11institutetext: University of Bucharest 11email<EMAIL_ADDRESS>
# Existence of equilibrium for multiobjective games in abstract convex spaces

Monica Patriche  University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, 010014 Bucharest, Romania

###### Abstract

In this paper we use the minimax inequalities obtained by S. Park (2011) to
prove the existence of weighted Nash equilibria and Pareto Nash equilibria of
a multiobjective game defined on abstract convex spaces.

###### Keywords:

minimax inequality, weighted Nash equilibria, Pareto Nash equilibria,
multiobjective game, abstract convex space.

2000 Mathematics Subject Classification: 47H10, 55M20, 91B50.

##  1 INTRODUCTION

Recently, in [8], S. Park introduced a new concept of abstract convex space
and several classes of correspondences having the KKM property. With this new
concept, the KKM type correspondences were used to obtain coincidence
theorems, fixed point theorems and minimax inequalities. S. Park generalizes
and unifies most of important results in the KKM theory on G-convex spaces,
H-spaces, and convex spaces (for example, see [8]-[13]).

For the history of KKM literature, we must remind Ky Fan [3], who extended the
original KKM theorem to arbitrarily topological vector space. The property of
close-valuedness of related KKM correspondences was replaced with more general
concepts. In [7], Luc and al. have introduced the concept of intersectionally
closed-valued correspondences and in [13], S. Park has obtained new KKM type
theorems for this kind of KKM correspondences.

In this paper we use the minimax inequalities obtained by S. Park in [13] to
prove the existence of weighted Nash equilibria and Pareto Nash Equilibria of
a multiobjective game defined on abstract convex spaces. For the history of
minimax theorems, I also must remind the name of Ky Fan (see [4]). Among the
authors who studied the existence of Pareto equilibria in game theory with
vector payoffs, I emphasize S. Chebbi [2], W. K. Kim [5], W. K. Kim, X. P.
Ding [6], H. Yu [16], J. Yu, G. X.-Z Yuan [17], X. Z. Yuan, E. Tarafdar [18].
A reference work is the paper of M. Zeleny [19]. The approaches of above-
mentioned authors deal with the Ky Fan minimax inequality, quasi-equilibrium
theorems or quasi-variational inequalities. We must mention the papers of P.
Borm, F. Megen, S. Tijs [1], who introduced the concept of perfectness for
multicriteria games and M. Voorneveld, S. Grahn, M. Dufwenberg [14], who
studied the existence of ideal equilibria. Ather authors, as H. Yu (see [16]),
obtained the existence of a solution of multiobjective games by using new
concepts of continuity and convexity.

The paper is organised as follows: In section 2, some notation, terminological
convention, basic definitions and results about abstract convex spaces and
minimax inequalities are given. Section 3 introduces the model, that is, a
multiobjective game defined on an abstract convex space and the concept of
weight Nash equilibrium. Section 4 contains existence results for weight Nash
equilibrium and Pareto Nash equilibrium.

##  2 ABSTRACT CONVEX SPACES AND MINIMAX INEQUALITIES

Let $A$ be a subset of a topological space $X$. $2^{A}$ denotes the family of
all subsets of $A$. $\overline{A}$ denotes the closure of $A$ in $X$ and
int$A$ denotes the interiorof $A$. If $A$ is a subset of a vector space, co$A$
denotes the convex hull of $A$. If $F$, $G:$ $X\rightarrow 2^{Y}$ are
correspondences, then co$G$, cl $G$, $G\cap F$ $:$ $X\rightarrow 2^{Y}$ are
correspondences defined by $($co$G)(x)=$co$G(x)$, $($cl$G)(x)=$cl$G(x)$ and
$(G\cap F)(x)=G(x)\cap F(x)$ for each $x\in X$, respectively. The graph of
$F:X\rightarrow 2^{Y}$ is the set Gr$(F)=\\{(x,y)\in X\times Y\mid y\in
F(x)\\}$ and $F^{-}:Y\rightarrow 2^{X}$ is defined by $F^{-}(y)=\\{x\in X:y\in
F(x)\\}$ for $y\in Y.$ Let $\tciFourier(A)$ be the set of all nonempty finite
subsets of a set $A.\vskip 6.0pt plus 2.0pt minus 2.0pt$

For the reader’s convenience, we review a few basic definitions and results
from abstract convex spaces.

