CHAPTER 5
SOME MINIMAX AND SADDLE POINT THEOREMS
5.1 INTRODUCTION
Fixed point theorems provide important tools in game theory which are used to prove the
equilibrium and existence theorems. For instance, the fixed point theorems by Brouwer,
Kakutani and Fan have been among the most used tools in economics and game theory, see
[4], [13], [161]. These theorems are further applied to prove the minimax theorems and the
existence of saddle points.
The first minimax theorem was given by von Neumann [157] in 1928. He established the
following.
Theorem 5.1.1 [188]. Let A be an m × n matrix and x and y be the sets of nonnegative row and
column vectors with unit sum. Then
min max x T Ay = max min x T Ay.
y∈Y
x∈ X
x∈ X
y∈Y
(5.1.1)
After nine years of this result von Neumann [186], showed that the bilinear character of
Theorem 5.1.1 was not needed. He extended his result by using Brouwer’s fixed point
theorem as follows.
129
Theorem 5.1.2 [186]. Let X and Y be nonempty compact, convex subsets of Euclidean spaces
and f : X × Y → R be jointly continuous. Suppose that f is quasiconcave on X and quasiconvex
on Y. Then
min max f ( x, y ) = max min f ( x, y ).
y∈Y
x∈ X
x∈ X
y∈Y
Thereafter, many papers appeared in the literature verifying the Neumann inequality. It is
noticed that majority of minimax theorems are proved by applying fixed point theorems under
different conditions on the spaces or on the mappings.
Sion [159] generalized von Neumann’s result on the basis of Knaster, Kuratowski and
Mazurkiewicz theorem [160], which was given in the year 1929 on the n-dimensional simplex
in the following way.
Theorem 5.1.3 [160]. Let D be the set of vertices of an n-simplex ∆ n and a multivalued map
G : D → ∆ n be a KKM map (that is, co( A) ⊂ G ( A) for each A ⊂ D ) with closed [resp.,
open] values. Then I z∈D G ( z ) ≠ φ .
This theorem is an equivalent formulation of the famous Brouwer fixed-point theorem.
At the beginning, the results of the KKM theory were established for convex subsets of
topological vector spaces mainly by Ky Fan [161]-[164]. A number of intersection theorems
and applications were followed in the convex subsets of topological vector spaces. In 1961,
Ky Fan generalized KKM theorem to infinite dimensional topological vector space, which is
known as Fan–Knaster-Kuratowski-Mazurkiewicz theorem (i.e. Fan KKM theorem or KKMF
theorem). Fan established following minimax inequality from the KKMF theorem.
Theorem 5.1.4 [162]. Let X be a nonempty compact convex set in a topological vector
space. If a function ϕ : X × X → (−∞, ∞) satisfies the following conditions
(i) for each x ∈ X , ϕ ( x, y ) is lower semicontinuous function of y on X,
(ii) for each y ∈ X , ϕ ( x, y ) is quasiconcave function of x on X.
Then the minimax inequality min sup ϕ ( x, y ) ≤ sup ϕ ( x, x) holds.
y∈ X x∈ X
x∈ X
130
Lassonde [184] extended the KKM theory to convex spaces by proving several
variants of KKM theorems for convex spaces and proposed a systematic development of the
method based on the KKM theorem. The KKM theorem was further extended to pseudoconvex spaces, contractible spaces and spaces with certain contractible subsets or H-spaces by
Horvath [166]-[167]. In these papers, most of the Fan’s results in the KKM theory are
extended to H-spaces by replacing the convexity condition by contractibility. With these
terminology, Park [168] established new versions of KKM theorems, minimax inequalities,
and others on H-spaces. Park and Kim [169]-[173], Park [174]-[178] and others extended
these results to more general spaces such as G-convex, abstract convex and KKM spaces.
The purpose of this chapter is to present a generalized version of the KKM theorem by
using the concept of Chang and Zhang [182]. As an application of it, a generalized minimax
inequality is obtained and an existence result for the saddle-point problem under general
settings is derived. Consequently, a saddle point theorem for two person zero sum parametric
