Existence and Generic Stability of Strong Noncooperative Equilibria of Vector-Valued Games

Abstract

In this paper, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals. We also obtain an existence theorem of strong equilibrium point of vector-valued games with single-leader–multi-follower framework by using the upper semicontinuous of parametric strong noncooperative equilibrium point set of the followers. Moreover, we obtain some results on the generic stability of general strong noncooperative equilibrium point vector-valued games.

1. Introduction

Nash [1,2] introduced the concept of equilibrium solution of noncooperative game theory and established existence theorems of noncooperative games by applying fixed point theorem. Since then, many authors have considered the existence and stability of the solution of general noncooperative games with real-valued payoff functions. Yu [3] investigated existence theorems of solutions of general noncooperative n-person games by using fixed point theorem and Ky Fan inequality, respectively. Simultaneously, the author obtained essential results of solutions for general noncooperative n-person games in the sense of Baire Category. By virtue of a Ky Fan minimax theorem, Yao [4] investigated an existence theorem of general Nash equilibrium points. Yang and Pu [5] obtained the existence and generic stability for minimax regret equilibria by using a fixed point theorem. Pang and Fukushima [6] introduced a class of multi-leader–follower game problems. Later, Yu and Wang [7] obtained an existence theorem of noncooperative equilibria for two-leader–two-follower games with the assumption of convexity on the set of parametric Nash equilibria of the followers. Ding [8] investigated some existence theorems of noncooperative equilibria for multi-leader–follower games in noncompact FC-spaces. Yang and Ju [9] investigated the existence and generic stability of cooperative equilibria for multi-leader–multi-follower games. Yang and Zhang [10] investigated the existence of the solution and essential components for population game problems. Park [11] obtained some of general fixed point theorems on topological vector space for set-valued mappings. Yu and Peng [12] obtained generic stability of Nash equilibria for noncooperative differential games. Voorneveld et al. [13] introduced the ideal Nash equilibria for finite-criteria games by the feasible-criterion mapping. Later, some authors [14,15] obtained some existence theorems for ideal Nash equilibria for finite-criteria games by applying a maximal element theorem.

In many practical problems, the players’ decisions are often guided by multiple goals, in which the players’ objective is represented by a partial order. For example, in population games, multiple goals are considered, such as individual payoff, social position, life satisfaction and so on. Specifically, Shapley and Rigby [16] first introduced the notion of Pareto–Nash equilibria in multicriteria games. Yang and Yu [17] obtained an essential component of the set of its weakly Pareto–Nash equilibrium points by a Ky Fan inequality of vector-valued functions. Jia et al. [18] obtained existence and stability of weakly Pareto–Nash equilibrium for generalized multiobjective multi-leader–follower games. Hung et al. [19] considered the generic stability of vector quasi-equilibrium problems on Hadamard manifolds. Note that these results require that the ordering cone has a nonempty interior, owing to the concept of Pareto solution. However, in many cases, the ordering cone has an empty interior. For example, in the normed space $l^p$ and $L^p(\Omega )$($1<p<\infty$), the positive cone has an empty interior. So, Gong [20,21] investigated some minimax theorem and existence theorems of Ky Fan points of strong solution for vector-valued mappings. Long et al. [22] obtained existence theorems and stability of solutions for generalized Ky Fan points of strong solution. Moreover, Li et al. [23,24] investigated some saddle point theorems and minimax theorems in lexicographic order. Recently, Zhang et al. [25] investigated existence theorems of general n-person noncooperative game problems and minimax regret equilibria problems with set payoff.

Motivated by these earlier work, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals by a partial order. We also obtain an existence theorem of strong equilibrium point of vector-valued games with single-leader–multi-follower framework by using the upper semicontinuous of parametric strong noncooperative equilibrium point set of the followers. Moreover, we obtain some results on the generic stability of general strong noncooperative equilibrium point of vector-valued games. We do not need the condition that the ordering cone has a nonempty interior. Hence, these results obtained are different from ones in the literature.

2. Preliminaries

We assume that $V,E,E_i,i=1,2,\cdots ,n$ are real locally convex Hausdorff topological vector spaces. Some basic concepts are presented as follows. Assume that S is a pointed closed convex cone in V.

Definition 1

([21,26]). Let A be a nonempty subset of V.

(i) 

A point $z\in A$ is said to be a strong maximal point of A if $z^{\prime }\in z-S,\forall z^{\prime }\in A$, and $Ma{x}_{s}A$ denotes the set of all strong maximal points of A with respect to the cone S.