Definition 1 [13]. Let $X$ be a topological space, $D$ be a nonempty set and
let $\Gamma:\tciFourier(D)\rightarrow 2^{X}$ be a correspondence with nonempty
values $\Gamma_{A}=\Gamma(A)$ for $A\in\tciFourier(D).$ The family
$(X,D;\Gamma)$ is called an abstract convex space.

Definition 2 [13]. For a nonempty subset $D^{\prime}$ of $D$, we define the
$\Gamma$-convex hull of $D^{\prime}$, denoted by co${}_{\Gamma}D^{\prime}$, as

co${}_{\Gamma}D^{\prime}$=$\cup\\{\Gamma_{A}:A\in\tciFourier(D^{\prime})\\}\subset
X.$

Definition 3 [13]. Given an abstract convex space ($X,D,\Gamma$), a nonempty
subset $Y$ of $X$ is called to be a $\Gamma$-convex subset of ($X,D,\Gamma$)
relative to $D^{\prime}$ if for any $A\in\tciFourier$($D^{\prime}$), we have
$\Gamma_{A}\subset Y$, that is, co${}_{\Gamma}D^{\prime}\subset Y.$

Definition 4 [13]. When $D\subset X$ in ($X,D,\Gamma$), a subset $Y$ of $X$ is
said to be $\Gamma$-convex if co${}_{\Gamma}(Y\cap D)\subset Y;$ in other
words, $Y$ is $\Gamma$-convex relative to $D^{\prime}=Y\cap D.$ In case $X=D,$
let $(X,\Gamma)=(X,X,\Gamma).$

Definition 5 [13]. The abstract convex space ($X,D,\Gamma$) is called compact
if $X$ is compact.

We have abstract convex subspaces as the following simple observation.

###### Proposition 1

For an abstract convex space ($X,D,\Gamma$) and a nonempty subset $D^{\prime}$
of $D$, let $Y$ be a $\Gamma$-convex subset of $X$ relative to $D^{\prime}$
and $\Gamma^{\prime}:\tciFourier$($D^{\prime}$)$\rightarrow 2^{Y}$ a
correspondence defined by

$\Gamma_{A}^{\prime}=\Gamma_{A}\subset X$ for $A\in\tciFourier$($D^{\prime}$).

Then ($Y,D^{\prime},\Gamma^{\prime}$) itself is an abstract convex space
called a subspace relative to $D^{\prime}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

The following result is known.

######  Lemma 1 (12)

Let ($X_{i},D_{i},\Gamma_{i}$)i∈I be any family of abstract convex spaces. Let
$X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i}$ be equipped with the
product topology and $D=\mathop{\textstyle\prod}\nolimits_{i\in I}D_{i}$. For
each $i\in I$, let $\pi_{i}:D\rightarrow D_{i}$ be the projection. For each
$A\in\tciFourier$($D$), define
$\Gamma(A)=\mathop{\textstyle\prod}\nolimits_{i\in I}\Gamma_{i}(\pi_{i}(A))$.
Then ($X,D,\Gamma$) is an abstract convex space.

Definition 6 [13]. Let ($X,D,\Gamma$) be an abstract convex space. Then
$F:D\rightarrow 2^{X}$ is called a KKM correspondence if it satisfies
$\Gamma_{A}\subset F(A):=\cup_{y\in A}F(y)$ for all $A\in\tciFourier$($D$).

Definition 7 [13]. The partial KKM principle for an abstract convex space
($X,D,\Gamma$) is the statement that, for any closed-valued KKM correspondence
$F:D\rightarrow 2^{X}$, the family $\\{F(z)\\}_{z\in D}$ has the finite
intersection property. The KKM principle is the statement that the same
property also holds for any open-valued KKM correspondence.

An abstract convex space is called a KKM space if it satisfies the KKM
principle.

###### Proposition 2

Let ($X,D,\Gamma$) be an abstract convex space and
($X,D^{\prime},\Gamma^{\prime}$) a subspace. If ($X,D,\Gamma$) satisfies the
partial KKM principle, then so does ($X$, $D^{\prime},\Gamma^{\prime}$).