game is also proved. Several existing well known results are obtained as special cases.
5.2 PRELIMINARIES
First we give basic definitions used in our results. We follow [160], [180], [182],[185], [193],
[195], [304] and [305] for notations and preliminaries.
Definition 5.2.1 [304]. A two-person, zero-sum game G is defined by G = ( A1 , A2 , f1 , f 2 )
where, for i = 1, 2, we have
(i) Ai is a finite set of player i’s actions,
(ii) f i : A → R, where A = A1 × A2 , is player i’s payoff function,
the player’s payoff functions satisfy
f1 (a ) + f 2 ( a ) = 0, for all a ∈ A.
If we consider two person zero sum game generated by function f : X × Y → R. This means
that the first player selects a point x from X and the second player selects a point y from Y .
Because of this choice, the second player pays the first one the amount f ( x, y ).
Definition 5.2.2 [305]. A point ( x * , y * ) ∈ X × Y is said to be saddle point of f in X × Y if
131
f ( x * , y) ≤ f ( x * , y * ) ≤ f ( x, y * ) for all ( x, y ) ∈ X × Y .
A point ( x * , y * ) ∈ X × Y is said to be a solution of the game if and only if it is a saddle point
of f in X × Y .
Definition 5.2.3 [193]. A two-person zero-sum parametric game (GPθ ) is defined by the
following form
( X , Y , f , g , θ , Fθ ),
where,
1.
X is a subset of a topological space E, which is called the strategy set of player 1,
2.
Y is a subset of a topological space E, which is called the strategy set of player 2,
3.
f : X × Y → R and g : X × Y → R+ , where R+ = (0, ∞),
4.
θ is a real number, which is called a parameter of the game,
5.
Fθ = f − θ g : X × Y → R, that is, for all ( x, y ) ∈ X × Y , Fθ ( x, y) = f ( x, y) − θ g ( x, y),
is a loss function of player 1 and − Fθ ( x, y ) is a loss function of player 2.
In general, Fθ = inf x∈X sup y∈Y Fθ ( x, y ) is called the minimal worst loss of player 1 and
Fθ = sup y∈Y inf x∈X Fθ ( x, y ) is called the maximal worst gain of player 2.
We illustrate the above concept of two person zero sum parametric game by following
example.
Example 5.2.1. Consider the sets
X = { x1 , x 2 , x 3 , x 4 }, Y = { y1 , y 2 , y3 , y 4 } and parameter θ = 2.
A function f : X × Y → R defined by the following payoff matrix
y1
y2
y3
y4
x1
7
8
3 -2
x2
4
7
4
8
x3
-3
4
3
6
5
-1
8
8
x4
132
and g : X × Y → R + by
y1
y2
y3
x1
2
2
1
1
x2
1
1
1
2
x3
1
1
1
3
2
1
3
2
y1
y2
y3
x1
3
4
1
-4
x2
2
5
4
x3
-5
2
2
1
0
x4
1
-3
2
4
x4
y4
When θ = 2, then Fθ : X × Y → R becomes
y4
and ( x2 , y3 ) is the saddle point of the function Fθ ( x, y ).
KKM mapping in Hausdorff topological vector space is defined as follows.
Definition 5.2.4 [160]. Let E be a Hausdorff topological vector space and X be a nonempty
subset of E. A multivalued mapping G : X → 2 E , that is, mapping with the values
n
G ( x) ⊂ E , for each x in X, is called a KKM mapping if co{x1 , x 2 , ... , x n } ⊂ U G ( xi ) for each
i =1
finite subset {x1 , x2 , ... , xn } ⊂ X , where co{x1 , x2 , ... , xn } denotes the convex hull of the set
{x1 , x2 , ... , xn }.
Definition 5.2.5 [182]. Let X be a nonempty subset of a topological vector space E. A
multivalued mapping G : X → 2 E is called a generalized KKM mapping, if for any finite set
133
{x1 , ... , xn } ⊂ X , there exists a finite subset { y1 , ... , y n } ⊂ E such that for any subset
{ y i1 , ... , y ik } ⊂ { y1 , ... , y n }, 1 ≤ k ≤ n, we have
k
co{ yi1 ,..., yik } ⊂ U G ( xi j ).
j =1
We extend this definition for two maps in the following manner.
Definition 5.2.6. Let X be a nonempty subset of a topological vector space E
and F , G : X → 2 E . Then G is said to be a generalized F-KKM mapping if for any finite set
{x1 , ... , xn } ⊂ X and each {i1 , ... , ik } ⊂ {1, 2, ... , n}, we have
k
k
j =1
j =1
co{ U F ( xi j )} ⊂ U G ( xi j ).
It is remarked that definition 5.2.6 becomes definition 5.2.5 when F : X → E and F ( xi ) = yi
for each i ∈ {1, 2, ... , n}.
In the following example we have shown that concept of F-KKM mapping is general than
KKM mapping.
Example 5.2.2. Let E = (−∞, ∞), X = [−2, 2]. Let F , G : X → 2 E be defined as
 
