Download An Approximate Equilibrium Theorem of the Generalized Game

An Approximate Equilibrium Theorem of the Generalized Game
or an Abstract Economy
Xu Yu-Guang1, Liu Xi-Biao2
1. Department of Mathematics, Kunming Teacher's College, Kunshi Road No: 2,
Kunming 650031, P. R. China, E-mail: [email protected]
2. Department of Finance, Yunnan University of Finance and Economics,
Kunming 650221, P. R. China
Abstract The purpose of this paper is to establish a sufficient condition for the existence of the
approximate equilibrium of a generalized game(or an abstract economy). The results presented in this
paper show that if F is almost lower semicontinuous then F has a continuous approximate
selection so that a new almost fixed point theorem is obtained and these results are applied to the
approximate equilibrium of game(or economy).
Key words abstract economy, generalized game, approximate equilibrium, continuous approximate
selection, almost fixed point
1. Introduction
A generalized game(or an abstract economy) Γ = X i , Ai , Bi , Pi
ordered quadruples
i∈I
is defined by a family of
X i , Ai , Bi , Pi , where I be a finite or infinite set of agents, X i is a
non-empty set of actions or a choice set(a topological space), Ai , Bi : X :=
constraint correspondences and Pi : X → 2
Xi
∏
j∈I
X j → 2 X i are
is a preference correspondence. An equilibrium for
Γ is a point x = ∏i∈I xi ∈ X such that xi ∈ Bi (x ) and Pi ( x ) Ι Ai ( x ) = φ . If Ai = Bi
and X i is a topological vector space for each i ∈ I then our definitions of an abstract economy
and an equilibrium coincide with the standard ones of Shafer-Sonnenschein[1].
Now, we describe a two-person game by Β = { X i , Fi , f i : i = 1, 2} , where X i is a strategy set,
f i : X → R is an utility function and Fi : X j → 2 X i ( i = 1, 2 and i ≠ j ) defined by
Fi ( x j ) = {xi ∈ X i : f i ( xi , x j ) = inf ui ∈X i f (u i , x j )} is said the constrained correspondence
for the i -th person.
Definition 1. Let Β = { X i , Fi , f i : i = 1, 2} is a two-person game. A point xi =
( x1 , x 2 ) ∈ X = X 1 × X 2 is called an ε -almost equlibrium point of game Β if there exists for
+
+
any given ε = (ε 1 , ε 2 ) ∈ R × R , V = V1 × V2 ∈ O (θ ) satisfied that
(2.1) xi ∈ Fi ( x j ) + Vi ;
(2.2) f i ( xi , x j ) − inf ui ∈X i f i ( ui , x ) < ε i
for all i (i = 1, 2) .
Throughout this paper, suppose that the topological spaces of all are Hausdorff and a subset of a
topological space is considered to have relative topology. O (x ) is denoted the collection of all
neighborhoods of x where x ∈ X or x ∈ Y .
* This work is supported by the National Natural Science Foundation of China(No.19961004) and the of
Foundation Yunnan Sci. Tech. commission(No. 2002A0058M)
The purpose of this paper is to establish a sufficient condition for the existence of the approximate
equilibrium of an abstract economy(or generalized game). The results presented in this paper show
that if F is almost lower semicontinuous then F has a continuous approximate selection so that a
new almost fixed point theorem is obtained and these results are applied to the approximate
equilibrium of game or economy.
2. The approximate equilibrium of the generalized game(or abstract economy)
In order to give our results, we need the following definitions.
Definition 2. Suppose that X is a nonempty subset of a topological space E , Y is a
topological vector space and F : X → 2 is a multivalued mapping with nonempty values. Let θ
be the origin of Y .
(1.1) F is called lower semicontinuous (abbreviated as 1.s.c.) at x ∈ X if for any open subset
V ⊂ Y with F ( x ) ∩V ≠ φ there exists a neighborhood N (x) of x such that
Y
F (u ) ∩ V ≠ φ for all u ∈ N ( x). F is called a 1.s.c. mapping if F is 1.s.c. at each
point of X .
(1.2) F is called almost lower semicontinuous (abbreviated as a.1.s.c.) at x ∈ X if for any open
N ( x) of x such that
subset V ∈ O ( x ) there exists a neighborhood
Ι {F (u ) + V : u ∈ N ( x )} ≠ φ .