Let ($X,D,\Gamma$) be an abstract convex space.

Definition 8 [13]. The function $f:X\rightarrow\overline{\mathbb{R}}$ is said
to be quasiconcave (resp. quasiconvex) if $\\{x\in X:f(x)>r\\}$ (resp.,
$\\{x\in X:f(x)<r\\}$ is $\Gamma$-convex for each
$r\in\overline{\mathbb{R}}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

In [7], Luc and al. have introduced the concept of intersectionally closed-
valued correspondences.

Definition 9. Let $F:D\rightarrow 2^{X}$ be a correspondence.

(i) [7] $F$ is intersectionally closed-valued if $\cap_{z\in
D}\overline{F(z)}=\overline{\cap_{z\in D}F(z)};$

(ii) $F$ is transfer closed-valued if $\cap_{z\in D}\overline{F(z)}=\cap_{z\in
D}F(z);$

(iii) [7] $F$ is unionly open-valued if Int$\cup_{z\in D}F(z)=\cup_{z\in
D}$Int$F(z);$

(iv) $F$ is transfer open-valued if $\cup_{z\in D}F(z)=\cup_{z\in
D}$Int$F(z);$

Luc at al. [7] noted that (ii)$\Rightarrow$(i$).$

######  Proposition 3 (7)

The correspondence $F$ is intersectionally closed-valued (resp. transfer
closed-valued) if only if its complement $F^{C}$ is unionly open-valued (resp.
transfer open-valued).

Definition 10 [13]. Let$Y$ be a subset of $X.$

(i) $Y$ is said to be intersectionally closed (resp. transfer closed) if there
is an intersectionally (resp., transfer) closed-valued correspondence
$F:D\rightarrow 2^{X}$ such that $Y=F(z)$ for some $z\in D.$

(ii) $Y$ is said to be unionly open (resp. transfer open) if there is an
unionly (resp., transfer) open-valued correspondence $F:D\rightarrow 2^{X}$
such that $Y=F(z)$ for some $z\in D.\vskip 6.0pt plus 2.0pt minus 2.0pt$

S. Park gives in [13] the concept of generally lower (resp. upper)
semicontinuous function.

Definition 11 [13]. The function $f:D\times X\rightarrow\overline{\mathbb{R}}$
is said to be generally lower (resp. upper) semicontinuous (g.l.s.c.) (resp.
g.u.s.c.) on $X$ whenever, for each $z\in D,$ $\\{y\in X:f(z,y)\leq r\\}$
(resp., $\\{y\in X:f(z,y)\geq r\\})$ is intersectionally closed for each
$r\in\overline{\mathbb{R}}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

The aim of this paper is to prove the existence of a weighted Nash equilibrium
for a multicriteria game defined in the framework of abstract convex spaces.
For our purpose, we need the following theorem (variant of Theorem 6.3 in
[13]).

###### Theorem 2.1

(Minimax inequality, [13]). Let ($X,D=X,\Gamma$) an abstract convex space
satisfying the partial KKM principle, $f,g:X\times
X\rightarrow\overline{\mathbb{R}}$ extended real-valued functions and
$\gamma\in\overline{\mathbb{R}}$ such that

(i) for each $x\in X,$ $g(x,x)\leq\gamma;$

(ii) for each $y\in X,$ $F(y)=\\{x\in X:f(x,y)\leq\gamma\\}$ is
intersectionally closed (respectiv, transfer closed);

(iii) for each $x\in X,$ co${}_{\Gamma}\\{y\in
X:f(x,y)>\gamma\\}\subset\\{y\in X:g(x,y)>\gamma\\};$

(iv) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:

there exists a nonempty compact subset $K$ of $X$ such that either

(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or

(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
X^{\prime}}\overline{F(y)}\neq\phi.$

Then

1) there exists a $x_{0}\in X$ $($resp., $x_{0}\in K)$ such that
$f(x_{0},y)\leq\gamma$ for all $y\in X;$

2) if $\gamma:=\sup_{x\in X}g(x,x)$, then we have

inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$

For the case when $X=D$ (we are concerned with compact abstract spaces
($X,\Gamma$) satisfying the partial KKM principle), we have the following
variants of the corollaries stated in [13].