x2 
x2 
  x
x
F ( x) =  − 1 +  , 1 +  and G ( x) = − 1 + , 1 + .
5 
5
3
  3
 
Since U G ( x) = [−9 / 5, 9 / 5], x ∉ G ( x), ∀x ∈ [−2, − 9 / 5) U (9 / 5, 2].
x∈ X
n
So co{x1 , ..., x n } ⊄ U G ( xi ) and G is not a KKM map. But it is a F-KKM map, since
i =1
k
k
 1 5
 1 5
U F ( x) =  − ,  and co{ U F ( xi j )} ⊂  − ,  = I G ( x) ⊂ U G ( xi j ).
j =1
j =1
x∈ X
 3 3
 3 5  x∈X
Definition 5.2.7 [182]. Let E be a topological vector space and X be a nonempty convex
subset of E. A function φ : X × X → (−∞, ∞) is said to be γ − generalized quasiconvex in y
for some γ ∈ (−∞, ∞) if, for any finite subset { y1 , y 2 , ... , y n } ⊂ X , there is a finite subset
134
{x1 , ... , xn } ⊂ X
such
that,
for
any
subset { x i1 , ... , x ik } ⊂ { x1 , ... , x n } and
any
x 0 ∈ co{ x i1 , ... , x ik }, we have γ ≤ maxφ ( x0 , yij ).
i ≤ j ≤k
Definition 5.2.8. Let E be a topological vector space and X be a nonempty convex subset of E.
Let γ ∈ (−∞, ∞) and F : X → 2Y , where, Y ∈ E. Then the function φ : X × X → (−∞, ∞) is
said
to
be
F −γ
generalized
quasiconvex
in
y
if
for
any
finite
subset
k
{ y1 , y 2 , ... , y n } ⊂ X , each {i1 , ... , ik } ⊂ {1, 2, ... , n}, and any x0 ∈ co{ U F ( xi j )} ⊂ Y and any
j =1
yi j ∈ F ( xi j ), we have γ ≤ max φ ( x0 , yi j ).
i≤ j ≤k
Notice that the function φ becomes γ − generalized quasi convex in y when F : X → Y and
F ( xi ) = yi for each i ∈ {1, 2, ... , n}.
Definition 5.2.9 [195]. Let X and Y be two topological spaces. A multivalued mapping
F : X → 2 Y is said to be transfer closed valued on X if,
I F ( x) = I F ( x).
x∈ X
x∈ X
Definition 5.2.10 [185]. Let X and Y be two topological spaces. A multivalued mapping
F : X → 2 Y is said to be intersectionally closed valued on Y if,
I F ( x) = I F ( x).
x∈ X
x∈ X
Definition 5.2.11 [180]. Let X , Y be two topological spaces. Then a function
f : X × Y → R = R ∪ {±∞} is said to be
(i)
quasiconcave in x if for each y ∈ Y and λ ∈ R, the set {x ∈ X : f ( x, y ) ≥ λ} is
convex,
(ii)
quasiconvex in y if for each x ∈ X and λ ∈ R, the set { y ∈ Y : f ( x, y ) ≤ λ} is
convex,
135
(iii)
upper semicontinuous (resp. generally upper semicontinuous) in x if for each
y ∈ Y and λ ∈ R, the set {x ∈ X : f ( x, y ) ≥ λ} is closed (resp. intersectionally
closed),
(iv)
lower semicontinuous (resp. generally lower semicontinuous) in y if for each
x ∈ X and λ ∈ R, the set { y ∈ Y : f ( x, y ) ≤ λ} is closed (resp. intersectionally
closed).
Definition 5.2.12 [180]. Let X , Y be two topological spaces. Then a function
f : X × Y → R = R ∪ {±∞} is said to be
(i)
transfer upper semicontinuous in x if for each λ ∈ R, and all x ∈ X ,
y ∈ Y with f ( x, y ) < λ there exists a neighbourhood V(x) of x and a point
y ′ ∈ Y such that f ( z, y ′) < λ for all z ∈ V (x).
(ii)
transfer lower semicontinuous in y if for each λ ∈ R, and all x ∈ X , y ∈ Y with
f ( x, y ) > λ there exists a neighbourhood V(y) of y and a point x′ ∈ X such that
f ( x ′, u ) > λ for all u ∈ V ( y ).
Definition 5.2.13 [305]. Let X and Y be two topological spaces. A function
φ : X × Y → (−∞, ∞) is said to be
(i)
γ − transfer lower semicontinuous function in x for some γ ∈ (−∞, ∞) if, for all
x ∈ X and y ∈ Y with φ ( x, y ) > γ , there exists some y ′ ∈ Y and some
neighborhood N(x) of x such that φ ( z , y ′) > γ for all z ∈ N (x).
(ii)
γ − transfer upper semicontinuous function in y for some γ ∈ (−∞, ∞) if, for all
x ∈ X and y ∈ Y with φ ( x, y ) < γ , there exists some x′ ∈ Y
and some
neighborhood N(y) of y such that φ ( x ′, u ) < γ for all u ∈ N ( y ).
Definition 5.2.14. Let X, Y be two nonempty sets. The mapping F : X → 2 Y is said to be
surjective if F ( X ) = U x∈X F ( x) = Y .
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5.3 GENERALIZED KKM THEOREMS AND THEIR APPLICATIONS
First we prove the main result of this section which will be used in the sequel.
Theorem 5.3.1. Let X be a nonempty convex subset of a Hausdorff topological vector space
E. Let G : X → 2 E be a multivalued mapping such that for each x ∈ X , G (x) is finitely closed.
Then the family of sets {G ( x) : x ∈ X } has a finite intersection property if and only if the
mapping G is a generalized F-KKM mapping for some mapping F : X → 2 E.
Proof. Let {G( x)}x∈X has finite intersection property. Then for any finite subset
{x1 , x2 , ... , xn } ⊂ X , I ni=1 G ( xi ) ≠ φ . Taking x* ∈ I ni=1 G ( xi ), and define F : X → 2 E by
F ( x) = {x*} for each x ∈ X . Then for each {i1 , ... , ik } ⊂ {1, 2, ... , n}, we have
k
k

