1
2.1 NTU Characteristic Function
2.2 The Core; Balanced Games and Scarf’s Theorem;
Market Games, The Gabage Dumping Game
2.3 The NTU Nucleolus
2.4 The λ-Transfer Value
2.5 NTU Core Existence Proof
NTU Coalitional Games
3
V(S ) ∩ V(T ) ⊆ V(S ∪ T ) ∪ V(S ∩ T ), ∀S , T ⊆ N.
Definition 2.1.3. An NTU coalitional game (N, F, V) is convex iff
V(S ) ∩ V(T ) ⊆ V(S ∪ T ), ∀S , T ⊆ N with S ∩ T = ∅.
Definition 2.1.2. An NTU coalitional game (N, F, V) is superadditive iff
Remark 2.1.1. The set-valued function V is called an NTU characteristic
function of a game (N, F, V). We will denote by (N, V) a game (N, F, V) where
F = V(N).
2
NTU Game Theory
NTU Characteristic Function
The Core
4
Then the core is the set {(1, 1, w), (1, w, 1), (w, 1, 1)}, which is of course not
convex.
V({i}) = {u ∈ N |ui ≤ w}, ∀i ∈ N.
V({i, j}) = {u ∈ N |ui ≤ 1, and u j ≤ 1}, ∀i, j ∈ N, (i j)
V(N) = {u ∈ N |u1 + u2 + u3 ≤ 2 + w}
Example 2.2.1. Let V be given by: N = {1, 2, 3}, 0 ≤ w < 1 and
S ∈N\{∅}
Definition 2.2.2. The core C(N, F, V) of a game (N, F, V) is the set of payoff
vectors in F that are not improved upon by any coalition, that is,
C(N, F, V) = F −
int V(S )
Definition 2.2.1. A coalition S can improve upon a payoff vector y iff there
is an x ∈ V(S ) such that xi > yi for all i ∈ S .
2.2
2
bounded subset relative to the subspace S ; that is, there is a number
M such that xi ≤ M for all i ∈ S and all x ∈ Q(S ).
ᲢeᲣF is closed, and ∀x ∈ V(N), ∃y ∈ F with x ≤ y.
ᲢdᲣThe set Q(S ) ={x|x ∈ V(S ), and x int V({i}) ∀i ∈ S } is a nonempty,
ᲢcᲣIf x ∈ V(S ) and xi = yi ∀i ∈ S , then y ∈ V(S )
ᲢbᲣV(S ) is comprehensive,i.e., if x ∈ V(S ) and y ≤ x then y ∈ V(S ).
ᲢaᲣV(S ) is a nonempty closed subset of N ; and V(∅) = ∅.
1. a set N of players, #N = n,
2. a set F of attainable outcomes, F ⊆ N , and
3. a correspondence V from N = 2N into N such that the followings are
satisfied for each S ∈ N.
Definition 2.1.1. An NTU coalitional game (N, F, V) is given by
2.1
Market Games
1
5
i∈S
i∈S
7
Here, Pco is the set of all convex preferences allowing continuous quasiconcave utility functions. The convex economy means such an economy. See
the next section .
Theorem 2.2.2. An NTU market game derived from a convex economy E :
N → Pco × m
+ is balanced.
where wi ∈ m
+ for all i ∈ N.
Definition 2.2.5. An NTU market game is a coalitional game (N, V) defined
as follows: For each S ⊆ N,
V(S ) = ū ∈ N ∃x = (x1, . . . , xn) ∈
m
+
i∈N
s.t.
xi =
wi, and ui(xi) ≥ ūi ∀i ∈ S
2.2.2
1
1
Balanced Games and Scarf’s Theorem
The core of a balanced game is
Hence, ū ∈ V(N).
