Download On Nash Equilibrium of the Abstract Economy or Generalized

On Nash Equilibrium of the Abstract Economy or Generalized
Game in H-Spaces
，
Xie Fang Xu Yuguang
Department of Mathematics, Kunming Teacher’s College,
Kunshi Road No. 2, Kunming 650031, P. R. China
Abstract The purpose of this paper is to introduce the H-space and to establish a theorem for
the Nash equilibrium of a generalized game (or an abstract economy). The result presented in this paper,
modify and extend the corresponding results given in the literatures, so that this new theorem are
applied to the equilibrium problems of the game (or the economy) in H-space.
Key words H-space; H-convex subset; H-quasi-convex function; H-quasi-concave function;
Nash equilibrium
1. Introduction and Preliminaries
A game is a situation in which several players each have partial control over some outcome and
generally have conflicting preferences over the outcome. The set of choices under player i ’s control is
denoted X i . Elements of X i are called strategies and X i is i ’s strategy set. Letting I denote
the set of players, X =
∏
i∈I
X i is the strategy vectors. We will describe an generalized qualitative
game or abstract economy by B = X i , Ai , f i : i ∈ I
Where I is finite or infinite ( countable or
uncountable ) set of players and agents, and for each i ∈ I , X i is the non-empty set of actions, the
Ai : X = ∏i∈I X i → 2 X i is the preference correspondence (set valued
strategy set or choice set;
mapping) and f i : X → R is the utility or pay off function. In generally, X i will be a subset of a
topological vector space for each i ∈ I .
i
A point x = ( xi , x ) ∈ X i ×
∏
i≠ j
X j is called an equilibrium point of B if
f i ( x ) = sup ui ∈Ai ( x ) f i (ui , x i ) ∀ i ∈ I .
The definitions of an abstract economy and an equilibrium coincide with the standard ones and for
further information of this topic, the reader is referred to Shafer-Sonnenschein[1].
In 1950, J. Nash proves the existence of equilibrium for games where the player’s preferences are
representable by continuous quasi-concave utilities and the strategy sets are simplexes. Debreu, in 1952,
he proves the existence of equilibrium for the generalized qualitative game or abstract economy. He
assumes that strategy sets are contractible polyhedral and that the feasibility correspondences have
closed graph, and the maximized utility is continuous and that the set of utility maximizes over each
constraint set is contractible.
In 1975, an important theorem is proved by Shafer and Sonnenschein as follows.
Theorem S-S.
Let B = X i , Ai , f i : i ∈ I
be an abstract economy. For each i ∈ I the
conditions as follows are satisfied
(i) X i ⊂ R is nonempty, compact and convex;
(ii) Ai : X → 2
Xi
is a continuous correspondence with nonempty compact convex values;
(iii) Gr f i is open in X × X i ;
(iv) xi ∉ co f i (x ) for all x ∈ X
then there is an equilibrium point of B .
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The purpose of this paper is to introduce the H-space and to establish an existence theorem for the
Nash equilibrium of a generalized game (or an abstract economy). The result, presented in this paper,
modify and extend the corresponding results given in the literatures so that it are applied to more
general equilibrium problems of the game (or the economy) in H-space.
To set the framework, we recall the notation of H-space introduced by C. Horvath[6] in 1987 as
follows.
Definition 1.1.
Let X be a topological space and F ( X ) a family of non-empty finite
subsets of X . Let {ΓA } be a family of non-empty contractible subsets of X
indexed by
A∈ F ( X ) such that ΓA ⊂ ΓA′ whenever A ⊂ A′ . The pair ( X , {ΓA }) is called an H-space.
Definition 1.2. Let {( X i , { Γ A i }) : i ∈ I } be a family of H-spaces where I is a
finite or infinite index set. It is Obvious that the Cartesian product X :=
still an H-space(see Lemma 1.1 in [3]).
Definition 1.3.
The set C ⊂ X is called H-convex in
∏{( X , {Γ
i
( X , {ΓA } )
non-empty finite subset A ⊂ C .
Definition 1.4.
Let ( X , {ΓA } ) be an H-space and Y
Ai
}) : i ∈ I } is
if ΓA ⊂ C for every
a topological space. Let
f : X × Y → R ∪ {± ∞} be a functional and λ ∈ R . For each fixed y ∈ Y , f ( x, y ) is called
H-quasi-convex (or H-quasi-concave) in X if and only if the set
{x ∈ X : f (x, y ) < λ} (or {x ∈ X : f (x, y ) > λ} )
is H-convex for any λ ∈ R .
Throughout this paper, all topological space are considered as Hausdorff spaces.
2. Nash equilibrium of the abstract economy or generalized game in
H-space
In order to prove our main theorem, we first state a lemma which had been proved in [3].
Lemma 2.1.
Let ( X α , {ΓAα } ) : α ∈ I
be a family of compact H-spaces, where
{
}
A∈ F ( X ) and I is an index set containing at least two elements. Denote
X = ∏ {X α : α ∈ I } and X α = ∏ { X β : β ∈ I , β ≠ α }.