co U F ( xi j ) = co{x*} = {x*} ⊂ U G ( xi j ).
 j =1

j =1
This implies that G : X → 2 E is a generalized F-KKM mapping.
Now consider G : X → 2 E to be a generalized F-KKM mapping, F : X → 2 E and suppose the
family {G ( x) : x ∈ X } does not have the finite intersection property, i.e., there exists some
finite subset {x1 , x2 , ... , xn } ⊂ X such that I ni=1 G ( xi ) = φ . Since G is a generalized F-KKM
k
k

mapping, then for each {i1 , ... , ik } ⊂ {1, 2, ... , n}, we have co U F ( xi j ) ⊂ U G ( xi j ). In
 j =1
 j =1
{
}
particular, co F ( xi ) ⊂ G ( xi ) for each i ∈ {1, 2, ... , n}.
Let S = co{ y1 , y 2 , ... , y n },
L = span{ y1 , y 2 , ... , y n }.
Then S ⊂ L. Since G is finitely closed, G( xi ) I L is a closed set. Let d be the Euclidean
metric on L. It is easy to see that
d ( x, L I G( xi )) > 0 ⇔ x ∉ L I G( xi )
Now we define a function f : S → [0, ∞ ) as
n
f (c ) = ∑ d (c, L I G ( xi )) y i
i =1
It follows from (5.3.1) and I ni=1 G ( xi ) = φ that for each c ∈ S , f (c) > 0. Let
137
(5.3.1)
n
R (c ) = ∑
i =1
1
d (c, L I G ( x i )) y i .
f (c )
(5.3.2)
Thus R is a continuous function from S into S. By the Brouwer fixed point theorem, there
exists an element c* ∈ S such that
n
c* = F ( c* ) = ∑
i =1
1
d (c* , L I G ( xi )) y i
f (c )
(5.3.3)
Denote
I = {i ∈ {1, 2, ... , n} : d (c* , G( xi ) I L > 0).
(5.3.4)
Then for each i ∈ I , c* ∉ G( xi ) I L. Since c* ∈ L, c* ∉ G( xi ), ∀i ∈ I , so we have
c* ∉ U G ( x i )
(5.3.5)
i∈I
From (5.3.3) and (5.3.4), we have
c* = ∑
i∈I
1
d (c* , L I G ( xi )) yi ∈ co( yi : i ∈ I }.
f (c* )
However, since G is a generalized F-KKM mapping from X into 2E , which follows that
k
k