S ∈B
i∈S
8
i∈N
S ∈B,S i
i∈N
which is a convex combination of f S (i), S ∈ B. By the convexity of preferences, we have ui( f (i) ≥ ūi for all i ∈ N. We have only to show that f is a
redistribution for E. But,
⎛
⎞
⎜⎜
⎟⎟
⎜
S
S
f (N) =
δS f (i) =
δS ⎜⎜⎝
f (i)⎟⎟⎟⎠
i∈N S ∈B,S i
i∈S
S ∈B
⎛
⎞
⎛
⎞
⎜⎜
⎜
⎟⎟⎟ ⎟
⎜⎜⎜ ⎟
⎜
⎟
δS ⎜⎜⎝ w(i)⎟⎟⎠ =
w(i) ⎜⎜⎝
δS ⎟⎟⎟⎠ =
w(i).
=
S ∈B,S i
Proof. Assume that ū ∈ S ∈B V(S ). Then, for each S ∈ B there is an allocation, say f S such that f S (S ) = w(S ) and ui( f S (i)) ≥ ū(i) for all i ∈ S . Define
the allocation
f (i) =
δS f S (i),
6
The proof will be given in the last subsection 2.5 (Go To p.38).
Theorem 2.2.1. (Scarf [34, 1967]).
nonempty.
S ∈B
Definition 2.2.4. A game (N, F, V) is balanced iff for every balanced family
B,
V(S ) ⊆ V(N).
S ∈B, S i
Definition 2.2.3. A family B of nonempty, proper subsets of N is balanced iff
there exist positive weights (balancing weights) δS for S ∈ B such that
δS = 1 f or all i ∈ N
2.2.1
Garbage Dumping Game
b(S ) :=
j∈S
9
b j ∈ m
+,
∀S ⊆ N.
=⇒
T ∩ Bm−1 ∅,
j∈Bm−1
11
Furthermore, ui ≤ ui(0) = ui(pi) for i {1, 2, . . . , k}.
Thus u ∈ V(N).
j∈N\T
then, m − 1 ∈ T for any T ∈ B with Bm ⊆ T , which, by the definition of Bi
above, implies that m − 1 ∈ Bm, a contradiction.
Thus, noting that u ∈ V(T ), Bm−1 ⊆ N \ T and that ui is strictly decreasing,
we have for each m = 1, 2, . . . , k that
b j ≤ um
b j = um(pm).
u m ≤ um
Bm ⊆ T
This must be so because, if it were true that
Bm ⊆ T and T ∩ Bm−1 = ∅.
By the definition of Bi and P, for each m = 1, 2, . . . , k, there exists a
coalition T ∈ B such that
where
Proposition 2.2.1. The NTU coalitional game V is balanced, where V is
defined with strictly decreasing function ui by
V(S ) = u ∈ n|ui ≤ ui(b(N \ S )) ∀i ∈ S , ∀S N,
⎫
⎧
⎪
⎪
∃t = (t1, . . . , tn) ∈ m×n
⎪
⎪
⎪
⎪
+
⎪
⎪
⎪
⎪
⎬
⎨
n
V(N) = ⎪
u ∈ s.t. j∈N t j = b(N),
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
ui ≤ ui(ti) ∀i ∈ N ⎪
The next NTU game is an NTU extension of the TU game with empty core
discussed by [42, Shapley and Shubik]. Here, it is shown that NTU makes
the game balanced without any convexity assumption [20, Hirai et al.].
2.2.3
S j,S ∈B
The NTU Nucleolus
*1
12
A generalization to NTU games is found in Shapley [38, 1969]; Axiomatization of the NTU Value is due to Aumann [3, 1985].
In this section, we review a theory of NTU nucleolus by Nakayama [30,
1982] which is a generalization of a TU nucleolus introduced as a point solution to coalitional games by Schmeidler [35, 1969]. Another well-known
point solution which is also generalized to NTU games is the Shapley value
(Shapley [36, 1953]). *1
The TU nucleolus always uniquely exists and it is by definition in the
nonempty core. In a sense, it can be viewed as a way of ultimate ‘downsizing’ of the core. Thus, it may serve as a reference point when it is necessary to single out a point from the core. It will turn out that our extension
preserves the existence and the inclusion in the NTU core. This is the reason
for reviewing the nucleolus here in the core analysis; the Shapley value will
be treated in a later proper occasion.
2.3
10
Thus, Bi = B j. Then, {Bi : i ∈ N} generates a partition P of N such that
P = {B1, B2, . . . , Bk } with k ≤ n.
