we have
0
X
X
s
s
s X
n0
X
X
n
µ(Tr ) =
ai0 ≤
|ai0 |
i0 =1
r=1
r=1
i0 =1
xi0 ∈Tr
0
=
n
X
r=1 i0 =1
xi0 ∈Tr
0
|ai0 | =
n
X
0
ai0 −
i0 =1
ai0 ≥0
i0 =1
n
X
ai 0
i0 =1
ai0 <0
0
= v(N ) − 2
n
X
ai0 ≤ v(N ) − 2v
i0 =1
[
n0
S i0
i0 =1
ai0 <0
ai0 <0
≤ v(N ) − 2L = R .
It follows that kµk ≤ R. To conclude, we have aTµ ∈ ba(A) such that it is a
solution to (9) and kµk ≤ R,Twhich means µ ∈ nj=1 FSj . Since FSj ⊆ F∅ for
j = 1, . . . , n, it holds µ ∈ nj=0 FSj . We have shown thus that the system
{FS }S∈A is centered.
As the closed R-ball BR is weakly* compact, we have
T
ba-core(v) = S∈A FS 6= ∅.
0
For any game v ∈ G A let
n
o
(A0 )
+
Kv =
A(λ) ,
c(λ)
: λ ∈ R+
(
)
X
X
(A0 )
=
λS χS ,
λS v(S) : λ ∈ R+
,
S∈A0
(10)
S∈A0
0
(A0 )
where R+ = λ ∈ R(A ) : λS ≥ 0 for all S ∈ A0 . It obviously holds
Kv+ ⊆ Kv , see (3).
0
Definition 16. We say that a game v ∈ G A is Schmeidler κ-balanced if z ≤
v(N ) for all z ∈ R such that (χN , z) ∈ Kv+ where the closure is understood in
the product of the weak topology with respect to baκ (A) and the Euclidean
topology on the space Λ(A) × R.
Remark 17. Notice that Schmeidler κ-balancedness is the same as κ-balancedness except that Kv is replaced by Kv+ in Definition 9. Moreover,
Schmeidler κ-balancedness and κ-balancedness are related by Lemma 13. It
says that a game v is Schmeidler κ-balanced if and only if it holds
z ≤ v(N ) .
sup
(A0 )
(λj )j∈J ⊆R∗
w
A(λj )−→χN
c(λj )−→z
lim inf λjN >−∞
17
The next example presents an unbounded below game without restricted
cooperation that is Schmeidler balanced, even Schmeidler κ-balanced, but its
ba-core is empty.
Example 18. Let the player set N = N and let A = { S ⊆ N : S is finite or
N \ S is finite }. Consider the game v ∈ G A defined as follows: for any S ∈ A
let

1
if S = {1},



 1+ 1
if S = {1, n} for n = 2, 3, . . . ,
v(S) =
P n 1
− n∈T n if S = N \ T for a finite T ∈ A,