co  U F ( xi j ) ⊂ U G ( xi j ). Therefore we have
 j =1
 j =1
c* ∈ co( yi : i ∈ I } ⊂ U G( xi ).
(5.3.6)
i∈I
This contradicts (5.3.5). Hence {G ( x) : x ∈ X } has the finite intersection property. This
completes the proof.
It is remarked that if F : X → E is defined as a single valued map with F ( xi ) = yi for each
i ∈ {1, 2, ... , n}, then above result implies the following results of Chang and Zhang [182,
Theorem 3.1].
Corollary 5.3.1 [182]. Let X be a nonempty convex subset of a Hausdorff topological vector
space E. Let G : X → 2 E be a multivalued mapping such that for each x ∈ X , G (x) is finitely
closed. Then the family of sets {G ( x) : x ∈ X } has the finite intersection property if and only if
the mapping G is a generalized KKM mapping.
Following is a generalized version of the KKM mapping theorem.
138
Theorem 5.3.2. Let Y be a nonempty convex subset of a Hausdorff topological vector space
E, φ ≠ X ⊂ Y and maps F , G : X → 2 E with G an intersectionally closed valued map of Y.
Furthermore, assume that there exists a nonempty subset X 0 ⊂ X , contained in some
precompact convex subset Y0 of Y, such that G( x0 ) is compact subset of Y for at least one
x0 ∈ X 0 and let G : X → 2 E be a generalized F-KKM mapping. Then I G( x) ≠ φ.
x∈ X
Proof. For any finite subset {x1 , x2 , ... , xn } of X, let X 1 = X 0 U {x1 , x2 , ... , xn }. Since Y0 is a
precompact convex subset of Y, the convex hull of Y0 U {x1 , x 2 , ... , x n } is also a compact
convex subset of Y, and denote it by K. For each y ∈ X 1 , let T ( y ) = G ( y ) I K . Since G(y) is
intersectionally closed valued map of Y and K is a compact subset of Y, each T(y) is also
compact. Further, since G : X → 2 E is defined by G ( x) = G( x) for each x ∈ X and G is a
generalized F-KKM mapping with closed values. So, we can easily show that T is also a FKKM map. Therefore, by Theorem 5.3.1, we have I T ( y ) ≠ φ . Hence we have
x∈ X 1
φ ≠ I T ( x) = I G ( x) I K
x∈ X 1
x∈ X 1
⊂ I G ( x0 ) I G ( x1 ) I ... I G ( x n ).
x0 ∈ X 0
= I G ( x0 ) I G ( x1 ) I ... I G ( x n )
x0 ∈ X 0
n
Let C denotes the compact set I G ( x 0 ). Then we have I G ( xi ) I C ≠ φ for every finite
i =1
x0 ∈ X 0
subset {x1 , x2 , ... , xn } ⊂ X . Since each G (x) is an intersectionally closed valued map of Y and
C is a compact subset of Y, each G ( x) I C is also a compact subset of Y. Since the family
{G ( x) I C | x ∈ X } has the finite intersection property, we have
I G ( x) I C = I G ( x) ≠ φ .
x∈ X
x∈ X
This completes the proof.
Following Corollary is the special case of Theorem 5.3.1.
Corollary 5.3.2. Let X be a nonempty convex subset of a Hausdorff topological vector space
E and maps F , G : X → 2 E with G an intersectionally closed valued map such that
139
G ( x 0 ) = K is compact for at least one x0 ∈ X and let G : X → 2 E be a generalized F-KKM
mapping. Then I G( x) ≠ φ .
x∈ X
Proof. Since G : X → 2 E is defined by G ( x) = G( x) for each x ∈ X , we have that G is a
generalized F-KKM mapping with closed values. By Theorem 5.3.1 the family of sets
{G ( x) : x ∈ X } has the finite intersection property. Since G( x0 ) is compact, we have
I G ( x) ≠ φ .
x∈ X
Since G is intersectionally closed valued mapping
I G ( x) = I G ( x) ≠ φ .
x∈ X
x∈ X
Notice that if F : X → E be a single valued map and F ( xi ) = yi for each i ∈ {1, 2, ... , n}, then
Corollary 5.3.1 contains the result of Ansari et al [179, Th. 2.1].
Corollary 5.3.3. [179]. Let X be a nonempty convex subset of a Hausdorff topological vector
space E and map G : X → 2 E with G a transfer closed valued map such that G ( x 0 ) = K is
compact for at least one x0 ∈ X and let G : X → 2 E be a generalized KKM mapping. Then
I G ( x) ≠ φ .
x∈ X
The relation between generalized F-KKM mappings and S − γ − generalized quasi-convexity
(quasi-concavity) is the following.
Theorem 5.3.3. Let X be a nonempty convex subset of a topological vector space E. Let
φ : X × X → (−∞, ∞), γ ∈ (−∞, ∞) and F : X → 2 X . Then the following are equivalent
(i)
The mapping G : X → 2 X defined by
G ( y ) = {x ∈ X : φ ( x, z ) ≤ γ , for all z ∈ F ( x )}
[resp. G ( y ) = {x ∈ X : φ ( x, z ) ≥ γ , for all z ∈ F ( x )}
for each y ∈ X is a generalized F-KKM mapping.
(ii)
φ ( x, y ) is
S − γ − generalized
quasi-concave,
quasiconvex] in y.
140
[resp.
S − γ − generalized
Proof. For the sake of simplicity, we prove the conclusion only for the first case in (i) and (ii).
The other case can be proved similarly.
(i ) ⇒ (ii ). Since G : X → 2 X is a generalized F-KKM mapping, for any finite set
{x1 , x2 , ... , xn } ⊂ X ,
and
{i1 , ... , ik } ⊂ {1, 2, ... , n},
each
k
we
have
k
k

k

co U F ( xi j ) ⊂ U G ( xi j ) and hence for any y 0 ∈ co U F ( xi j ), y 0 ∈ U G ( xi j ). So there
 j =1
 j =1
 j =1

j =1
exists some m ∈ {1, 2, ... , k}, such that y 0 ∈ G ( y im ), so we have ϕ ( y 0 , z im ) ≤ γ for all
z im ∈ F ( x im ). Therefore, we have
min ( x0 , z i j ) ≤ γ for all z im ∈ F ( x im ).
1≤ j ≤ k
i.e. ϕ is F- γ − generalized quasi-concave in y.
(ii ) ⇒ (i ). Since ϕ is F- γ − generalized quasi-concave in y, for any finite set
k
{x1 , x2 , ... , xn } ⊂ X , each {i1 , ... , ik } ⊂ {1, 2, ... , n}, and any y0 ∈ co  U F ( xi j ),
 j =1