######  Corollary 1 (13)

Let $f,$ $g:X\times X\rightarrow\mathbb{R}$ be real-valued functions and
$\gamma\in\mathbb{R}$ such that

(i) for each $x,y\in X,$ $f(x,y)\leq g(x,y)$ and $g(x,x)\leq\gamma;$

(ii) for each $y\in X,$ $\\{x\in X:f(x,y)>\gamma\\}$ is unionly open in $X$;

(iii) for each $x\in X,$ $\\{y\in X:g(x,y)>\gamma\\}$ is $\Gamma$-convex on
$X;$

Then

1) there exists a $x_{0}\in X$ such that $f(x_{0},y)\leq\gamma$ for all $y\in
X;$

2) if $\gamma:=\sup_{x\in X}g(x,x)$, then we have

inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$

######  Corollary 2 (13)

Let $f,$ $g:X\times X\rightarrow\mathbb{R}$ be functions such that

(i) for each $x,y\in X,$ $f(x,y)\leq g(x,y)$ and $g(x,x)\leq\gamma;$

(ii) for each $y\in X,$ $f(\cdot,y)$ is g.l.s.c on $X$;

(iii) for each $x\in X,$ $f(x,\cdot)$ is quasiconcave on $X;$

Then we have

inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$

##  3 MULTIOBJECTIVE GAMES

Now we consider the multicriteria game (or multiobjective game) in its
strategic form. Let $I$ be a finite set of players and for each $i\in I,$ let
$X_{i}$ be the set of strategies such that
$X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i}$ and ($X_{i}$,
$D_{i},\Gamma_{i}$ for each $i\in I$) is an abstract convex space with
$D_{i}\subset X_{i}$. Let $T^{i}:X\rightarrow 2^{\mathbb{R}^{k_{i}}}$, where
$k_{i}\in\mathbb{N}$, which is called the payoff function (or called
multicriteria). From Lemma 1, we also have that $(X,D,\Gamma)$ is an abstract
convex space, where $X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i},$
$D=\mathop{\textstyle\prod}\nolimits_{i\in I}D_{i}$ and
$\Gamma(A)=\mathop{\textstyle\prod}\nolimits_{i\in I}\Gamma_{i}(\pi_{i}(A))$
for each $A\in\tciFourier$($D$).

Definition 12. The family $G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ is
called multicriteria game.

If an action $x:=(x_{1},x_{2},...,x_{n})$ is played, each player $i$ is trying
to find his/her payoff function
$T^{i}(x):=(T_{1}^{i}(x),...,T_{k_{i}}^{i}(x)),$ which consists of
noncommensurable outcomes. We assume that each player is trying to minimize
his/her own payoff according with his/her preferences.

In order to introduce the equilibrium concepts of a multicriteria game, we
need several necessary notation.

Notation. We shall denote by

$\mathbb{R}_{+}^{m}:=\\{u=(u_{1},u_{2},...u_{m})\in\mathbb{R}^{m}:u_{j}\geq 0$
$\forall j=1,2,...,m\\}$ and

int$\mathbb{R}_{+}^{m}:=\\{u=(u_{1},u_{2},...u_{m})\in\mathbb{R}^{m}:u_{j}>0$
$\forall j=1,2,...,m\\}$

the non-negative othant of $\mathbb{R}^{m}$ and respective the non-empty
interior of $\mathbb{R}_{+}^{m}$ with the topology induced in terms of
convergence of vector with respect to the Euclidian metric.

Notation. For each $i\in I,$ denote
$X_{-i}:=\mathop{\textstyle\prod}\nolimits_{j\in I\setminus\\{i\\}}X_{j}.$ If
$x=(x_{1},x_{2},...,x_{n})\in X,$ we denote
$x_{-i}=(x_{1},...,x_{i-1},x_{i+1},...,x_{n})\in X_{-i}.$ If $x_{i}\in X_{i}$
and $x_{-i}\in X_{-i}$, we shall use the notation
$(x_{-i},x_{i})=(x_{1},...,x_{i-1},x_{i},x_{i+1},...,x_{n})=x\in X.$

Notation. For each $u,v\in\mathbb{R}^{m}$, $u\cdot v$ denote the standard
Euclidian inner product.

