Weak local Nash equilibrium - part II
In the paper “Weak local Nash equilibrium” we define a concept of local equilibrium to non-cooperative games and we prove its existence applying the Lefschetz fixed point theorem. We was inspired by the original Nash’s theorem and his proof.
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2013
|
| Online Zugang: | https://trim.imath.kiev.ua/index.php/trim/article/view/307 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Завантажити файл: |  |
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552811951030272 |
|---|---|
| author | Biasi, C. Monis, T. F. M. Biasi, C. Monis, T. F. M. |
| author_facet | Biasi, C. Monis, T. F. M. Biasi, C. Monis, T. F. M. |
| author_institution_txt_mv | [
{
"author": "C. Biasi",
"institution": "Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo."
},
{
"author": "T. F. M. Monis",
"institution": "Instituto de Geociências e Ciências Exatas, Univ Estadual Paulista"
}
] |
| author_sort | Biasi, C. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | In the paper “Weak local Nash equilibrium” we define a concept of local equilibrium to non-cooperative games and we prove its existence applying the Lefschetz fixed point theorem. We was inspired by the original Nash’s theorem and his proof. |
| first_indexed | 2026-08-04T01:05:22Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 209–223
Carlos Biasi , Thaís F. M. Monis
Weak local Nash equilibrium - part II
biasi@icmc.usp.br , tfmonis@rc.unesp.br
In the paper “Weak local Nash equilibrium” we define a concept of local
equilibrium to non-cooperative games and we prove its existence applying
the Lefschetz fixed point theorem. We was inspired by the original Nash’s
theorem and his proof.
1 Introduction
In the paper “Weak local Nash equilibrium” we define a concept of
local equilibrium to non-cooperative games and we prove its existence
applying the Lefschetz fixed point theorem. We was inspired by the
original Nash’s theorem and his proof.
The concept of Nash equilibrium says that an equilibrium for payoff
functions
p1, p2, . . . , pn : S = S1 × S2 × · · · × Sn → R
is a point s̃ = (s̃1, s̃2, . . . , s̃n) ∈ S such that, for each i ∈ {1, 2, . . . , n},
pi(s̃1, . . . , s̃i−1, si, s̃i+1, . . . , s̃n) ≤ pi(s̃), for all si ∈ Si.
Nash proved that:
Theorem 1.1 (Nash’s Theorem). Let S1, . . . , Sn be compact convex sub-
sets of an Euclidean space. Suppose that p1, . . . , pn : S = S1×· · ·×Sn →
R are maps such that, for each i = 1, . . . , n, pi(s1, . . . , sn) is linear
(afim) as a function of si. Then there exists at least one equilibrium
to p1, . . . , pn.
c© Carlos Biasi , Tháis F. M. Monis , 2013
210 Weak local Nash equilibrium - part II
The proof is the following: let Si ⊂ Rdi , where di is the dimension of
Si. Thus, S ⊂ Rd, where d = d1 + · · · + dn. From the hypothesis, the
payoff functions are of the type
pi(s) = vi(s) · si + ui(s)
where vi : S → Rdi and ui : S → R are maps which don’t depend on the
coordinate si, i = 1, . . . , n. Let v : S → Rd be the vector field defined
by v(s) = (v1(s), . . . , vn(s)). Let r : Rn → S be the natural retraction
that assigns each point p ∈ Rn to the point r(p) ∈ S which realizes the
distance of p to S. Finally, let f : S → S be defined by f(s) = r(s+v(s)).
Then, one can shown that s̃ ∈ S is a Nash equilibrium to p1, . . . , pn if
and only if s̃ is a fixed point of f . Note that the existence of a fixed point
to f is assured by Brouwer’s fixed point theorem.
Based on the above proof, we investigated the existence of equilib-
rium in the context that the spaces of strategies are compact ENR’s, not
necessarily convex. This means that each space Si is a subset of some eu-
clidean space Rdi and there is an open neighborhood Vi of Si in Rdi and
a retraction ri : Vi → Si. From this research, the following definitions
arise.