(ii) 

A point $z\in A$ is said to be a strong minimal point of A if $z^{\prime }\in z+S,\forall z^{\prime }\in A$, and $Mi{n}_{s}A$ denotes the set of all strong minimal points of A with respect to the cone S.

Clearly, if $Ma{x}_{s}A\ne \varnothing (Mi{n}_{s}A\ne \varnothing )$, the $Ma{x}_{s}A\left(Mi{n}_{s}A\right)$ is a singleton set.

Definition 2

([20,21]). Let A be a nonempty subset of E and $f:A\to V$ be a vector-valued mapping.

(i) 

f is said to be lower semicontinuous on A, if for any $z\in Z$, the level set $\{x\in A:f(x)\in z-S\}$ is closed;

(ii) 

f is said to be upper semicontinuous on A, if for any $z\in Z$, the level set $\{x\in A:f(x)\in z+S\}$ is closed;

Note that if f is S-lower semicontinuous (see Definition 1 in [20]), then f is lower semicontinuous, but not vice versa (see Lemma 1 in [20]).

Definition 3

([21,27]). Let $A\subset E$ be a nonempty convex subset and $f:A\to V$ be a vector-valued mapping.

(i) 

f is said to be S-quasiconcave on A, if for any $z\in V$, the level set

$$Le{v}_f\left(z\right):=\{x\in X_0:f\left(x\right)\in z+S\}$$

is convex. f is said to be S-quasiconvex if $-f$ is S-quasiconcave.

(ii) 

f is said to be properly S-quasiconcave on $X_0$, if for any $x_1,x_2\in A$ and $l\in [0,1]$,

$$f(l{x}_1+(1-l)x_2)\in f\left(x_1\right)+Sorf(l{x}_1+(1-l)x_2)\in f\left(x_2\right)+S.$$

f is said to be properly S-quasiconvex if $-f$ is properly S-quasiconcave.

Remark 1.

If f is a real-valued function and $S\equiv R_{+}$, then S-quasiconcave and properly S-quasiconcave reduce to the quasiconcave of real-valued functions. However, for vector-valued mapping f, these ones are different. Clearly, if f is properly S-quasiconcave, then f is also S-quasiconcave, but not vice versa. The following example explains this case.

Example 1.

Let $A=[0,1]\subset R$ and $S=R_{+}^2$. Define vector-valued mapping $f:A\to R^2$ as follows:

$$f\left(x\right)=(x,{(x-1)}^2),x\in A.$$

Clearly, f is S-quasiconcave. Nevertheless, take $x_1=0,x_2=1,\overline{l}=\frac{1}{2}$; we have

$$f(\overline{l}{x}_1+(1-\overline{l})x_2)=(\frac{1}{2},\frac{1}{4})\notin f\left(x_1\right)+R_{+}^2=(0,1)+R_{+}^2$$

and

$$f(\overline{l}{x}_1+(1-\overline{l})x_2)=(\frac{1}{2},\frac{1}{4})\notin f\left(x_2\right)+R_{+}^2=(1,0)+R_{+}^2.$$

Hence, f is not properly $R_{+}^2$-quasiconcave.

The following assumption plays a very important role on the existence of a strong solution for vector-valued mappings.

Definition 4

([20,21,28]). Let A be a nonempty subset of V and $f:A\to V$ be a vector-valued mapping. f is said to be downward directed on A, if for each $x_1,x_2\in A$ such that $\overline{x}\in A$

$$f\left(\overline{x}\right)\in f\left(x_1\right)-S,f\left(\overline{x}\right)\in f\left(x_2\right)-S.$$

Note that if f is lower semicontinuous and properly S-quasiconvex, then f is downward directed (see [21]).

Lemma 1

([21]). Let A be a nonempty compact subset of V and $f:A\to V$ be a vector-valued mapping. If the following conditions hold:

(i) 

f is upper semicontinuous on A;

(ii) 

$-f$ is downward directed on A;

then $Ma{x}_s{\bigcup }_{x\in A}\ne \varnothing$.

Definition 5

([29]). Let $F:E\to 2^V$ be a set-valued mapping.

(i) 

The nonempty compact-valued set-valued map F is said to be upper semicontinuous (u.s.c.) at $x_0\in E$ if for any net $\left\{x_{\alpha }\right\}\subset E$ with $x_{\alpha }\to x_0$ and for any $y_{\alpha }\in F\left(x_{\alpha }\right)$, there exist $y_0\in F\left(x_0\right)$ and a subnet $\left\{y_{\beta }\right\}$ of $\left\{y_{\alpha }\right\}$, such that $y_{\beta }\to y_0$.