Let $\widehat{x}=(\widehat{x}_{1},\widehat{x}_{2},...,\widehat{x}_{n})\in X.$
Now we have the following definitions.

Definition 13. A strategy $\widehat{x}_{i}\in X_{i}$ of player $i$ is said to
be a Pareto efficient strategy (resp., a weak Pareto efficient strategy) with
respect to $\widehat{x}\in X$ of the multiobjective game
$G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if there is no strategy
$x_{i}\in X_{i}$ such that

$T^{i}(\widehat{x})-T^{i}(\widehat{x}_{-i},x_{i})\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$
(resp.,
$T^{i}(\widehat{x})-T^{i}(\widehat{x}_{-i},x_{i})\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}).\vskip
6.0pt plus 2.0pt minus 2.0pt$

###### Remark 1

Each Pareto equilibrium is a weak Pareto equilibrium, but the converse is not
always true.

Definition 14. A strategy $\widehat{x}\in X$ is said to be a Pareto
equilibrium (resp., a weak Pareto equilibrium) of the multiobjective game
$G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if for each player $i\in I$,
$\widehat{x}_{i}\in X_{i}$ is a Pareto efficient strategy (resp., a weak
Pareto efficient strategy) with respect to $\widehat{x}.\vskip 6.0pt plus
2.0pt minus 2.0pt$

Definition 15. A strategy $\widehat{x}\in X$ is said to be a weighted Nash
equilibrium with respect to the weighted vector $W=(W_{i})_{i\in I}$ with
$W_{i}=(W_{i,1},W_{i,2},...,W_{i,k_{i}})\in\mathbb{R}_{+}^{k_{i}}$ of the
multiobjective game $G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if for each
player $i\in I$, we have

(i) $W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\};$

(ii) $W_{i}\cdot T^{i}(\widehat{x})\leq W_{i}\cdot
T^{i}(\widehat{x}_{-i},x_{i}),$ $\forall x_{i}\in X_{i},$where $\cdot$ denotes
the inner product in $\mathbb{R}^{k_{i}}.$

###### Remark 2

In particular, if $W_{i}\in\mathbb{R}_{+}^{k_{i}}$ with
$\mathop{\textstyle\sum}\nolimits_{j=1}^{k_{i}}W_{i,j}=1$ for each $i\in I,$
then the strategy $\widehat{x}\in X$ is said to be a normalized weighted Nash
equilibrium with respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  4 EXISTENCE OF WEIGHTED NASH EQUILIBRIUM AND PARETO NASH EQUILIBRIUM

Now, as an application of Theorem 1, we have the following existence theorem
of weighted Nash equilibria for multiobjective games.

###### Theorem 4.1

Let $I$ be a finite set of indices, let ($X_{i},D_{i}=X_{i},\Gamma_{i}$)i∈I be
any finite family of abstract convex spaces such that the product space
($X,\Gamma$) satisfies the partial KKM principle. If there is a weighted
vector $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:

(i) for each $y\in X,$ $F(y)=\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
0\\}$ is intersectionally closed (respectiv, transfer closed);

(ii) there exists $g:X\times X\rightarrow\overline{\mathbb{R}}$ extended real-
valued function such that for each $x\in X,$ $g(x,x)\leq 0$ and for each $x\in
X,$ co${}_{\Gamma}\\{y\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}\subset\\{y\in
X:g(x,y)>0\\};$

(iii) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:

there exists a nonempty compact subset $K$ of $X$ such that either

(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or

(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
x^{\prime}}\overline{F(y)}\neq\phi;$

then there exists $\widehat{x}\in K$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof. Define the function $f:X\times X\rightarrow\mathbb{R}$ by
$f(x,y)=\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i})),$
$(x,y)\in X\times X.$ By Theorem 1, we have that inf${}_{x\in X}\sup_{y\in
X}f(x,y)\leq\sup_{x\in X}g(x,x)=0.$ It follows that there exists an
$\widehat{x}\in K$ such that $f(\widehat{x},y)\leq 0$ for any $y\in X.$ That
is
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i})\leq
0$ for any $y\in X.$ For any given $i\in I$ and any given $y_{i}\in X_{i},$
let $y=(\widehat{x}_{-i},y_{i}).$ Then we have