Now, define (pi)i∈N as follows:
pm = i∈Bm−1 bi for m = 1, 2, . . . , k, with m − 1 = k for m = 1
pm = 0 for m {1, 2, . . . k}.
Note that i∈N pi = i∈N bi.
S i,S ∈B
Suppose that j ∈ Bi. Then, for all S ∈ B, if i is in S then j is also in S . By
the balancedness,
λS =
λS = 1, and ∀S ∈ B, λS > 0.
S i,S ∈B
Proof. (Proposition 6). Let B be a balanced collection and let u ∈ ∩S ∈BV(S ).
Define, for all i ∈ N,
Bi :=
S.
Nucleolus Share Ratios
n
15
Definition 2.3.3. A share ratio a∗ ∈ AIR is said to be a nucleolus share ratio
θ(a) = (θ1(a), . . . , θ2n (a))
where
θ j(a) ≥ θk (a) if j < k.
IR
For each a ∈ A , let θ(a) be the 2 -dimensional vector of excesses arranged
in the nonincreasing order, i.e.,
where e(a, S ) = 0 for S the empty set.
i∈S
Definition 2.3.2. The excess of coalition S under share ratio a ∈ AIR is given
by
e(a, S ) =
h(a, S ) − h(a, N) ai
a ∈ AIR and S ⊆ N.
13
∃u ∈ V(S ) ∀i ∈ S ui ≥ hai
maximize h
subject to the condition that
Maximization Problem P(a,S) given a ∈ A
i∈N
Let A be an (n − 1)-simplex; namely
ai = 1, ai ≥ 0 ∀i ∈ N .
A = a ∈ N Let (N, F, V) be an NTU coalitional game with F = V(N). For each i ∈ N,
we assume without loss of generality that the set V({i}) is given by V({i}) =
{x ∈ N |xi ≤ wi] with wi > 0.
2.3.1
*2
Existence of Nucleolus Share Ratios
For uniqueness, see
*2
In NTU games, the uniqueness is not guaranteed.
16
Proof. It is clear that the function mini∈S { uaii } is continuous on V(S ) × A◦, and
Lemma 2.3.2. For each S ⊆ N, the function h(·, S ) is continuous on the
interior A◦ of A.
2.3.2
Proof. Existence is proved in the next subsection.
Schmeidler’s paper.
Theorem 2.3.1. (Schmeidler [35, 1969]) Every TU coalitional game has a
unique nucleolus.
where a∗ is a nucleolus share ratio is called an NTU nucleolus.
(h(a∗, N)a∗1, . . . , h(a∗, N)a∗n)
if it minimizes θ(a) in the lexicographical order. The payoff vector
14
Since wi > 0 for all i ∈ N by assumption, h(a, S ) is well-defined for all
a ∈ AIR = {a ∈ A | h(a, N)ai ≥ wi ∀i ∈ N}.
Definition 2.3.1. A share ratio a ∈ A is individually rational iff
Proof. The set {h ≥ 0| ∃u ∈ V(S ) ∀i ∈ S ui ≥ hai} is nonempty and compact.
Lemma 2.3.1. For each S ⊆ N, P(a, S ) has the optimal solution iff ai > 0
for some i ∈ S .
For each S ⊆ N, let h(a, S ) be the maximum of h if it exists. The payoff
to each player i ∈ N is given by h(a, N)ai via the problem P(a, N). A payoff
vector (h(a, N)a1, . . . , h(a, N)an) is individually rational if h(a, N)ai ≥ wi for
all i ∈ N. A point a ∈ A will be called a share ratio.
min e(a, S ) | S ∈ F F ⊆ 2N , |F| = k
19
It is enough to show that A2n is nonempty. First, since θ1(·) is continuous on
AIR and AIR is compact, the closed subset A1 of AIR is compact and nonempty.
Similarly, since θ2(·) is continuous on A1 and A1 is compact, the closed subset
A2 of A1 is also compact and nonempty. Continuing this finitely many times,
we will arrive at the conclusion that A2n is nonempty
Ak = {a ∈ Ak−1| θk (a) ≤ θk (ā), ∀ā ∈ Ak−1}, k = 1, 2, ..., 2n.