(ii) 

The set-valued map F is said to be lower semicontinuous (l.s.c.) at $x_0\in E$, if for any net $\left\{x_{\alpha }\right\}\subset E$ with $x_{\alpha }\to x_0$ and for any $y_0\in F\left(x_0\right)$, there exists $y_{\alpha }\in F\left(x_{\alpha }\right)$ such that $y_{\alpha }\to y_0$.

(iii) 

The set-valued map F is said to be continuous at $x_0\in E$, if F is both u.s.c. and l.s.c. at $x_0$.

Theorem 1

([30,31]). Let E be a real locally convex Hausdorff topological vector space. Let $X_0$ be a nonempty compact convex subset of E. If $T:X_0\to 2^{X_0}$ is u.s.c., and for any $x\in X_0$, $T\left(x\right)$ is a nonempty, closed and convex set, then T has a fixed point.

3. Existence

In this section, we first investigate the existence results of the strong noncooperative equilibrium point of vector-valued games.

Next we depict general n-person strong noncooperative games with vector payoffs.

Let $N=\{1,2,\cdots ,n\}$ be a set of players, and $f_i:X={\prod }_{i=1}^n{X}_i\to V$ be the ith player’s vector payoff function. For each $i\in N$, let $X_{-i}:={\prod }_{j\in N\backslash \left\{i\right\}}{X}_j$ and $K_i:X_{-i}\to 2^{X_i}$ be nonempty set-valued mapping with respect to the ith player. A strategy $x_i^{*}\in K_i\left(x_{-i}^{*}\right)$ is called a strong noncooperative equilibrium point vector-valued game if the following system holds: For each $i\in N$

$$f_i(x_i^{*},x_{-i}^{*})=Ma{x}_s\bigcup _{x_i\in K_i\left(x_{-i}^{*}\right)}{f}_i(x_i,x_{-i}^{*}),$$

where $x_{-i}^{*}=(x_1^{*},\cdots ,x_{i-1}^{*},x_{i+1}^{*},\cdots x_n^{*})\in X_{-i}$.

Remark 2.

(i) In this game, every player maximizes all his goals. If $V=R$, $S=R_{+}$ and the set-valued mapping $K_i\equiv X_i$, the above game problem reduces to the well-known Nash equilibrium problem of the real-valued function (see [1,2]).

(ii) 

If $N=\{1,2\},f_2=-f_1$, and the constraint correspondence $K_i\equiv X_i$, the strong Nash equilibrium point with vector payoff $x^{*}=(x_1^{*},x_2^{*})$ reduces to the strong saddle point in [21].

(iii) 

If $V=R^n$, $N=\{1,2\},f_2=-f_1$, the set-valued mapping $K_i\equiv X_i$ and S is the lexicographic cone (not closed; see [23]), the strong Nash equilibrium point with vector payoff $x^{*}=(x_1^{*},x_2^{*})$ reduces to the lexicographic saddle point in [23,24].

Lemma 2.

For $i\in N$, let $f_i:X={\prod }_{i=1}^n{X}_i\to V$ be a vector-valued mapping and $K_i:X_{-i}\to 2^{X_i}$ be a set-valued mapping. Let ${\Phi }_i\left(x_{-i}\right):=Ma{x}_s{\bigcup }_{x_i\in K_i\left(x_{-i}\right)}{f}_i(x_i,x_{-i})\ne \varnothing$.

(i) 

If $f_i$ is lower semicontinuous and $K_i$ is l.s.c., then ${\Phi }_i\left(x_{-i}\right)$ is lower semicontinuous;

(ii) 

If $f_i$ is upper semicontinuous and $K_i$ is u.s.c. compact valued, then ${\Phi }_i\left(x_{-i}\right)$ is upper semicontinuous;

Proof. 

(i) We only need to verify that for any $z\in V$, the level set

$$Le{v}_{⪯}\left({\Phi }_i\right):=\{x_{-i}\in X_{-i}:{\Phi }_i\left(x_{-i}\right)\in z-S\}$$

is closed. Let the net $\left\{x_{\alpha }^{-i}\right\}\subset Le{v}_{⪯}\left({\Phi }_i\right)$ and $x_{\alpha }^{-i}\to x_0^{-i}$. Suppose that $x_0^{-i}\notin Le{v}_{⪯}\left({\Phi }_i\right)$. There exists $x_0^i\in K_i\left(x_0^{-i}\right)$ such that

$$f_i(x_0^i,x_0^{-i})\notin z-S.$$

For the above $x_0^i\in K_i\left(x_0^{-i}\right)$, by the l.s.c. of $K_i$, there exists $x_{\alpha }^i\in K_i\left(x_{\alpha }^{-i}\right)$ satisfying $x_{\alpha }^i\to x_0^i$. Since the net $\left\{x_{\alpha }^{-i}\right\}\subset Le{v}_{⪯}\left({\Phi }_i\right)$,

$$f_i(x_{\alpha }^i,x_{\alpha }^{-i})\in z-S.$$

By lower semicontinuous of $f_i$,

$$f_i(x_0^i,x_0^{-i})\in z-S.$$

This is a contradiction and hence ${\Phi }_i$ lower semicontinuous.