0
otherwise.
First, we show that ba-core(v) = ∅.Suppose for
contradiction that there
exists a µ ∈ ba-core(v). Then µ {1} ≥ v {1} = 1 and µ N \ {1} ≥
v N \{1} = −1. Since µ {1} + µ N \ {1} = µ(N ) = v(N ) = 0, we have
µ {1} = 1. As µ {1, n} ≥ v {1, n} = 1 + 1/n, it follows
µ {n} ≥ 1/n
for all n = 2, 3, . . . Summing up, it follows µ {1, . . . , n} ≥ ln(n + 1), so
µ∈
/ ba(A) because µ is not bounded. It follows ba-core(v) = ∅.
Next, we show that v is Schmeidler κ-balanced (so it is Schmeidler balanced too). Choose a real number z such that (χN , z) ∈ Kv+ . We shall show
that z ≤ v(N ) + ε for all ε > 0.
Given an ε > 0, find a natural number m such that 1/m < ε. Define the
sequence (ai )∞
i = 0 if
i=1 as follows: let ai = 1/i for i = 1, . . . , m, and let aP
i > m. Then define the function µ ∈ ba(A)
as
follows:
let
µ(S)
=
i∈S ai
P
if S ∈ A is finite, and let µ(S) = 1/m − i∈T ai if S ∈ A is infinite, which
means T = N \ S is finite. Observe that µ(S) ≥ v(S) for all S ∈ A, but
µ∈
/ ba-core(v) because µ(N ) = 1/m > 0 = v(N ).
Recall that a µ ∈ ba(A) induces the linear functional µ0 on Λ(A), see (2).
Now, it holds for any µ ∈ ba(A) and for any t ∈ R that: if µ0 (f ) + tα ≥ 0 for
all (f, α) ∈ Kv+ , then µ0 (χN ) + tz ≥ 0. Otherwise, the point (χN , z) would
be strictly separated from the closed cone Kv+ , contrary to the hypothesis
defined µ and t = −1, and
that (χN , z) ∈ Kv+ . Hence, considering the
above
+
considering the pairs (f, α) = χS , v(S) ∈ Kv for S ∈ A, we conclude that
ε > 1/m = µ(N ) = µ0 (χN ) ≥ z. Since the ε > 0 can be arbitrary small, it
follows that z ≤ 0 = v(N ). That is, the game v is Schmeidler κ-balanced.
Remark 19. Consider the game v of Example 18. Although both Schmeidler
balancedness and Schmeidler κ-balancedness fail to reveal the ba-core of the
game is empty, we can use Theorem 12 and Definition 9 to disclose that.
(A)
i
Take the sequence (λi )∞
i=1 , with λ ∈ R∗ , defined as follows: for any
18
i ∈ N and for any S ∈ A let


−(i − 1)

 i
i
λS =

i



0
if S = N ,
if S = {1, i} for i = 2, 3, . . . ,
if S = N \ {1},
otherwise.
P
w
Then S∈A λiS χS = χN + χ{i} −→ χN , where
P the weak convergence in the
space Λ(A) is with respect to ba(A), and S∈A λiS v(S) = 1 > 0 = v(N).
This shows that (χN , 1) ∈ Kv , therefore v is not κ-balanced. By Theorem 12,
the ba-core of the game is empty.
∞
Notice that the sequence (λiN )∞
i=1 = (1 − i)i=1 is unbounded below. Indeed, if (λiN )∞
i=1 were bounded below, then by Lemma 13 the game would be
Schmeidler κ-balanced, contrary to Example 18.
Remark 20. Example 18 shows that, if one wants to keep Schmeidler balancedness, then Theorem 15 for games without restricted cooperation cannot
be generalized further.
Combining the previous results, Theorem 12 and Lemma 13 with Theorem 15 for games without restricted cooperation, we obtain the following
corollary.
Corollary 21. For a below bounded game v ∈ G A , where A is a field of
sets, the notion of κ-balancedness is equivalent with Schmeidler balancedness.
Moreover, the game v is Schmeidler balanced if and only if
X
X
(A)
λS χS = χN , λ ∈ R∗
λS v(S) :
≤ v(N ) .
sup
S∈A
S∈A
Remark 22. Considering Schmeidler balancedness and applying Theorem 15,
the next result also follows. Let v ∈ G A be a below bounded game without
restricted cooperation. For an ε > 0, define the game vε ∈ G A as follows: let
vε (N ) = v(N ) + ε and vε (S) = v(S) for all S ∈ A \ {N }. If ba-core(vε ) 6= ∅
for all ε > 0, then ba-core(v) 6= ∅.
Until now, we have considered the additive core case for games without
restricted cooperation. The next example presents a Schmeidler balanced,
even Schmeidler κ-balanced, non-negative game with restricted cooperation,
but its ba-core is empty. That is, Theorem 15 does not hold for games with
restricted cooperation. Despite that, we show in Remark 25 how to apply
Theorem 15 to below bounded Schmeidler balanced games with restricted
cooperation.
19
0
Example
23.
Let
the
player
set
N
=
N
and
let
A
=
∅ ∪ {1, i} : i = 1, 2,
0
0
3, . . . ∪ N \ {1} ∪ N . Consider the game v ∈ G A defined as follows:

2
if S = {1},



 2 + 1 if S = {1, n} for n = 2, 3, . . . ,
n
v 0 (S) =

0
if S = N \ {1} or S = ∅,



1
if S = N .
Notice that the game v 0 is analogous to the game v defined in Example 18.
The field hull of A0 is A = { S ⊆ N : S is finite or N \ S is finite }. To show
that ba-core(v 0 ) = ∅ and that the game v 0 is Schmeidler κ-balanced (therefore
Schmeidler balanced too), it is enough to follow the arguments presented in
Example 18.
Remark 24. By following the arguments given in Remark 19, one can see
that the game v 0 of Example 23 is not κ-balanced for κ = ℵ0 (Definition 9),
hence ba-core(v 0 ) = ∅ by Theorem 12.
The following remark can also be used to show that in Example 23
ba-core(v 0 ) = ∅.
0
Remark 25. Consider a game v ∈ G A , where ∅, N ∈ A0 ⊆ P(N ), and let
A be the field hull of A0 . We are interested to know whether ba-core(v) 6= ∅.
We show how to apply Theorem 15 even in this restricted cooperation case.
Assume that the game v is Schmeidler balanced, see (6), and that the
game v is bounded below. (If the game is not Schmeidler balanced, then
ba-core(v) = ∅. If the game is not bounded below, then Theorem 15 cannot
be applied even if the game is without restricted cooperation, that is, if
A0 = A is a field of sets, see Example 18, but we have A0 ⊆ A now; we have
to apply Theorem 12.) Then the following result is easily obtained from
Theorem 15.
If the game v with restricted cooperation is Schmeidler balanced and
bounded below, then ba-core(v) 6= ∅ if and only if there exists a finite constant L̃ ∈ R, which can be negative, such that
X
X
X
(A)
sup
λS v(S) +
λS L̃ :
λS χS = χN , λ ∈ R+
≤ v(N ) . (11)
S∈A0
S∈A\A0
S∈A
To see the “only if” part, assume that there exists a µ ∈ ba-core(v). Since
µ is a bounded function, there exists a finite L̃ ∈ R such that µ(S) ≥ L̃ for
all S ∈ A. Define a new game ṽ ∈ G A by ṽ(S) = v(S) for S ∈ A0 and
by ṽ(S) = L̃ for S ∈ A \ A0 . Then µ ∈ ba-core(ṽ), whence the game ṽ is
Schmeidler balanced, which is (11).
20
To see the “if” part, let (11) hold for some finite constant L̃ ∈ R. Consider
the game ṽ ∈ G A defined above. Then the game ṽ is Schmeidler balanced
by (11) and is bounded below too. By Theorem 15 and obviously, we have
∅=
6 ba-core(ṽ) ⊆ ba-core(v).
Regarding the game v 0 of Example 23, there is no finite constant L̃ ∈ R
to satisfy (11) for v = v 0 , hence ba-core(v 0 ) = ∅.
4.3
The σ-additive case
In this subsection let κ = ℵ1 . Then baκ (A) = ca(A), the space of all bounded
A0
countably additive functions on
A. Given a game v ∈ G , its κ-core is the
σ-additive core
ca-core(v) = µ ∈ ca(A) : µ(N ) = v(N ) and µ(S) ≥ v(S)
0
for all S ∈ A .
In Example 18 we presented a Schmeidler κ-balanced unbounded below
game without restricted cooperation having an empty ba-core for κ = ℵ0 . By