A1 = {a ∈ AIR| θ1(a) ≤ θ1(ā), ∀ā ∈ AIR}
Then, θk (·) is continuous on AIR, since it is defined by min and max of a finite
number of continuous functions.
Now, define
θk (a) = max
each k = 1, 2, ..., 2n that
17
Problem 2.3.1. Show the example you think is simplest, in which the func-
• h(·, S ) is lower semicontinuous at a◦ if for any real number r, h(a◦, S ) > r
implies that for some neighborhood U(a◦, δ) of a◦, h(a, S ) > r whenever
a ∈ U(a◦, δ).
• h(·, S ) is upper semicontinuous at a◦ if for any real number r, h(a◦, S ) < r
implies that for some neighborhood U(a◦, δ) of a◦, h(a, S ) < r whenever
a ∈ U(a◦, δ).
ui .
h(a, S ) = max min
i∈S
u∈V(S )
ai
Then, it will be easy to see that the function h(·, S ) is both upper semicontinuous and lower semicontinuous directly from the definitions:
h(a, S ) can be represented by
Inclusion in the Core
C.Berge, Topological Spaces, Macmillan, New York, 1963
18
20
so that h(a, S ) > h(a, N), a contradiction.
ũi = h(a, S )ai > h(a, N)ai ∀i ∈ S
Proof. (sufficiency). Suppose that there was an S ⊆ N that has a payoff vector
ū ∈ V(S ) such that ūi > h(a, N)ai for all i ∈ S . Since ai > 0 for all i ∈ S ,
there is a ũ ∈ V(S ) satisfying
h(a, N) = max{h(a, S )| S ⊆ N}.
Lemma 2.3.3. A payoff vector u ∈ V(N) is in the core of the NTU game iff
there exists an a ∈ AIR such that ui = h(a, N)ai ∀i ∈ N and that
Recall that a payoff vector u ∈ V(N) is in the core of an NTU game iff no
coalition S has a payoff vector ū ∈ V(S ) satisfying ūS > uS .
2.3.3
*3
Proof. Note first that AIR is nonempty and compact. Nonemptiness follows
from the definition of an NTU game. It must be compact because AIR ⊆ A
and h(·, N) is continuous on A, and A is compact. Since h(·, S ) is continuous
on AIR for each S ⊆ N, so is e(·, S ) continuous on AIR for each S ⊆ N.
We may now follow the proof due to Schmeidler [35, 1969]. First, note for
Theorem 2.3.2. There exists a nucleolus share ratio.
Remark 2.3.1. For a formal proof of the above lemma, we may apply the
Berge maximum theorem *3 , by noting that h(a, S ) = max{h | h ∈ U(a, S )},
where U(a, S ) = {h ≥ 0 | ha ∈ V(S )}. But, showing the continuity of the
correspondence U(·, S ) is almost equivalent to showing the continuity of the
very h(·, S ).
tion h(·, S ) can be discontinuous on the boundary of A.
> ui, ∀i ∈ S ,
The λ-Transfer Value
*4
An NTU game (N, F, V) with F V(N).
23
That is, x is an NTU value if it can be attained in the grand coalition N,
and if there exists a vector of weights λ such that in a TU game vλ where
utilities are transferable at the ratios given by the weights, the value of the
game vλ, called the λ-transfer value, coincides with the “λ -transfer payoff
Definition 2.4.1. (Shapley [38, 1969]). A payoff vector x is said to be an
NTU value if x ∈ V(N) and there exists a nonnegative vector λ ∈ +N \ {0}
such that λi xi = (φvλ)i for all i ∈ N.
i∈S
Given an NTU game (N, V) *4 and a vector of nonnegative weights λ =
(λ1, . . . , λn) 0, let us define the TU game vλ as follows:
vλ(S ) = max{ λi xi | x ∈ V(S )}, ∀S ⊆ N.