minϕ ( y0 , z i j ) ≤ γ for all z im ∈ F ( x im ).
1≤ j ≤ k
Hence there exists some m : 1 ≤ m ≤ k , such that ϕ ( y 0 , z im ) ≤ γ for all z im ∈ F ( x im ). This
k
shows that y 0 ∈ G ( x im ) and hence y 0 ∈ U G ( xi j ).
j =1
k
By the arbitrariness of y 0 ∈ co  U F ( xi j )  , we have
 j =1

k
k

co U F ( xi j ) ⊂ U G ( xi j ).
j
=
1

 j =1
This implies that G : X → 2 X is a generalized F-KKM mapping.
Let
X =E
be
a
G ( y ) = { x ∈ E : φ ( x, y ) ≤ γ }
topological
vector
space,
G : X → 2E
(resp. G ( y ) = {x ∈ E : φ ( x, y ) ≥ γ }) for
each
defined
y ∈ E be
by
a
generalized KKM mapping, ψ ( x, y ) is a γ − generalized quasiconcave (resp., convex) function
in y and F : X → E be a single valued map, F ( xi ) = yi for each i ∈ {1, 2, ... , n}, then Theorem
5.3.2 implies Proposition 3.1 of Ansari et al [179].
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Corollary 5.3.4 [179]. Let X be a nonempty convex subset of a topological vector space E.
Let φ : X × X → (−∞, ∞) and γ ∈ (−∞, ∞). Then the following are equivalent.
The mapping G : X → 2 X defined by
(ii)
G ( y ) = {x ∈ X : φ ( x, y ) ≤ γ } [resp. G ( y ) = {x ∈ X : φ ( x, y ) ≥ γ } ]
is a generalized KKM mapping.
φ ( x, y ) is γ − generalized quasi-concave, [resp. γ − generalized quasiconvex] in y.
(iii)
Some Applications
As applications of the above results we present minimax and saddle point results.
First we state and prove the following general minimax inequality.
Theorem 5.3.4. Let X be a nonempty closed convex subset of a Hausdorff topological vector
space
E
and
F : X → 2X.
Let
γ ∈ (−∞, ∞) be
given
number
and
maps
φ , ψ : X × X → (−∞, ∞) satisfy the following conditions.
(i)
For any fixed y ∈ X , φ ( x, y ) is a γ − generally lower semicontinuous function in
x.
(ii)
For any fixed x ∈ X , ψ ( x, y ) is a F- γ − generalized quasiconcave function in y.
(iii)
φ ( x, y ) ≤ ψ ( x, y ) for all ( x, y ) ∈ X × X .
(iv)
The set {x ∈ X : φ ( x, y0 ) ≤ γ } is precompact for at least one y0 ∈ X . Then there
exists x ∈ X such that inf sup φ ( x , z ) ≤ γ .
x∈ X z∈F ( x )
Proof. Define maps T , G : X → 2 X by
T ( y ) = {x ∈ X : ψ ( x, z ) ≤ γ for all z ∈ F ( x)} and
G ( y ) = {x ∈ X : φ ( x, z ) ≤ γ for all z ∈ F ( x)}.
Condition (i) implies that G is an intersectionally closed-valued mapping on X. Indeed, if
x ∉ G ( y ), then φ ( x, z ) > γ . Since φ ( x, y ) is γ − generally lower semicontinuous function in x,
there exists a y′ ∈ X and a neighborhood N(x) of x such that
φ (r , y ′) > γ , for all r ∈ N (x).
Then G ( y′) ⊂ X \ N ( x ). Hence x ∉ G ( y ′). Thus G is intersectionally closed-valued. From
condition (ii) and Theorem 5.3.1, T is a generalized F-KKM mapping. From condition (iii), we
142
have
T ( y ) ⊂ G ( y ), and hence G is also generalized F-KKM mapping. So G is also
generalized F-KKM mapping. By condition (iv) G( y0 ) is precompact. Hence, G( y0 ) is
compact.
Now by Theorem 5.3.2., it follows that I G( x) ≠ φ . Therefore, there exists x ∈ X such that
x∈ X
φ ( x , z ) ≤ γ , for all z ∈ F ( X ). In particular, we have inf sup φ ( x , z ) ≤ γ .
x∈ X z∈F ( x )
This completes the proof.
It is remarked that If F : X → X , F ( xi ) = yi for each i ∈ {1, 2, ... , n} and ψ ( x, y ) is a
γ − generalized quasiconcave function in y. Then Theorem 5.3.3 implies Theorem 3.1 of
Ansari et al [179].
Corollary 5.3.5 [179]. Let X be a nonempty closed convex subset of a Hausdorff topological
vector space E. Let γ ∈ (−∞, ∞) be given number, and maps φ , ψ : X × X → (−∞, ∞) satisfy
the following conditions.
(v)
For any fixed y ∈ X , φ ( x, y ) is a γ − transfer lower semicontinuous function in x.
(vi)
For any fixed x ∈ X , ψ ( x, y ) is a γ − generalized quasiconcave function in y.
(vii)
φ ( x, y ) ≤ ψ ( x, y ), for all ( x, y ) ∈ X × X .
(viii)
The set {x ∈ X : φ ( x, y 0 ) ≤ γ } is precompact for at least one y0 ∈ X . Then there
exists x ∈ X such that
inf sup φ ( x , y ) ≤ γ , for all y ∈ X .
x∈ X z∈F ( x )
We now present the following saddle point results. First, we present a saddle point theorem
for γ − generally lower semicontinuous (upper) function in x and F- γ − generalized
quasiconcave (convex) function in a Hausdorff topological vector space E using Theorem
5.3.4.
Theorem 5.3.5. Let X be a nonempty closed convex subset of a Hausdorff topological vector
space E and F : X → 2 X . Let γ ∈ (−∞, ∞) be given number, F : X → 2 X a surjective mapping
and let φ : X × X → (−∞, ∞) satisfy the following conditions.
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(i) φ ( x, y ) is a γ − generally lower semicontinuous function in x and F- γ − generalized
quasiconcave function in y.
(ii) φ ( x, y ) is a γ − generally upper semicontinuous function in y and F- γ − generalized
quasiconvex function in x.
(iii) There exists x1 , y1 ∈ X such that the sets
{x ∈ X : φ ( x, y1 ) ≤ γ } and { y ∈ X : φ ( x1 , y ) ≥ γ } are precompact.
Then, there exists a saddle point of φ ( x, y ); that is, there exists ( x , y ) ∈ X × X such that
φ ( x , y ) ≤ φ ( x , y ) ≤ φ ( x, y ), for all x, y ∈ X .
Moreover, we also have
inf sup φ ( x, y ) = sup inf φ ( x, y ) = γ .
x∈ X y∈X
y∈ X x∈ X