the next example we demonstrate that there exists a Schmeidler κ-balanced
non-negative game without restricted cooperation having an empty ca-core
for κ = ℵ1 .
Example 26. Let the player set N = N, the system of coalitions A = P(N),
and the game v be defined as follows: for any S ∈ A let
(
1 if #(N \ S) ≤ 1,
v(S) =
0 otherwise.
We show that ca-core(v) = ∅. If µ ∈ ca-core(v),
then
µ
N
\
{n}
P∞ ≥ v N \
{n} = 1 and v(N ) = 1, whence µ {n} ≤ 0. So 0 ≥
n=1 µ {n} =
µ(N ) = v(N ) = 1, a contradiction.
Now we show that, if (χN , z) ∈ Kv+ , see (10), where the closure of the cone
is understood in the product of the weak topology with respect to ca(A) and
the Euclidean topology on the space Λ(A) × R, then z ≤ 1 = v(N ). We have
(χN , z) ∈ Kv+ if and only if each neighborhood of the point (χN , z) intersects
the cone Kv+ . In particular, if (χN , z) ∈ Kv+ , then for any natural number m
and for any ε > 0 there exists a point (f, t) ∈ Kv+ such that f belongs to the
weak neighborhood
f ∈ Λ(A) : δi0 (f ) − 1 < ε for i = 1, . . . , m ,
where δi0 is the continuous linear functional induced by the Dirac
measure δi
concentrated
at
i,
see
(2),
and
t
belongs
to
the
neighborhood
t ∈ R : |t −
z| < ε . Hence, we have a natural number n, some distinct sets S0 , S1 , . . . ,
21
Sn ∈ A, and some non-negative λS0 , λS1 , . . . , λSn such that f = λS0 χS0 +
λS1 χS1 + · · · + λSn χSn and
n
X
λSj − 1 < ε
for i = 1, . . . , m
(12)
X
X
n
n
< ε.
λ
v(S
)
−
z
=
λ
−
z
S
j
S
j
j
(13)
j=0
Sj 3i
with
j=0
j=0
#(N \Sj )≤1
We can assume w.l.o.g. that S0 = N , and #(N \ Sj ) = 1 for j = 1, . . . , n1
and #(N \ Sj ) > 1 for j = n1 + 1, . . . , n, where n1 ≤ n.
T 1
Everything is clear if there exists an i ∈ {1, . . . , m} such that i ∈ nj=1
Sj .
Then by (12)
n1
n
n
X
X
X
λSj =
λSj ≤
λSj < 1 + ε ,
j=0
#(N \Sj )≤1
j=0
j=0
Sj 3i
whence z < 1 + 2ε by (13).
In the other case we have m ≤ n1 and, because the sets S0 , S1 , . . . , Sn are
pairwise distinct, for i = 1, . . . , m we can assume w.l.o.g. that Si = N \ {i}.
By (12)
n1
X
j=0
λSj − λSi =
n1
X
λSj ≤
j=0
j6=i
n
X
λSj < 1 + ε
for i = 1, . . . , m .
j=0
Sj 3i
Pn1
Pm
Pn1
λ
<
m+mε,
whence
m
Summing
up,
we
get
m
λ
−
S
S
j
i
j=0
i=1
j=0 λSj −
Pn1
λ
<
m
+
mε.
It
then
follows
j=0 Si
n
X
λSj =
j=0
#(N \Sj )≤1
n1
X
λSj <
j=0
m
(1 + ε) .
m−1
Taking (13) into account, we obtain
z<
m
(1 + ε) + ε .
m−1
(14)
Since 1 + 2ε < (1 + ε)m/(m − 1) + ε, inequality (14) holds in both cases.
By that m ≥ 2 and ε > 0 can be arbitrary, we conclude that z ≤ 1.
22
Remark 27. Consider the game v of Example 26. Since ca-core(v) = ∅, the
game is not κ-balanced. To see that, consider the sequence (λi )∞
i=1 , with
(A)
i
λ ∈ R∗ , defined as follows: for any i ∈ N and for any S ∈ A let