2.4
21
which contradicts the assumption that u is in the core.
u◦i
This implies that there is a payoff vector u ∈ V(S ) such that
◦
j∈N
u j ai = ui ∀i ∈ S .
h(a, S )ai > h(a, N)ai ≥
(necessity). Suppose that u ∈ V(N) is in the core. Then, since ui ≥ wi > 0
for all i ∈ N, letting
ui
ai = j∈N u j
we have, by definition, that h(a, N) ≥ j∈N u j; and hence, that a ∈ AIR.
Now, suppose that for some S N we had h(a, S ) > h(a, N). Then,
so that
e(a∗, S ) ≤ 0 ∀S ⊆ N.
Existence of NTU Values
i∈S
22
24
Assumption 2.4.1. The payoff vector φ(λ) is continuous in λ, Pareto efficient
and individually rational.
and let φ(λ) be the value of the TU game vλ.
F(λ) = {(λ1 x1, . . . , λn xn) | λ ∈ Λ and x ∈ F},
Further, define
Given an NTU game (N, V), let F V(N) be the set of attainable payoff vectors, which is a compact convex subset of +N . Let
Λ = {λ ∈ +N | i∈N λi = 1}, and for each λ ∈ Λ let vλ be again the
game defined by
⎧
⎫
⎪
⎪
⎪
⎪
⎨
⎬
vλ(S ) = max ⎪
λi xi | x ∈ V(S )⎪
, ∀S ⊆ N.
⎪
⎪
⎩
⎭
2.4.1
vector” (λ1 x1, . . . , λn xn).
which completes the proof.
h(a∗, N) ≥ h(a∗, S ) ∀S ⊆ N,
Since a∗i > 0 for all i ∈ N, we may rewrite this as follows:
θ1(a∗) ≤ θ1(a) ≤ 0
Proof. By the previous lemma, there is an a ∈ AIR such that h(a, N) ≥ h(a, S )
for all S ⊆ N. Hence e(a, S ) ≤ 0 for all S ⊆ N. Then, letting a∗ ∈ AIR be any
nucleolus share ratio, we have by definition that
Theorem 2.3.3. If an NTU game has a nonempty core, the NTU nucleolus is
in the core.
27
For an excellent discussion to motivate the NTU value from interpersonal
utility comparisons, see the original Shapley’s paper [38, 1969]. Sixteen
years after this paper, Aumann [3, 1985] succeeded in axiomatizing the NTU
value.
The NTU value may not be unique. In the two-person case with a strictly
individually rational portion in the Pareto frontier, the NTU value is unique
and coincides with the Nash bargaining solution.
25
Let A be a simplex in the hyperplane {α | i∈N αi = 1}, that is large enough
to contain all sets T (λ), λ ∈ Λ and Λ itself. This is possible because of the
upper hemicontinuity of T ; so that T (Λ) becomes compact.
T (λ) = λ + P(λ) = {λ + π | π ∈ P(λ)}.
Define the set-valued function T by
Then, P(λ) is nonempty, convex, compact for each λ ∈ Λ; and is upperhemicontinuous in λ.
i∈N
Proof. Let P(λ) be the set of vectors π satisfying
πi = 0 and φ(λ) − π ∈ F(λ).
Theorem 2.4.1. (Shapley [38, 1969]). There exists λ ∈ Λ such that φ(λ) ∈
F(λ).
max(0, αi)
.
j∈N max(0, α j )
NTU Core Existence Proof
28
Before starting the proof, we prepare some notations and remarks. For the
ordeing of vectors in N , x ≥ y iff xi ≥ yi for all i ∈ N, x > y iff x ≥ y but
x y, and x y iff xi > yi for all i ∈ N. For each S ⊆ N, let eS ∈ N be the
characteristic vector of S , that is, eSi = 1 if i ∈ S and eSi = 0, otherwise. We
write e for eN , and ei for e{i}.
V({i}) = {x ∈ N | xi ≤ wi}, where wi > 0 ∀i ∈ N.
In the following, we shall prove the fundamental theorem that a balanced
NTU game has a nonempty core. Contrary to the original proof by Scarf
[34], the proof below due to Shapley and Vohra [43] uses the Kakutani’s
fixed point theorem that is familiar to economists.