$W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))=$

$=\mathop{\textstyle\sum}\nolimits_{j=1}^{n}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))-\mathop{\textstyle\sum}\nolimits_{j\neq
i}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))$

$=\mathop{\textstyle\sum}\nolimits_{j=1}^{n}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))\leq
0.$

Therefore, we have
$W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))$
$\leq 0$ for each $i\in I$ and $y_{i}\in X_{i},$ that is $\widehat{x}\in K$ is
a weighted Nash equilibrium of the game $G$ with respect to $W.\vskip 6.0pt
plus 2.0pt minus 2.0pt$

We obtain the following corollaries for the compact games when $X=D$.

###### Corollary 3

Let $I$ be a finite set of indices, let ($X_{i},\Gamma_{i}$)i∈I be any finite
family of abstract convex spaces such that the product space ($X,\Gamma$)
satisfies the partial KKM principle. If there is a weighted vector
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:

(i) there exists $g:X\times X\rightarrow R$ such that for each $x,y\in X,$
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
g(x,y)$ and $g(x,x)\leq 0;$

(ii) for each $y\in X,$ $\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}$
is unionly open in $X$;

(iii) for each $x\in X,$ $\\{y\in X:g(x,y)>0\\}$ is $\Gamma$-convex on $X;$

then there exists $\widehat{x}\in X$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$

###### Corollary 4

Let $I$ be a finite set of indices, let ($X_{i},\Gamma_{i}$)i∈I be any finite
family of abstract convex spaces such that the product space ($X,\Gamma$)
satisfies the partial KKM principle. If there is a weighted vector
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:

(i) there exists $g:X\times X\rightarrow R$ such that for each $x,y\in X,$
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
g(x,y);$

(ii) for each fixed $y\in X,$ the function
$x\rightarrow\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))$
is g.l.s.c on $X$;

(iii) for each fixed $x\in X,$ the function
$y\rightarrow\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))$
is quasiconcave on $X$;

then there exists $\widehat{x}\in X$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$

In order to prove an existence theorem of Pareto equilibria for multiobjective
games, we need the following lemma.

######  Lemma 2 (15)

Each normalized weighted Nash equilibrium $\widehat{x}\in X$ with a weight
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ (resp.,
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\})$ and
$\mathop{\textstyle\sum}\nolimits_{j=1}^{k_{i}}W_{i,j}=1$ for each $i\in I,$
for a multiobjective game $G=(X_{i},T^{i})_{i\in I}$ is a weak Pareto
equilibrium (resp. a Pareto equilibrium) of the game $G.$

###### Remark 3

The conclusion of Lemma2 still holds if $\widehat{x}\in X$ is a weighted Nash
equilibrium with a weight $W=(W_{1},W_{2},...,W_{n})$,
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I$(resp.,
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I)$ of the
game $G.$

###### Remark 4

A Pareto equilibrium of $G$ is not necessarily a weighted Nash equilibrium of
the game $G.\vskip 6.0pt plus 2.0pt minus 2.0pt$

###### Theorem 4.2

Let $I$ be a finite set of indices, let ($X_{i},D_{i}=X_{i},\Gamma_{i}$)i∈I be
any finite family of abstract convex spaces such that the product space
($X,\Gamma$) satisfies the partial KKM principle. If there is a weighted
vector $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:

(i) for each $y\in X,$ $F(y)=\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
0\\}$ is intersectionally closed (respectiv, transfer closed);

(ii) there exists $g:X\times X\rightarrow\overline{\mathbb{R}}$ extended real-
valued function such that for each $x\in X,$ $g(x,x)\leq 0$ and for each $x\in
X,$ co${}_{\Gamma}\\{y\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}\subset\\{y\in
X:g(x,y)>0\\};$