(ii) We only need to verify that for any $z\in V$, the level set

$$Le{v}_{⪰}\left({\Phi }_i\right):=\{x_{-i}\in X_{-i}:{\Phi }_i\left(x_{-i}\right)\in z+S\}$$

is closed. Let the net $\left\{x_{\alpha }^{-i}\right\}\subset Le{v}_{⪰}\left({\Phi }_i\right)$ and $x_{\alpha }^{-i}\to x_0^{-i}$. Since $\left\{x_{\alpha }^{-i}\right\}\subset Le{v}_{⪰}\left({\Phi }_i\right)$, there exists $\overline{x_{\alpha }^i}\in K_i\left(x_{\alpha }^{-i}\right)$ such that

$$f_i(\overline{x_{\alpha }^i},x_{\alpha }^{-i})\in z+S.$$

For the above $\overline{x_{\alpha }^i}\in K_i\left(x_{\alpha }^{-i}\right)$, by the u.s.c. of $K_i$, there exist $\overline{x_0^i}\in K_i\left(x_0^{-i}\right)$ and a subnet $\left\{\overline{x_{\beta }^i}\right\}$ of $\left\{\overline{x_{\alpha }^i}\right\}$ satisfying $\overline{x_{\beta }^i}\to \overline{x_0^i}$. By the upper semicontinuous of $f_i$,

$$f_i(\overline{x_0^i},x_0^{-i})\in z+S.$$

Thus,

$${\Phi }_i\left(x_0^{-i}\right)\in z+S.$$

Namely, $x_0^{-i}\in Le{v}_{⪰}\left({\Phi }_i\right)$ and hence ${\Phi }_i$ upper semicontinuous. This completes the proof. □

First, we give an existence result for strong noncooperative equilibrium point of vector-valued games.

Theorem 2.

For each $i\in N$, let $E_i$ be a real locally convex Hausdorff topological vector space and $X_i\subset E_i$ be a nonempty compact convex subset. Let $f_i:{\Pi }_{i=1}^n{X}_i\to V$ be the ith player’s vector payoff function and $K_i:X_{-i}\to 2^{X_i}$ be the set-valued mapping with respect to the ith player. The following conditions are satisfied:

(i) 

$\forall x_{-i}\in X_{-i}$, $-f_i(·,x_{-i})$ is downward directed on $K_i\left(x_{-i}\right)$;

(ii) 

$f_i$ is continuous on ${\Pi }_{i=1}^n{X}_i$;

(iii) 

$K_i$ is continuous with nonempty convex compact valued;

(iv) 

$\forall x_{-i}\in X_{-i}$, $f_i(·,x_{-i})$ is S-quasiconcave.

Then, there is at least a general strong Nash equilibrium point with vector payoff relative to ${\left\{(K_i,f_i)\right\}}_{i\in N}$.

Proof. 

For each $i\in N$, define the set-valued mapping $G_i:X_{-i}\to 2^{X_i}$ as follows:

$$G_i\left(x_{-i}\right)=\{y_i\in K_i\left(x_{-i}\right):f_i(x_i,x_{-i})\in f_i(y_i,x_{-i})-S,\forall x_i\in K_i\left(x_{-i}\right)\},x_{-i}\in X_{-i}.$$

By assumptions and Lemma 1, for each $x_{-i}\in X_{-i}$, there exists $\overline{y_i}\in X_i$ such that

$$f_i(\overline{y_i},x_{-i})=Ma{x}_s\bigcup _{x_i\in K_i\left(x_{-i}\right)}{f}_i(x_i,x_{-i}).$$

So, for any $i\in N$ and $x_{-i}\in X_{-i}$, $G_i\left(x_{-i}\right)$ is nonempty.

For each $i\in N$ and $x_{-i}\in X_{-i}$, let a net $\{y_{\alpha }^i:\alpha \in I\}\subset G_i\left(x_{-i}\right)$ and $y_{\alpha }^i\to y_0^i$. We have for each $x_i\in K_i\left(x_{-i}\right)$,

$$f_i(y_{\alpha }^i,x_{-i})\in f_i(x_i,x_{-i})+S.$$

By the continuity of $f_i$, for each $x_i\in K_i\left(x_{-i}\right)$,

$$f_i(y_0^i,x_{-i})\in f_i(x_i,x_{-i})+S.$$

Thus, $y_0^i\in G_i\left(x_{-i}\right)$ and hence $G_i\left(x_{-i}\right)$ is a closed set.