Definition 1. Let (S1, d1), . . . , (Sn, dn) be metric spaces and p1, . . . , pn :
S1 × · · · × Sn → R real functions. We say that s̃ = (s̃1, . . . , s̃n) ∈ S is
a weak local equilibrium (abbrev., w.l.e.) for p1, . . . , pn if for all
ε > 0 there exists δ > 0 such that
pi(s̃1, . . . , s̃i−1, si, s̃i+1, . . . , s̃n) ≤ pi(s̃) + εdi(si, s̃i),
for every si ∈ B(s̃i, δ), i = 1, 2, . . . , n, where B(s̃i, δ) denotes the open
ball with center in s̃i and radius δ > 0 in (Si, di).
Definition 2. We say that a subset X of Rm has the property of
convenient retraction (abbrev., p.c.r.) if there exists a retraction
r : V → X, where V is an open neighborhood of X in Rm, satisfying:
given x0 ∈ V and ε > 0, there exists δ > 0 such that
〈x0 − r(x0), x− r(x0)〉 ≤ ε‖x− r(x0)‖,
for all x ∈ X with ‖x− r(x0)‖ < δ, where 〈 , 〉 is the usual inner product
in Rm and ‖ · ‖ is the norm induced by it. In this case, we say that
r : V → X is a convenient retraction.
Carlos Biasi , Thaís F. M. Monis 211
Example 1. Every closed convex subset K of Rm has the p.c.r.. In fact,
there is a natural retraction r : Rm → K such that to each x ∈ Rm
assigns the point r(x) ∈ K which realizes the distance of x to K. This
retraction satisfies 〈x0 − r(x0), x − r(x0)〉 ≤ 0 for every x0 ∈ Rm and
x ∈ K.
Example 2 ([3], Proposition 4.3). Every submanifold M of Rn, of class
C2, with or without boundary, has the p.c.r..
Let X be a closed subset of the Euclidean space Rn and let V be an
open neighborhood of X in Rn. A map r : V → X is called a proximative
retraction (or metric projection) if
‖r(y)− y‖ = dist(y,X), for every y ∈ V,
where
dist(y,X) = inf{‖x− y‖ | x ∈ X}
is the distance of y to X.
Evidently, every proximative retraction is a retraction map but not
conversely.
A compact subset K ⊂ Rn is called a proximative neighborhood re-
tract (written K ∈ PANR) if there exists an open neighborhood V of K
in Rn and a proximative retraction r : V → K.
We have the following statement:
Example 3 ([2]). Let K be a compact subset of Rn. If K ∈ PANR then
K is an ENR with the p.c.r..
In the previous paper, we was able to prove the following result.
Theorem 1.2 ([2]). Let p1, . . . , pn : S1 × . . . × Sn → R be maps,
where each Si ⊂ Rmi is a compact ENR with the p.c.r.. Also, suppose
pi(s1, . . . , sn) continuously differentiable in a neighborhood of si when
the other variables are kept fixed, i = 1, 2, . . . , n. If χ(Si) 6= 0 for
i = 1, 2, . . . , n then p1, p2, . . . , pn have at least one w.l.e..
Our goal in this paper is to prove a more general version of Theo-
rem 1.2 changing the hypothesis of the continuously differentiable on the
payoffs by a weaker hypothesis.
212 Weak local Nash equilibrium - part II
2 Preliminaires
In this section, we define a concept of an upper semi differen-
tiable(u.s.d.) function.
The open ball in Rn with center in x0 and radius r > 0 will be denoted
by B(x0, r).
Definition 3. Let f : A → R be a function, where A is an open non-
empty subset of Rn. Given x0 ∈ A, we say that f is upper semi differ-
rentiable(u.s.d.) at x0 if there exists at least one point v ∈ Rn together
with a function r : B(0, ε)→ R such that lim
h→0
r(h)
‖h‖
= 0 and
f(x0 + h) ≤ f(x0) + v · h+ r(h)
for every h such that x0 + h ∈ A.