Proof. By Theorem 5.3.4 with φ = ψ , there exists x ∈ X such that
φ ( x , z ) ≤ γ , for all z ∈ F ( y ).
Since F is surjective, then
sup inf φ ( x , z ) = sup inf φ ( x , y ) and hence
y∈ X Z ∈F ( y )
y∈ X x∈ X
φ ( x , y ) ≤ γ , for all y ∈ X .
(5.3.7)
Let f : X × X → (−∞, ∞) be defined as f ( y, x) = −φ ( x, y ). By assumption (ii) f ( x, y ) is
γ − generally lower semicontinuous function in x and F − γ generalized quasiconcave
function in y. Therefore, again by Theorem 5.3.4, there exists y ∈ X such that
sup inf φ ( z , y ) = − inf sup f ( y , z )
y∈ X z∈F ( x )
z∈F ( x ) y∈ X
Note that F is surjective. Then
inf sup φ ( z , y ) = inf sup φ ( x, y )
z∈F ( x ) y∈ X
x∈ X y∈ X
and hence
φ ( x, y ) ≥ γ , for all x ∈ X .
(5.3.8)
Combining (5.3.7) and (5.3.8) we have φ ( x , y ) = γ and
φ ( x , y ) ≤ φ ( x , y ) ≤ φ ( x, y ), for all x, y ∈ X .
Finally, again by (5.3.7) and (5.3.8), we get
sup inf φ ( x, y ) ≤ inf sup φ ( x, y )
y∈ X x∈ X
x∈ X y∈X
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(5.3.9)
≤ sup φ ( x , y ) ≤ φ ( x , y ) ≤ inf φ ( x, y ) by (5.3.9)
x∈ X
y∈ X
≤ sup inf φ ( x, y ).
y∈ X x∈ X
Consequently,
inf sup φ ( x, y ) = sup inf φ ( x, y ) = γ .
x∈ X y∈X
y∈ X x∈ X
This completes the proof.
Let φ ( x, y ) be a quasiconcave function in y and γ − generalized quasiconvex function in x,
F : X → X , F ( xi ) = yi for each i ∈ {1, 2, ... , n}. Then Theorem 5.3.4 implies Theorem 3.2 of
Ansari et al [179].
Corollary 5.3.6 [179]. Let X be a nonempty closed convex subset of a Hausdorff topological
vector space E. Let γ ∈ (−∞, ∞) be given number. Suppose that φ : X × X → (−∞, ∞) satisfy
the following conditions.
(i) φ ( x, y ) is a γ − transfer lower semicontinuous function in x and γ − generalized
quasiconcave function in y.
(ii) φ ( x, y ) is a γ − transfer upper semicontinuous function in y and γ − generalized
quasiconvex function in x.
(iii) There exists x1 , y1 ∈ X such that the sets
{x ∈ X : φ ( x, y1 ) ≤ γ } and { y ∈ X : φ ( x1 , y ) ≥ γ } are precompact.
Then, there exists a saddle point of φ ( x, y ); that is, there exists ( x , y ) ∈ X × X such that
φ ( x , y ) ≤ φ ( x , y ) ≤ φ ( x, y ), for all x, y ∈ X .
Moreover, we also have
inf sup φ ( x, y ) = sup inf φ ( x, y ) = γ .
x∈ X y∈X
y∈ X x∈ X
Now we present the saddle point theorem for parametric games in topological vector spaces.
Theorem 5.3.6. Let E and F be two Hausdorff topological vector space and let X ⊂ E and
Y ⊂ F be two nonempty closed convex subsets. Let γ ∈ (−∞, ∞) be given number, F be a
surjective mapping and let Gθ = f − θ g : X × Y → R satisfy the following conditions.
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(i) f ( x, y ) is a γ − generally lower semicontinuous function in x and F- γ − generalized
quasiconvex function in y.
(ii) f ( x, y ) is a γ − generally upper semicontinuous function in y and F- γ − generalized
quasiconcave function in x.
(iii) g ( x, y ) is a γ − generally lower semicontinuous function in y and F- γ − generalized
quasiconvex function in x.
(iv) g ( x, y ) is a γ − generally upper semicontinuous function in x and F- γ − generalized
quasiconcave function in y.
(v) There exists x1 , y1 ∈ X such that the sets
{x ∈ X : Gθ ( x, y1 ) ≤ γ } and { y ∈ X : Gθ ( x1 , y) ≥ γ } are precompact.
Then, there exists a saddle point of Gθ ( x, y); that is, there exists ( x , y ) ∈ X × X such that
Gθ ( x , y ) ≤ Gθ ( x , y ) ≤ Gθ ( x, y ), for all x, y ∈ X .
Moreover, we also have
inf sup Gθ ( x, y ) = sup inf Gθ ( x, y ) = γ .
x∈ X y∈ X
y∈ X x∈X
Proof. Since θ ≥ 0, from (ii) and (iii) in the theorem, it follows that the function Gθ ( x, y) is
γ − generally upper semicontinuous function in y and F- γ − generalized quasiconcave
function in x. Similarly Gθ ( x, y) is γ − generalized lower semicontinuous function in x and F-
γ − generalized quasiconvex function in y. Then from Theorem 5.4.5, Gθ ( x, y) has a saddle
point ( x , y ) ∈ X × Y .
The following result is derived from Theorem 5.3.5.
Corollary 5.3.7. Let E and F be two Hausdorff topological vector spaces and let X ⊂ E and
Y ⊂ F be two nonempty closed convex subsets. Let γ ∈ (−∞, ∞) be given number and let
Gθ = f − θ g : X × Y → R satisfy the following conditions.
(i) f ( x, y ) is a γ − transfer lower semicontinuous function in x and γ − generalized
quasiconvex function in y.
(ii) f ( x, y ) is a γ − transfer upper semicontinuous function in y and γ − generalized
quasiconcave function in x.
(iii) g ( x, y ) is a γ − transfer lower semicontinuous function in y and γ − generalized
quasiconvex function in x.
146
(iv) g ( x, y ) is a γ − transfer upper semicontinuous function in x and γ − generalized
quasiconcave function in y.
(v) There exists x1 , y1 ∈ X such that the sets
{x ∈ X : Gθ ( x, y1 ) ≤ γ } and { y ∈ X : Gθ ( x1 , y) ≥ γ } are precompact.
Then, there exists a saddle point of Gθ ( x, y); that is, there exists ( x , y ) ∈ X × X such that
Gθ ( x , y) ≤ Gθ ( x , y ) ≤ Gθ ( x, y ), for all x, y ∈ X .
Moreover, we also have
inf sup Gθ ( x, y ) = sup inf Gθ ( x, y ) = γ .
x∈ X y∈ X
y∈ X x∈X
As an illustration we give the following example.
Example 5.3.1. Consider the following payoff matrix
 3 1