(iii) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:

there exists a nonempty compact subset $K$ of $X$ such that either

(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or

(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
x^{\prime}}\overline{F(y)}\neq\phi;$

then there exists $\widehat{x}\in K$ such that $\widehat{x}$ is a weak Pareto
equilibrium of the game $G=((X_{i},D_{i}=X_{i},\Gamma_{i}),T^{i})_{i\in I}.$
In addition, if $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in$int$R_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I,$ then $G$ has at
least a Pareto equilibrium point $\widehat{x}\in X.$

Proof. By Theorem 2, $G$ has at least weighted Nash equilibrium point
$\widehat{x}\in K$ with respect of the weighted vector $W.$ Lemma 2 and Remark
3 shows that $\widehat{x}$ is also a weak Pareto equilibrium point of $G,$ and
a Pareto equilibrium point of $G$ if $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for each $i\in I.$

Acknowledgment: This work was supported by the strategic grant
POSDRU/89/1.5/S/58852, Project ”Postdoctoral programme for training scientific
researchers” cofinanced by the European Social Found within the Sectorial
Operational Program Human Resources Development 2007-2013.

The author thanks to Professor João Paulo Costa from the University of Coimbra
for the fruitfull discussions and for the hospitality he proved during the
visit to his departament.

## References

  * (1) P. Borm, F. Megen, S. Tijs, A perfectness concept for multicriteria games. Math. Meth. Oper. Res. 49 (1999), 401-412.
  * (2) S. Chebbi, Existence of Pareto equilibria for non-compact constrained multi-criteria games. J. Appl. An. 14 (2008), 2, 219-226.
  * (3) K. Fan, A generalization of Tyhonoff’s fixed point theorem. Math. Ann. 142 (1961), 305-310.
  * (4) K. Fan, A minimax inequality and applications in: O. Shisha (Ed.), Inequalities III, Academic Press, New York, 1972, pp. 103-113.
  * (5) W. K. Kim, Weight Nash equilibria for generalized multiobjective games. J. Chungcheong Math. Soc. 13 (2000), 1, 13-20.
  * (6) W. K. Kim, X. P. Ding, On generalized weight Nash equilibria for generalized multiobjective games. J. Korean Math. Soc. 40 (2003), 5, 883-899.
  * (7) D. T. Luc, E. Sarabi and A. Soubeyran, Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364 (2010), 544-555.
  * (8) S. Park, On generalizations of the KKM principle on abstract convex spaces. Nonlinear Anal. Forum 11 (2006), 1, 67–77.
  * (9) S. Park, Elements of the KKM theory on abstract convex spaces. J. Korean Math. Soc. 45 (2008), 1, 1–27.
  * (10) S. Park, Generalizations of the Nash EquilibriumTheorem in the KKM Theory. Fixed Point Theory Appl. doi:10.1155/2010/234706
  * (11) S. Pak, The KKM principle in abstract convex spaces: equivalent formulations and applications. Nonlinear Anal. 73 (2010), 1028-1042.
  * (12) S. Park, Generalizations of the Nash Equilibrium Theorem in the KKM Theory. Fixed Point Theory Appl., doi:10.1155/2010/234706.
  * (13) S. Park, New generalizations of basic theorems in the KKM theory. Nonlinear Anal. 74 (2011), 3000-3010.
  * (14) M. Voorneveld, S. Grahn, M. Dufwenberg, Ideal equilibria in noncooperative multicriteria games. Math. Meth. Oper. Res. 52 (2000), 65-77.
  * (15) S. Y. Wang, Existence of a Pareto equilibrium, J. Optim. Theory Appl. 95 (1997), 373-384.
  * (16) H. Yu, Weak Pareto Equilibria for Multiobjective Constrained games. Appl. Math. Let. 16 (2003), 773-776.
  * (17) J. Yu, G. X.-Z Yuan, The study of Pareto Equilibria for Multiobjective games by fixed point and Ky Fan Minimax Inequality methods. Computers Math. Applic.  35, (1998), 9, 17-24.
  * (18) X. Z. Yuan, E. Tarafdar, Non-compact Pareto equilibria for multiobjective games. J. Math. An. Appl. 204 (1996),156-163.
  * (19) M. Zeleny, Game with multiple payoffs. Internat. J. Game Theory 4 (1976), 179-191.