We denote by DSf(x0) the set of such vectors v.
Example 4. If f : A→ R is differentiable at x0 then f is u.s.d.. More-
over, DSf(x0) = {f ′(x0)}. In fact, suppose v ∈ Rn and r : B(0, ε)→ R
such that lim
h→0
r(h)
h
= 0 and f(x0 + h) ≤ f(x0) + v · h+ r(h) for every h.
Thus, for 0 < t < ε,
f(x0 + tei)− f(x0)
t
≤ v · ei +
r(tei)
t
.
It follows that
∂f
∂xi
(x0) = lim
t→0+
f(x0 + tei)− f(x0)
t
≤ v · ei.
On the other hand, for −ε < t < 0,
f(x0 + tei)− f(x0)
t
≥ v · ei +
r(tei)
t
.
It follows that
∂f
∂xi
(x0) = lim
t→0−
f(x0 + tei)− f(x0)
t
≥ v · ei.
Therefore,
∂f
∂xi
(x0) = v · ei.
Thus, v = f ′(x0) =
(
∂f
∂x1
(x0), . . . ,
∂f
∂xn
(x0)
)
.
Carlos Biasi , Thaís F. M. Monis 213
The next result shows that the set DSf(x0) is convex.
Theorem 2.1. If f is u.s.d. at x0 then DSf(x0) is a convex subset of
Rn.
Proof. Let v1, v2 ∈ DSf(x0) be arbitraires and let r1, r2 : B(0, ε) → R
be such that
f(x0 + h) ≤ f(x0) + v1 · h+ r1(h)
f(x0 + h) ≤ f(x0) + v2 · h+ r2(h)
with lim
h→0
r1(h)
‖h‖
= lim
h→0
r2(h)
‖h‖
= 0.
Let v = αv1 + (1− α)v2, with α ∈ (0, 1). We have
f(x0 + h) = αf(x0 + h) + (1− α)f(x0 + h)
≤ αf(x0) + αv1 · h+ αr1(h) + (1− α)f(x0)
+(1− α)v2 · h+ (1− α)r2(h)
= f(x0) + v · h+ αr1(h) + (1− α)r2(h).
Since
lim
h→0
αr1(h) + (1− α)r2(h)
‖h‖
= α lim
h→0
r1(h)
‖h‖
+ (1− α) lim
h→0
r2(h)
‖h‖
= 0,
it follows that v ∈ DSf(x0).
Therefore, DSf(x0) is convex.
In the next theorems, we give conditions to DSf(x0) be compact.
Theorem 2.2. Let f : J → R be a function, where J ⊂ R is open
interval, and let x0 ∈ J . Suppose the existence of the right and left-hand
limits
c = lim
h→0+
f(x0 + h)− f(x0)
h
and
d = lim
h→0−
f(x0 + h)− f(x0)
h
Then, f is u.s.d. if and only if c ≤ d. Moreover, DSf(x0) = [c, d].
214 Weak local Nash equilibrium - part II
Proof. Suppose f u.s.d. at x0 and let v ∈ DSf(x0). If 0 < h < ε, we
have
f(x0 + h)− f(x0)
h
≤ v +
r(h)
h
,
following that c = lim
h→0+
f(x0 + h)− f(x0)
h
≤ v.
Analogously, if −ε < h < 0, we have
f(x0 + h)− f(x0)
h
≥ v +
r(h)
h
,
following that d = lim
h→0−
f(x0 + h)− f(x0)
h
≥ v.
Therefore, c ≤ d.
On the other hand, suppose c ≤ d. Note that, above, we show that
DSf(x0) ⊂ [c, d]. Now, to conclude thatDSf(x0) = [c, d], sinceDSf(x0)
is convex, it is sufficient to show that c, d ∈ DSf(x0).