F = 
 2 4
Let the strategies p and q of the players ‘1’ and ‘2’ be given respectively by p = ( x, 1 − x) and
q = ( y, 1 − y ), where x, y ∈ [0, 1] . Then we have
3 1   y 
 

F ( p, q ) = ( x, 1 − x ) 
 2 4  1 − y 
= 3 xy + x(1 − y ) + 2(1 − x ) y + 4(1 − x )(1 − y )
= 4 xy − 3 x − 2 y + 4 = f ( x, y ).
Now we have function f ( x, y ) = 4 xy − 3x − 2 y + 4, x, y ∈ [0, 1], whose graph is given like
this:
4
3
1
2
0.8
1
0
0.6
0.4
0.2
0.4
0.2
0.6
0.8
1 0
147
This function is continuous so both upper and lower semicontinuous and convex so both
quasiconvex and quasiconcave.
∂

∂
f ( x, y ),
f ( x, y )  = (4 y − 3, 4 x − 2), which is 0 for
Now the slope of this function is 
∂y
 ∂x

x = 1 / 2 and
y = 3 / 4. Now
f (1 / 2 + ε , 3 / 4 + δ ) = 5 / 2 + 4εδ
yields that the point
(1 / 2, 3 / 4) is a saddle point and the value of the game is F ( p, q) = 5 / 2.
148