For each
α ∈ I , tα ∈ R , let fα : X → R
be a functional such that
x = (xα , x )∈ (X α , X α ) , t h e r e i s a yα ∈ X α s u c h t h a t
α
( 2.1. 1) for ea ch
f α ( yα , x α ) > tα ;
(2.1.2)
f α (•, y α ) is H-quasi-concave for each fixed y α ∈ X α ;
(2.1.3)
f α ( xα , • ) is lower semi-continuous for each fixed xα ∈ X α .
(
Then there exists a x = xα , x
Lemma 2.2.
elements.
{
Suppose
α
)∈ X such that f (x ) > t
α
α
for each
α∈I .
Let X be a topological space and I an index set containing at least two
that
{( X i , {ΓAi }) : i ∈ I }
f i : X = Π j∈I X j → R : i ∈ I
}
is
a
family
of
compact
is a family of continuous functional. If
H-quasi-concave in xi for each i ∈ I , then there exists a point x ∈ X such that
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H-spaces
and
f i (xi , x i ) is
(
f i ( x ) = max f i ui , x i
ui ∈ X i
for all i ∈ I .
Proof. Setting
)
Fi ( x ) = f i ( x ) − max f i (ui , x i )
(
i
ui ∈X i
)
for all i ∈ I and all x = xi , x ∈ X , then, for each i ∈ I by the uniform continuity of f i on X ,
(
we can easily see that max ui ∈ X i f i ui , x
i
) is a continuous function on
i
X . Consequently, Fi ( x ) is
(
i
continuous on X . Now, for each i ∈ I and for any fixed y ∈ X , we have f i xi , y
(
H-quasi-concave in xi . Hence, Fi xi , y
i
)
i
)
is
is H-quasi-concave as well. In view of the compactness of
X i , there exist an ui ∈ X i such that max ui ∈X i f i (ui , y i ) = f i (ui , y i ) . Consequently, we obtain
(
i
)
that Fi ui , y > −ε for each
ε > 0 . By virtue of Lemma 2.1, there exists an x ∈ X
Fi ( x ) > −ε ∀i ∈ I .
such that
Putting
Ei (ε ) = {x ∈ X : Fi ( x ) ≥ −ε }
for all i ∈ I , and all ε > 0 , from the continuity of Fi , we know that Ei (ε ) is a non-empty
compact subset of X . If ε1 < ε 2 , then it is clear that E (ε1 ) ⊂ E (ε 2 ) . Therefore, we have
I {E (ε ) : ε ∈ R + }≠ φ . It follows that there exsits x = (xi , x i )∈ X such that Fi (x ) ≥ 0 for
each i ∈ I , i.e.,
f i ( x ) = max f i (ui , x i )
∀i ∈ I .
□
ui ∈X i
This completes the proof.
Theorem 2.3.
Let B = { X i , Ai , f i : i ∈ I } be an abstract economy or generalized
qualitative game and I an index set containing at least two elements. For each i ∈ I , if the following
conditions hold :
(2.3.1)
( X , {Γ } ) is a compact H-space;
i
i
A
(2.3.2) Ai ( x ) = X i ,
∀x ∈ X ;
(2.3.3) f i : X → R is continuous on X and H-quasi-concave on X i ;
then there exists an equilibrium point of B .
Proof.
The conclusion of Theorem 2.3 comes from Lemma 2.2. In fact, X =
∏X
i
also is a
i∈I
compact H-space, { Ai : i ∈ I } is a family of continuous correspondence with nonempty compact
(
convex values and f i xi , x
(
i
)
is H-quasi-concave in xi for each i ∈ I . It follows Lemma 2.2 that
i
)
there exists a point x = xi , x ∈ X such that
f i ( x ) = max f i (ui , x i )
ui ∈X i
∀i ∈ I .
I.e., x is an equilibrium point of B = { X i , Ai , f i : i ∈ I } .
□
This completes the proof.
Remark. Theorem 2.3 is an existence theorem of the equilibrium with infinte agents on the strategy
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spaces without linear structure. Hence, it extend the corresponding results of J.P.Aubin [4] and Ky Fan
[5].
References
[1]
W. Shafe and H. Sonnenschein, Equilibrium in abstract economics without ordered
preferences, J. Math. Economics, 1975, 2: 345 348
[2] C. Horvath, Some results on multi-valued mappings and inequalities without convexity in
Nonlinear and Convex Analysis, Lecture Notes in Pure and Appl. Math., Series V.107.
Springer Verlag, 1987: 1257
.
[3] E. Tarafdar, A fixed point theorem in H-space and related results, Bull. Austral. Math.
Soc. , 1990, 42: 133 140
[4] J. P. Aubin, Mathematical methods of game and economic theory, North-Holland Publishing
Company, Oxford, 1979: 263 292
[5] Fan, K., Applications of a theorem concerning sets with convex sections, Math Annalen.,
1966, 163: 189 203
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