Define r(h) =
f(x0 + h)− f(x0)− ch se h > 0
0 se h = 0
f(x0 + h)− f(x0)− dh se h < 0
Then lim
h→0
r(h)
h
= 0. Moreover, for h > 0, we have
f(x0 + h) = f(x0) + ch+ f(x0 + h)− f(x0)− ch
and, for h < 0, we have
f(x0 + h) = f(x0) + ch+ f(x0 + h)− f(x0)− ch ≤ f(x0) + ch+
+f(x0 + h)− f(x0)− dh
Therefore, c ∈ DSf(x0).
Analogously, for h > 0, we have
f(x0 + h) = f(x0) + dh+ f(x0 + h)− f(x0)− dh ≤ f(x0) + dh+
+f(x0 + h)− f(x0)− ch
Carlos Biasi , Thaís F. M. Monis 215
and for h < 0,
f(x0 + h) = f(x0) + dh+ f(x0 + h)− f(x0)− dh
Therefore, d ∈ DSf(x0).
Example 5. Let f : R → R be defined by f(x) =
{
x, if x < 0
−x, if x ≥ 0
.
The function f is u.s.d. at 0. In fact, we have
lim
h→0+
f(h)− f(0)
h
= −1 < 1 = lim
h→0−
f(h)− f(0)
h
Then, by Theorem 2.2, f is u.s.d. at 0 and DSf(0) = [−1, 1].
Notation: Let f : A → R be a map, where A is an open subset of
Rn. Let x0 ∈ A. We denote the right-hand partial derivatives and the
left-hand partial derivatives, respectively, by
∂f+
∂xi
(x0) = lim
t→0+
f(x0 + tei)− f(x0)
t
and
∂f−
∂xi
(x0) = lim
t→0−
f(x0 + tei)− f(x0)
t
i = 1, . . . , n
Theorem 2.3. Let f : A → R be a map, A ⊂ Rn open. Suppose well
defined the right-hand and the left-hand partial derivatives of f at every
x0 ∈ A. Also, suppose the functions
∂f+
∂xi
,
∂f−
∂xi
: A→ R
continuous and that
∂f+
∂xi
(x0) ≤ ∂f−
∂xi
(x0), ∀ x0 ∈ A,
i = 1, . . . , n. Then, f is u.s.d. and
DSf(x0) = [a1, b1]× [a2, b2]× · · · × [an, bn],
216 Weak local Nash equilibrium - part II
where ai =
∂f+
∂xi
(x0), bi =
∂f−
∂xi
(x0), i = 1, . . . , n. Thus, DSf : A( Rn
is an u.s.c. multivalued map with convex compact values.
Proof. Given x0 ∈ A, let ai =
∂f+
∂xi
(x0), bi =
∂f−
∂xi
(x0), i = 1, . . . , n. The
technique used to prove that
DSf(x0) ⊂ [a1, b1]× [a2, b2]× · · · × [an, bn]
is the same used in Theorem 2.2: let v = (v1, . . . , vn) ∈ DSf(x0) arbi-
trary. Thus,
f(x0 + h) ≤ f(x0) + v · h+ r(h),
with lim
h→0
r(h)
‖h‖
= 0. In particular, if h = tei then
f(x0 + tei) ≤ f(x0) + tv · ei + r(tei),
with lim
h→0
r(tei)
t
= 0. It follows that, for every t > 0,
f(x0 + tei)− f(x0)
t
≤ vi +
r(tei)
t
.
Therefore
ai =
∂f+
∂xi
(x0) ≤ vi.
Also, for every t < 0, we have
f(x0 + tei)− f(x0)
t
≥ vi +
r(tei)
t
.
Therefore,
bi =
∂f−
∂xi
(x0) ≥ vi.
Hence, v ∈ [a1, b1]× [a2, b2]× · · · × [an, bn].
Since DSf(x0) is convex, in order to prove the equality
DSf(x0) = [a1, b1]× [a2, b2]× · · · × [an, bn]
it is sufficient to show that each vertex of that parallelepiped is contained
in DSf(x0).