F is called an a.1.s.c. mapping if F is a.1.s.c. at each point of X .
(1.3) A continuous function f : X → Y is called a continuous approximate selection of F if
f ( x ) ∈ F ( x) for all x ∈ X .
*
*
*
(1.4) For any given V ∈ O(θ ), If there exists an x ∈ X such that F ( x ) Ι ( x + V ) ≠ φ
*
then x is said a V -almost fixed point of F . If there exists for any given V ∈ O(θ ), an
V -almost fixed point then F has the almost fixed points.
Remark 1. In 1956, E. Michael[2,3] proved well-known Michael selection theorem: every lower
semicontinuous multivalued mapping F admits a continuous selection. But the lower semicontinuity
of F is not a necessary condition for the existence of continuous selection. In fact, it is often absent
in many given problems. This has forced some researchers to establish new continuous selection
theorems in more general settings(see, for example [4-7]). Sometimes, there is not any continuous
selection if F is not lower semicontinuous, but F may have a continuous approximate selection. For
R
example, we define F : R → 2 by
if x is rational ;
 [−1, 0)
F ( x) = 
if x is irrational.
 (0, 1]
It is clear that F is a.l.s.c. but F is not l.s.c. on R . Furthermore, F has not any continuous
selection but f ≡ 0 is a continuous approximate selection of F .
Lemma 1. Let X be a paracompact subset of the topological space E and Y a locally
Y
convex topological vector space. Suppose that F , G : X → 2 are two multivalued mappings
satisfied that G (x) ≠ φ and coG ( x ) ⊂ F ( x) for each x ∈ X . If G is a.l.s.c. then F has
a continuous approximate selection.
Proof.
For any V ∈ O (θ ) ( θ ∈ Y ), there is a convex neighborhood V ∈ O(θ ) such that
*
V * ⊂ V . Since G is a.l.s.c., for each x ∈ X there exists an U ∈ O (x) such that
Ι {G ( z ) + V * : z ∈ U } ≠ φ .
To pick out an U ( x ) ∈ O ( x ) for each x ∈ X , the family of subsets {U ( x ) : x ∈ X } is an
open cover of X . By the paracompactness of X there is a locally finite refinement
Ω := {U α : α ∈ I } where I is an index set and a partition of unity {g α : α ∈ I } subordinated
to Ω such that
(a) g α : X → [0, 1] is continuous for each α ∈ I ;
α ∈I ;
(b) {x ∈ X : gα ( x ) > 0} ⊂ U α , for each
(c)
∑α
∈I
gα ( x ) = 1 for each x ∈ X .
Since Ω is a refinement, there exists an xα ∈ X such that U α ⊂ U ( xα ) for each
α ∈I .
Now, we can choose a yα ∈ Ι {G ( z ) + V * : z ∈ U ( x)} to define a mapping f : X → Y by
f ( x) = ∑ g α ( x) yα
∀x∈ X .
α ∈I
Since Ω is locally finite, there are at most finitely many g α ( x) ≠ 0 , hence f is continuous. For
each
x∈ X
and any
α ∈I
g α ( x) ≠ 0 then x ∈ U α ⊂ U ( xα ) . Consequently,
, if
yα ∈ G ( x ) + V . It follows that
*
f ( x ) ∈ co[G ( x ) + V * ] ⊂ coG( x) + coV * ⊂ F ( x ) + V * ⊂ F ( x) + V ,
i.e., F has a continuous approximate selection. This completes the proof.□
Lemma 2. Suppose that X and Y are as in Lemma 1. Let F : X → 2 is a multivalued
mapping with nonempty convex values. If F is a.l.s.c. then F has a continuous approximate
selection.
Proof. Taking F = G in Lemma 1.□
Remark 2. We like to point out that the proof of Lemma 1 do not depend on any metric
topological, therefore, one has improved the condition of Lemma 4.1 in Michael[2] by the almost
lower semicontinuity of F instead of the lower semicontinuity of F .
Lemma 3. Let X be a compact convex subset of a locally convex topological vector space E .
Y
If F : X → 2 is an almost lower semicontinuous multivalued mapping with nonempty convex
values then F has the almost fixed points.
Proof. From the Lemma 2, for each U ∈ O(θ ) there is a continuous mapping f : X → X
X
such that f ( x ) ∈ F ( x ) + U for all x ∈ X . By virtue of the Tychonoff's fixed point Theorem,
there exists an x ∈ X such that f ( x ) = x . Consequently,
*
*
*
x* ∈ F (x* ) + U .