We maintain the assumptions on V(S ) stated in the beginning, and the one
put in the NTU nucleolus that each V({i}) contains a positive portion, that is,
2.5
26
Then, by the Kakutani’s theorem, there is a fixed point α∗ satisfying α∗ ∈
T (α∗). Let us write λ∗ for f (α∗).
Now, suppose that α∗ λ∗. Then, by the definition of f , we have α∗ ∈
A \ Λ, and λ∗i = 0 > α∗i for some i.
But, that α∗ ∈ T (λ∗) = λ∗ + P(λ∗) and that λ∗ = f (α∗) ≥ 0 ∈ +N imply that
π∗i < 0 for some π∗ ∈ P(λ∗).
Since φi(λ∗) ≥ 0 by individual rationality, in the feasible payoff vector
φ(λ∗) − π∗ ∈ F(λ∗) player i obtains a positive amount φi(λ∗) − π∗i > 0.
But this is impossible without sidepayments, since all i’s payoffs in vλ∗ are
zero because λ∗i = 0.
Therefore we conclude that α∗ = λ∗, so that λ∗ ∈ T (λ∗) implying that
0 ∈ P(λ∗). Hence, φ(λ∗) ∈ F(λ∗), which completes the proof.
T (α) = T ( f (α)), where fi(α) = Extend the definition of T to A by
i
i
eS
m = .
|S |
S
(1)
S ∈N
⎞
⎛
⎟⎟⎟
⎜⎜⎜ ⎜
V(S )⎟⎟⎟⎠ ∩ Q.
W = ⎜⎜⎝
31
u ∈ ∂W and v u =⇒ v W
(2)
Since all V(S ) and Q are comprehensive, so is W. Then, denoting by ∂W the
boundary of W, it will be obvious that
Define
x ∈ V(S ) and xS ≥ 0 =⇒ xi < q ∀i ∈ S .
In the proof below, a modification of the function h(·, S ) from A to the
Pareto frontier is used, as we have employed in the NTU nucleolus.
Let Q = {x ∈ N | x ≤ qe} with q > 0 be the set including every V(S )
properly relative to S under the assumption stated in Definition 2.1.1.(3)(c),
that is, for all S ∈ N,
29
mN ∈ co{mS | S ∈ B}.
Lemma 2.5.1. B is balanced if and only if
Then, we note the following fact.
Now, for each S ∈ N, let
S ∈B
Recall that a collection B ⊆ N of nonempty subsets of N is balanced if
there exist nonnegative weights δS for each S ∈ B such that
δS eS = eN .
Then, AN is the unit symplex A in N .
S
A = convex hull of {e | i ∈ S } := co{e | i ∈ S }.
Define, for each nonempty S ⊆ N,
δS
S ∈B
1 |S |
1
δS
δS = 1
=
=
|N|
|N| i∈N |N| S i
i∈S
S ∈B
30
mN ∈ co{mS | S ∈ B}.
u ∈ ∂W and (∃ j ∈ N) u j = 0 =⇒ (∃i ∈ N) ui = q
⇐⇒
N
|S | eS e
αS mS
δS
mN =
=
=
|N| S ∈B
|N| |S | S ∈B
S ∈B
δS |S |
.
|N|
(3)
V({1})
x2
V({2})
V({1, 2})
Q
32
x1
This is because any u with u j = 0 belongs, by assumption, to V({ j}), which
implies that u = (q, .., q, 0, q, .., q) whenever u ∈ ∂W.
Moreover,
⇐⇒
S ∈B
αS =
δS eS
S ∈B
⇐⇒
eN =
Then,
αS =
Proof. Let αS ≥ 0 and δS ≥ 0 satisfy for all S ∈ B that
S ∈N V(S ) and f is nonempty.
35
Since x p → x, f is continuous and V(S ) is closed, we have f (x) ∈ V(S ),
which means that y ∈ G(x).
Since G(x) is a finite set, there exists p̄ such that for all p ≥ p̄, y p = y.