Carlos Biasi , Thaís F. M. Monis 217
To elucidate, we will write the proof to the case n = 2 and for the
vertex (a1, a2). The general case is analogous.
Let x0 = (x1, x2) and h = (h1, h2). We need to show that
f(x1 + h1, x2 + h2)− f(x1, x2)− h1a1 − h2a2 ≤ r(h)
with lim
h→0
r(h)
‖h‖
= 0.
Consider the functions g(y) = f(x1 +h1, y) and l(x) = f(x, x2). Note
that
∂g+
∂y
(x2) =
∂f+
∂x2
(x1 + h1, x2)
∂g−
∂y
(x2) =
∂f−
∂x2
(x1 + h1, x2)
∂l+
∂x
(x1) =
∂f+
∂x1
(x1, x2)
∂l−
∂x
(x1) =
∂f−
∂x1
(x1, x2)
>From Theorem 2.2, g and l are u.s.d.. Moreover,
DSg(x2) =
[
∂f+
∂x2
(x1 + h1, x2),
∂f−
∂x2
(x1 + h1, x2)
]
and
DSl(x1) =
[
∂f+
∂x1
(x1, x2),
∂f−
∂x1
(x1, x2)
]
.
Thus,
g(x2 + h2)− g(x2)− h2
∂f+
∂x2
(x1 + h1, x2) ≤ r1(h2)
l(x1 + h1)− l(x1)− h1
∂f+
∂x1
(x1, x2) ≤ r2(h1)
with lim
x→0
r2(x)
x
= lim
y→0
r1(y)
y
= 0.
218 Weak local Nash equilibrium - part II
We have that
f(x1 + h1, x2 + h2)− f(x1, x2)− h1a1 − h2a2 =
g(x2 + h2)− g(x2)− h2
∂f+
∂x2
(x1 + h1, x2) + l(x1 + h1)− l(x1)−
−h1
∂f+
∂x1
(x1, x2) + h2
[
∂f+
∂x2
(x1 + h1, x2)− ∂f+
∂x2
(x1, x2)
]
≤ r(h)
where r(h) = r1(h2) + r2(h1) + h2
[
∂f+
∂x2
(x1 + h1, x2)− ∂f+
∂x2
(x1, x2)
]
.
Now, it is easy to see that lim
h→0
r(h)
‖h‖
.
3 The main theorem
In this section, we will stablish a generalization of the Theorem 1.2.
It is the following:
Theorem 3.1. Let p1, . . . , pn : S1 × . . . × Sn → R be maps, where
each Si ⊂ Rmi is a compact ENR with the p.c.r.. Also, suppose that
pi(s1, . . . , si, . . . , sn) as a function of si = (s11, . . . , s
mi
1 ) satisfies:
• The map xi 7−→ p(s−i, xi) can be continuously defined on a
neighborhood Vi of Si. The symbol (s−i, xi) denotes the point
(s1, . . . , si−1, xi, si+1, . . . , sn).
• pi(s−i, ) : Vi → R has continuous lateral partial derivatives
∂pi
+
∂xji
(s−i, ),
∂pi
−
∂xji
(s−i, ) : Vi → R
j = 1, . . . ,mi and
•
∂pi
+
∂xji
(s−i, xi) ≤
∂pi
−
∂xji
(s−i, xi), ∀ xi ∈ Vi
With these assumptions, if χ(Si) 6= 0 for i = 1, 2, . . . , n then
p1, p2, . . . , pn have at least one w.l.e..
Carlos Biasi , Thaís F. M. Monis 219
The proof of Theorem 3.1 is an application of a fixed point theorem
of multivalued maps.
3.1 The Lefschetz Fixed Point Theorem for Admis-
sible Multivalued Mappings
The spaces considered here are metric. Also, we are considering the
C̆ech homology functor with compact carriers and with coefficients in Q.
A proper map f : X → Y is a map such that, for all K ⊂ X compact,
f−1(K) is compact.