−(i − 2) if S = N ,
i
λS =
1
if S = N \ {n} for n = 1, . . . , i,


0
otherwise.
P
w
Then S∈A λiS χS = 2χN − χ{1,...,i} −→ χN , where
weak convergence in
P the
i
the space Λ(A) is with respect to ca(A), and S∈A λS v(S) = 2 > 1 = v(N).
∞
Notice again that the sequence (λiN )∞
i=1 = (2−i)i=1 is unbounded below. If
(λiN )∞
i=1 were bounded below, then by Lemma 13 we would get a contradiction
with Example 26.
(A)
Example 26 demonstrates that it is not sufficient to use R+ and Kv+ in
the definition of κ-balancedness, that is, Schmeidler κ-balancedness is unable
to reveal that the ca-core is empty even for non-negative games without
restricted cooperation. The net (λiN )i∈I must be let unbounded below.
Remark 28. Reconsidering Schmeidler balancedness and Theorem 15 for the
additive case, it is somehow tempting to ask whether the following “σ-extension” of condition (6) could lead to a similar result in the σ-additive case
too:
X
X
[A0 ]
sup
λS v(S) :
λS χS = χN , λ ∈ R+
≤ v(N ) ,
(15)
S∈A0
S∈A0
[A0 ]
[A0 ]
A0
0
where
R
=
λ
∈
R
:
#{
S
∈
A
:
λ
and
R
=
S 6= 0 } ≤ ℵ0
+
0
λ ∈PR[A ] : λS ≥ 0 for all S ∈ A0 . Moreover, the convergence of the
sum S∈A0 λS χS is understood pointwise; in this case it is equivalent to say
that P
the convergence is weak in the space Λ(A) with respect to ca(A). If the
sum
S∈A0 λS v(S) is convergent, but not absolutely convergent, then we put
P
S∈A0 λS v(S) := +∞.P
P
Denoting A(λ) =
S∈A0 λS χS and c(λ) =
S∈A0 λS v(S), we can also
consider the following generalization of (15). Let z ≤ v(N ) whenever there
w
[A0 ]
exists a net (λi )i∈I ⊆ R+ such that A(λi ) −→ χN and c(λi ) −→ z where
(A0 )
z is finite. Then for each i ∈ I there exists a sequence (λin )∞
such
n=1 ⊆ R+
w
in
i
in
i
that A(λ ) −→ A(λ ) and c(λ ) −→ c(λ ). Consequently, there exists a net
w
(A0 )
(λj )j∈J ⊆ R+ such that A(λj ) −→ χN and c(λj ) −→ z. In other words,
Schmeidler κ-balancedness covers extensions of condition (6) like (15).
In addition, in Example 26 we presented a non-negative Schmeidler κbalanced game with an empty ca-core. Therefore, the game in Example 26
23
is balanced according to (15) and its generalization, although its ca-core is
empty.
Kannai (1992) gave the following result for non-negative games without
restricted cooperation (Kannai (1992), Theorem 3.2, p. 368).
Proposition 29. For a non-negative game v ∈ G A , where A is a field of sets,
it holds that ca-core(v) 6= ∅ if and only if there exists another non-negative
game without restricted cooperation w ∈ G A such that for each decreasing
sequence (Si )∞
i=1 of elements of A with empty intersection we have w(Si ) −→
0, and
X
m
n
X
sup
λi v(Si ) −
µj w(Tj ) ≤ v(N ) ,
i=1
j=1
n
where the supremum is taken over all finite sequences (λi )m
i=1 and (µj )j=1 of
n
positive numbers, and (Si )m
i=1 , (Tj )j=1 of elements of A, such that
m
n
X
X
λi χSi −
µj χTj ≤ 1.
i=1
j=1
Combining Definition 9, Proposition 11, and Theorem 12, we obtain the
following result.
0
Proposition 30. For any game v ∈ G A it holds ca-core(v) 6= ∅ if and only
if, for any ε > 0, there exist a δ > 0, a natural number n and µ1 , . . . ,
µn ∈ ca(A) such that
!
X
0
)
∀λ ∈ R(A
:
λS µi (S) − µi (N ) < δ for i = 1, . . . , n
∗
S∈A0
X
=⇒
λS v(S) < v(N ) + ε .
S∈A0
It follows that, if the game without restricted cooperation v ∈ G A is nonnegative, then Kannai’s (1992) condition in Proposition 29 is equivalent with
the condition in Proposition 30. However, the condition in Proposition 30
can be applied to a broader class of games.
5
Conclusion
We have generalized the Bondareva-Shapley Theorem to TU games with
arbitrarily many players and with various types of cores. In the particular
24
case of below bounded TU games without restricted cooperation we have
proved that the additive core of the TU game is not empty if and only if
the game is Schmeidler balanced. In the general case we have proved for an
arbitrary infinite cardinal κ that κ-core of a TU game is not empty if and
only if the TU game is κ-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 not sign
restricted. Moreover, as in the finite case, in the infinite case the proper notion of balancedness (κ-balancedness) is closely related to the strong duality
theorem.
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