Then, for all p > p̄, we have y ∈ G(x p); that is, there exists S ∈ N such that
y = mS and f (x p) ∈ V(S ).
Let x p → x, x p ∈ A for all p, y p ∈ G(x p) for all p and y p → y. We must
show that y ∈ G(x) (Note: A is compact and G(x) is finite (compact), so that
G is upper hemicontinuous iff G has a closed graph).
Claim 2. G is upper hemicontinuous.
G is nonempty, since for all x ∈ A, f (x) ∈
G(x) = {mS | S ∈ N, S ∅ and f (x) ∈ V(S )}.
Now define a set valued function G : A → A by
33
f (x) = {y ∈ ∂W | y = tx ∃t ≥ 0}.
To show the existence of such u∗ ∈ ∂W, let us define f : A → W by
To see that u∗ is in the core, let us suppose the contrary. Then, there exists
S ∈ N and v ∈ V(S ) such that vS u∗S . Since u∗ qe, there exists v̄ u∗
such that v̄ qe and v̄ ∈ V(S ), that is, v̄ ∈ W. But, this contradicts the fact
that u∗ ∈ ∂W.
Since the game is balanced, it follows that u∗ ∈ V(N).
S ∈T
Proof. We shall show that there exist a balanced collection T and u∗ ∈ ∂W ∩
+N such that
V(S ).
u∗ ∈
Theorem 2.5.1. A balanced game has a nonempty core.
xi + max(gi − 1/n, 0)
1 + j∈N max(g j − 1/n, 0)
∀i ∈ N,
36
is one that satisfies the condition of Kakutani’s fixed point theorem (correspondence being compact, convex-valued; and, the range being compact,
convex).
h × co(G) : A × A → A × A
Then, the correspondence
which is clearly a well defined, continuous function.
hi(x, g) =
Finally, for any x, g ∈ A, let h : A × A → A be defined by
34
Finally, f is single-valued. For, otherwise, there exist x ∈ A, y ∈ f (x) and
ȳ ∈ f (x) with y = tx, ȳ = t¯x and t¯ > t. If x 0 and ȳ y, this contradicts
(4). If, on the other hand, xk = 0 for some k ∈ N, then let K = {k ∈ N | xk = 0}
and I = {i ∈ N | xi > 0}. Then, since t¯xi ≤ q for all i ∈ N, we have txi < q for
all i ∈ I and txk = 0 for all k ∈ K. This contradicts (5).
Secondly, f is easily seen to be upper hemicontinuous given that for any
x ∈ A f is compact-valued and that the range is also compact by {t ∈ + | tx ∈
∂W} ⊆ [0, n(q + 1)].
Firstly, f is nonempty, since 0 ∈ W and n(q + 1)x W where x ∈ A, which
must be so because i∈N n(q + 1)xi = n(q + 1) > i∈N q. Hence, there exists
t ∈ [0, n(q + 1)] with tx ∈ ∂W.
Claim 1. f is a well defined continuous function.
xi∗ + max(g∗i − 1/n, 0))
1 + j∈N max(g∗j − 1/n, 0)
37
(4)
39
Let u∗ = f (x∗). By the definition of G, it follows that u∗ ∈ S ∈T V(S ). Since
the game is balanced, we have that u∗ ∈ V(N), which completes the proof.
Since G(x∗) = {mS | S ∈ T } and mN ∈ co(G(x∗)), it follows from Lemma
2.5.1 that T is balanced.
Now, define T = {S ∈ N | f (x∗) ∈ V(S )}.
j∈N
∀i ∈ N.
∀i ∈ N.
⎞
⎛
⎟⎟⎟
⎜⎜⎜
xi∗ ⎜⎜⎜⎝ max(g∗j − 1/n, 0)⎟⎟⎟⎠ = max(g∗i − 1/n, 0)
Or, equivalently,
xi∗ =
Let (x∗, g∗) ∈ A × A be a fixed point of h × co(G) such that x∗ = h(x∗, g∗)
and g∗ ∈ co(G(x∗)). Then
40
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Since g∗ ∈ co(G(x∗)), it follows from the construction of G and g∗i > 0 ∀i ∈ I
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