A compact space X is called acyclic if H0(X) = Q and Hq(X) = 0
for q > 0.
A map p : (X,X0) → (Y, Y0) is called a Vietoris map if p : X → Y
is proper, p−1(Y0) = X0 and p−1(y) is acyclic, for every y ∈ Y . Symbol:
p : (X,X0)⇒ (Y, Y0).
Theorem 3.2 (Vietoris Mapping Theorem). If p : (X,X0)⇒ (Y, Y0) is
a Vietoris map then p∗ : H∗(X,X0)→ H∗(Y, Y0) is an isomorphism.
Let X and Y be two spaces and assume that for each point x ∈ X a
nonempty closed subset ϕ(x) of Y is given; in this case, we say that ϕ is
a multivalued map from X into Y and we write ϕ : X ( Y .
A multivalued map ϕ : X ( Y is called upper semicontinuous (u.s.c.)
if for every open subset U of Y the set ϕ−1(U) = {x ∈ X | ϕ(x) ⊂ U} is
an open subset of X.
An u.s.c. multivalued map ϕ : X ( Y is called acyclic if for every
x ∈ X the set ϕ(x) is an acyclic subset of Y .
An u.s.c. multivalued map ϕ : X ( Y is called admissible if there
exists a space Γ and mappings p : Γ⇒ X, q : Γ→ Y such that:
• p is a Vietoris map,
• q(p−1(x)) ⊂ ϕ(x), for every x ∈ X.
(p, q) is called a selected pair of ϕ (written (p, q) ⊂ ϕ).
Let ϕ : X ( Y be an admissible multivalued map. The set {ϕ}∗ of
linear induced mappings is defined by
{ϕ}∗ = {q∗p−1∗ : H∗(X)→ H∗(Y ) | (p, q) ⊂ ϕ}
220 Weak local Nash equilibrium - part II
Two admissible multivalued maps ϕ,ψ : X ( Y are called homotopic
(written ϕ ∼ ψ) if there exists an admissible multivalued map χ : X×[0, 1]
such that:
χ(x, 0) ⊂ ϕ(x) and χ(x, 1) ⊂ ψ(x) for every x ∈ X
Theorem 3.3 ([5], Theorem (40.11)). Let ϕ : X ( Y be two admissible
multivalued maps. Then ϕ ∼ ψ implies that there exists selected pairs
(p, q) ⊂ ϕ and (p̄, q̄) ⊂ ψ such that
q∗p
−1
∗ = q̄∗p̄
−1
∗
Let X be a compact ANR and let ϕ : X ( X be an admissible
multivalued map. Then, it is well defined the Lefschetz set Λ(ϕ) of ϕ by
putting
Λ(ϕ) = {Λ(q∗p
−1
∗ ) =
∑
i
(−1)itracei(q∗p−1∗ ) | (p, q) ⊂ ϕ}
Theorem 3.4 (Lefschetz fixed point theorem for admissible multivalued
mappings). Let X be a compact ANR and ϕ : X ( X be a compact
admissible multivalued map. If Λ(ϕ) 6= {0} then Fix(ϕ) 6= ∅.
3.2 Proof of Theorem 3.1
In order to prove Theorem 3.1 we will define an admissible multivalued
map F : S ( S and we will prove that if s̃ ∈ F (s̃) then s̃ is an w.l.e.
for p1, . . . , pn. The conclusion of the proof will follow from the Lefschetz
fixed point theorem for admissible multivalued mappings. First, we need
the following lemma.
Lemma 1. Let X be a compact subset of Rm and let V be an open
neighborhood of X in Rm. Then, given a multivalued map ϕ : X ( Rm
u.s.c. with compact values, there exists t1 > 0 such that x + tv ∈ V for
all x ∈ X, v ∈ ϕ(x) and t ∈ [0, t1].