I.e., F has the almost fixed points.□
Remark 3. In Lemma 3, E is a topological vector space with uniform neighborhood system. In
fact, a topological vector space E may be equipped with uniform neighborhood system of origin of
E by means of a natural way. If we define
U = {( x, y ) : x − y ∈ V } ∀V ∈ O ( x) and ∀ x ∈ E
then {U : ∀V ∈ O( x), ∀ x ∈ E} is a filter on E × E . It lead to an uniformity structure on E
and so E has become an uniform space.
Lemma 4. Let X 1 and X 2 be compact locally convex topological vector space and
Fi : X j → 2 X i ( i = 1, 2 and i ≠ j ) is an almost lower semicontinuous mapping with
nonempty convex values. If defined F : X = X 1 × X 2 → 2
F ( x1 , x 2 ) = F1 ( x 2 ) × F2 ( x1 )
X
by
∀ ( x1 , x 2 ) ∈ X
then F has the almost fixed points.
Proof. It is easy to verify that F is an a..l.s.c. multivalued mapping with nonempty convex
values and X is a compact locally convex topological vector space. From the Lemma 3, F has
the almost fixed points.□
Theorem 5. Let Β = { X i , Fi , f i : i = 1, 2} be a two-person game. If the following conditions
are satisfied
(5.1) X 1 and X 2 are two compact locally convex topological vector spaces;
(5.2) F1 : X 2 → 2
F2 : X 1 → 2 X 2 are almost lower semicontinuous;
(5.3) f1 , f 2 : X → R are continuous, f1 (⋅, x 2 ) and f 2 ( x1 , ⋅) are quasiconvex.
+
+
Then Β has the ε -almost equilibrium point for any given ε = (ε 1 , ε 2 ) ∈ R × R .
Proof. From (5.1) and (5.3) we know that X is a compact space with the uniformity structure
+
+
and f i (i ∈ I ) is uniformly continuous. For any given ε = (ε 1 , ε 2 ) ∈ R × R and every
( p, q ) ∈ X × X , there exists a V = V1 × V2 ∈ O (θ ) such that p − q ∈ V implies
| f i ( p ) − f i (q ) | < ε i ( i ∈ I ). Setting
M i ( x j ) = {xi ∈ X i : ∃ qi ∈ Fi ( x j ) such that ( xi , x j ) − (qi , x j ) ∈ V }
X1
and
where i, j ∈ I and i ≠ j . If pi ∈ Fi ( x j ) + Vi , we may choose a qi ∈ Fi ( x j ) such that
pi − q i ∈ Vi ,. It follows from ( pi , x j ) − (qi , x j ) ∈ V
that
pi ∈ M i ( x j ) . hence,
Fi ( x j ) + Vi ⊂ M i ( x j ) so that
Fi ( x j ) = {xi ∈ X i : f i ( xi , x j ) ≤ inf ui ∈X i f i (u i , x j )} (i, j ∈ I , i ≠ j )
is nonempty convex. By the Theorem 4, we have a V -almost fixed point x ∈ X such that
( x + V ) Ι F ( x ) ≠ φ , which implies (2.1). Since xi ∈ M i ( x j ) , it turns out that (2.2) is true.
Hence, x is a
ε -almost equilibrium point of Β .□
References
[1] W. Shafer and H. Sonnenschein. Equilibrium in abstract economies without ordered preference. SIAM. J. Math. Anal., 1983, 14:185-194
[2] Michael E. Continuous selections I, Ann. Math. 1956. 63(2):361-381
[3] Michael E. Selected selection theorems, Amer. Math. Monthly. 1956. 63:233-237
[4] Deutsch H. and Kenderov P. Continuous selections and approximate selections for set-va
lued mappings and applications to metric projections, SIAM. J. Math. Anal., 1983, 14:18
5-194
[5] Hadzic O. Almost fixed point and best approximations theorems in H-spaces, Bull. Aust
ral. Math. Soc., 1996. 53:447-454
[6] Wu X. and Li F.Approximate selection theorems in H-spaces with applications, J. Math.
Anal. Appl. 1999. 231:118-132
[7] Xu YuGuang. A note on a continuous approximate selection theorem, J. Math. Appr.
Theory, 2001. 113:324-325