Proof. Let ϕ : X ( Rm be an u.s.c. multivalued map with compact
values. If ϕ(x) = {0} for every x ∈ X, there is nothing to prove. Suppose
Carlos Biasi , Thaís F. M. Monis 221
ϕ(x) 6= {0} for some x ∈ X. Since X is compact and ϕ is u.s.c. with
compact values, the image ϕ(X) =
⋃
x∈X
ϕ(x) is also compact. Then,
the real number u = max
v∈ϕ(X)
{‖v‖} is a finite positive number. For every
x ∈ X, there is εx > 0 such that B(x, εx) ⊂ V . Since X is compact, we
obtain a finite open subcover
{
B
(
xi,
εxi
4
)}l
i=1
with
X ⊂
l⋃
i=1
B
(
xi,
εxi
4
)
⊂
l⋃
i=1
B(xi, εxi
) ⊂ V.
Let ε = min
1≤i≤l
{εxi
4
}
and t1 =
ε
u
. Thus, x+tv ∈ V for all x ∈ X, v ∈ ϕ(x)
and t ∈ [0, t1]. In fact, given x ∈ X, we have x ∈ B
(
xi,
εxi
4
)
for some
xi. If v = 0 the conclusion is obvious. If v 6= 0 then, given t ∈ [0, t1], we
have
t ≤ t1 =
ε
u
≤ εxi
4u
≤ εxi
4‖v‖
.
It follows that
‖x+ tv − xi‖ ≤ ‖x− xi‖+ t‖v‖ ≤ εxi
4
+
εxi
4‖v‖
‖v‖ =
εxi
2
< εxi .
Therefore, x+ tv ∈ B(xi, εxi
) ⊂ V .
Hence, for all x ∈ X, v ∈ ϕ(x) and t ∈ [0, t1].
Proof of Theorem 3.1. Since S1 ⊂ Rm1 , . . . , Sn ⊂ Rmn are compact
ENR’s with the p.c.r., the product S = S1×· · ·×Sn ⊂ Rm is also a space
with the p.c.r, m = m1 + · · ·+mn. Thus, let r : V → S be a convenient
retraction.
Let ϕ : S ( Rm be the multivalued map defined by
ϕ(s) = ϕ1(s)× · · · × ϕn(s)
where ϕi(s) = DSpi(s−i, si).
>From Lemma 1, there exists t1 > 0 such that s + tv ∈ V for all
s ∈ S, t ∈ [0, t1] and v ∈ V (s).
Finally, we define F : S ( S by
F (s) = {r(s+ t1v) | v ∈ ϕ(s)}.
222 Weak local Nash equilibrium - part II
As defined, F is a compact admissible multivalued map. Moreover, F is
homotopic to the identity map via homotopy ψ : S× [0, t1]→ S given by
ψ(s, t) = {r(s + tv) | v ∈ ϕ(s)}. Thus, by Theorema 3.3, there exists a
selected pair (p, q) ⊂ F such that
Λ(q∗p
−1
∗ ) = Λ(idS) = χ(S) = χ(S1) · · ·χ(Sn).
If χ(Si) 6= 0, i = 1, . . . , n, then Λ(F ) 6= {0}. It follows, from Theorem
3.4, that F has a fixed point, ie, a point s̃ ∈ S such that s̃ ∈ F (s̃).
We affirm that a such fixed point s̃ is a w.l.e. for p1, . . . , pn. In fact, if
s̃ ∈ F (s̃) then s̃ = r(s̃ + t1v) for some v ∈ ϕ(s̃). Since r is a convenient
retraction, given ε > 0, there exists δ > 0 such that
‖x− r(s̃+ t1v)‖ = ‖x− s̃‖ < δ
implies that
〈s̃+ t1v − r(s̃+ t1v), x− r(s̃+ t1v)〉 = t1〈v, x− s̃〉
≤ t1ε
2
‖x− s̃‖.
Moreover, from the definition of ϕ, we can assume that if ‖s̃ − s‖ < δ
then
pi(s̃1, . . . , s̃i−1, si, s̃i+1, . . . , s̃n) ≤ pi(s̃) + 〈vi, si − s̃i〉+
ε
2
‖si − s̃i‖,
1 ≤ i ≤ n. It follows that, if s ∈ S and ‖s− s̃‖ < δ then
pi(s̃1, . . . , s̃i−1, si, s̃i+1, . . . , s̃n) ≤ pi(s̃) + ε‖si − s̃i‖,
1 ≤ i ≤ n.
Hence, s̃ is a w.l.e. for p1, . . . , pn.
References
[1] Alós-Ferrer, C., Ania, A.B., Local equilibria in economic games.
Econom. Lett., 70, no. 2, 165-173 (2001).
[2] Biasi, C., Monis, T.F.M., Weak local Nash equilibrium. Topological
Methods in Nonlinear Analysis 41, no. 2, 409-419 (2013).
Carlos Biasi , Thaís F. M. Monis 223
[3] Biasi, C., Mendes Monis, T. F., Some coincidence theorems and its
applications to existence of local Nash equilibrium. JP J. Fixed Point
Theory Appl. 5, no. 2, 81-102 (2010).
[4] Eilenberg, S., Montgomery, D., Fixed point theorems for multi-valued
transformations. Amer. J. Math., 68, 214-222 (1946).
[5] Górniewicz, L., Topological fixed point theory of multivalued map-
pings. Second edition. Topological Fixed Point Theory and Its Ap-
plications, 4. Springer, Dordrecht, 2006.
[6] Milnor, J., A nobel prize for John Nash. Math. Intelligencer, 17, no.
3, 11-17 (1995).
Carlos Biasi
Instituto de Ciências Matemáticas e de Computação. Universidade
de São Paulo.
e-mail: biasi@icmc.usp.br
Tháis Fernanda Mendes Monis
Instituto de Geociências e Ciências Exatas. Univ Estadual Paulista.
email: tfmonis@rc.unesp.br
|
| id | oai:trim.imath.kiev.ua:article-307 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:22Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/b5/8d47a6c65018bd8a58604191b97205b5.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3072018-02-10T20:56:26Z Weak local Nash equilibrium - part II Слабка локальна точка рівноваги Неша - частина II Biasi, C. Monis, T. F. M. Biasi, C. Monis, T. F. M. In the paper “Weak local Nash equilibrium” we define a concept of local equilibrium to non-cooperative games and we prove its existence applying the Lefschetz fixed point theorem. We was inspired by the original Nash’s theorem and his proof. У роботі "Слабка локальна точка рівноваги Неша" ми визначаємо поняття локальної рівноваги для некооперативних ігор і доводимо його існування, застосовуючи теорему про нерухому точку Лефшеца. Нас надихнула оригінальна теорема Неша та його доказ. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/307 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 209-223 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 209-223 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 209-223 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/307/287 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Biasi, C. Monis, T. F. M. Biasi, C. Monis, T. F. M. Weak local Nash equilibrium - part II |
| title | Weak local Nash equilibrium - part II |
| title_alt | Слабка локальна точка рівноваги Неша - частина II |
| title_full | Weak local Nash equilibrium - part II |
| title_fullStr | Weak local Nash equilibrium - part II |
| title_full_unstemmed | Weak local Nash equilibrium - part II |
| title_short | Weak local Nash equilibrium - part II |
| title_sort | weak local nash equilibrium - part ii |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/307 |
| work_keys_str_mv | AT biasic weaklocalnashequilibriumpartii AT monistfm weaklocalnashequilibriumpartii AT biasic weaklocalnashequilibriumpartii AT monistfm weaklocalnashequilibriumpartii AT biasic slabkalokalʹnatočkarívnovaginešačastinaii AT monistfm slabkalokalʹnatočkarívnovaginešačastinaii AT biasic slabkalokalʹnatočkarívnovaginešačastinaii AT monistfm slabkalokalʹnatočkarívnovaginešačastinaii |