Let us next prove that for each $i\in N$, $G_i$ is convex valued. Let $y_1^i,y_2^i\in G_i\left(x_{-i}\right)$ and $t\in [0,1]$. By the definition of $G_i$, for any $x_i\in K_i\left(x_{-i}\right)$,

$$y_1^i\in K_i\left(x_{-i}\right),f_i(y_1^i,x_{-i})\in f_i(x_i,x_{-i})+S$$

and

$$y_2^i\in K_i\left(x_{-i}\right),f_i(y_2^i,x_{-i})\in f_i(x_i,x_{-i})+S.$$

Since $K_i$ is convex valued and $f_i(·,x_{-i})$ is S-quasiconcave, we have for any $x_i\in K_i\left(x_{-i}\right)$,

$$t{y}_1^i+(1-t)y_2^i\in K_i\left(x_{-i}\right),f_i(t{y}_1^i+(1-t)y_2^i,x_{-i})\in f_i(x_i,x_{-i})+S.$$

Namely, $t{y}_1^i+(1-t)y_2^i\in G_i\left(x_{-i}\right)$ and hence $G_i\left(x_{-i}\right)$ is a convex set.

Let us now prove that for each $i\in N$, $G_i$ is u.s.c. on $X_{-i}$. Since $X_i$ is compact, it is sufficient to show that $G_i$ is a closed map (see [29]). Let a net

$$\left\{(x_{\alpha }^{-i},y_{\alpha }^i)\right\}\subset Graph{G}_i:=\{(x_{-i},y_i)\in \prod _{i=1}^n{X}_i:f_i(y_i,x_{-i})=Ma{x}_s\bigcup _{x_i\in K_i\left(x_{-i}\right)}{f}_i(x_i,x_{-i})\}$$

and $(x_{\alpha }^{-i},y_{\alpha }^i)\to (x_0^{-i},y_0^i)$. By the definition of $Graph{G}_i$,

$$f_i(y_{\alpha }^i,x_{\alpha }^{-i})=Ma{x}_s{f}_i(K_i\left(x_{\alpha }^{-i}\right),x_{\alpha }^{-i}).$$

By the continuity of $f_i$ and Lemma 2,

$$f_i(y_0^i,x_0^{-i})=Ma{x}_s{f}_i(K_i\left(x_0^{-i}\right),x_0^{-i}).$$

Namely, $(x_0^{-i},y_0^i)\in Graph{G}_i$ and hence $G_i$ is u.s.c..

Let $G={\prod }_{i=1}^n{G}_i$. Then, by Kakutani–Fan–Glicksberg fixed point theorem, there exists $x^{*}=(x_1^{*},x_2^{*},\cdots ,x_n^{*})\in {\prod }_{i=1}^n{X}_i$ such that

$$x^{*}\in G\left(x^{*}\right);$$

i.e., for each $i\in N$

$$f_i(x_i^{*},x_{-i}^{*})=Ma{x}_s\bigcup _{x_i\in K_i\left(x_{-i}^{*}\right)}{f}_i(x_i,x_{-i}^{*}).$$

This completes the proof. □

We give the following example to explain that the assumption (i) in Theorem 2 is essential.

Example 2.

Let $N=\{1,2\}$, $X_1=X_2=[\frac{1}{2},1]$ and $S=\{(s_1,s_2)\in R^2:s_1\ge 0,s_2\ge 0|\}$. Let $K_1\equiv X_1,K_2\equiv X_2$. Let $f_1,f_2:X_1\times X_2\to R^2$ be player’s vector payoffs functions:

$$f_1(x_1,x_2)=x_2(x_1,\sqrt{1-x_1^2})$$

and

$$f_2(x_1,x_2)=x_1(x_2,1-x_2).$$

Clearly, all conditions of Theorem 2 hold except assumption (i). Indeed, letting $x_2=1$, $-f_1(x_1,1)$ is not downward directed on $X_1$. Let $x_1^{\prime }=\frac{1}{2},x_1^{″}=1$. There is no $\overline{x}\in X_1$ such that

$$f_1(\overline{x},1)\in f_1(\frac{1}{2},1)+S,f_1(\overline{x},1)\in f_1(1,1)+S.$$

We claim that the general strong Nash equilibrium point set is empty. In fact, for each $x_1\in X_1,x_2\in X_2$,

$$Ma{x}_S\bigcup _{x_1\in X_1}{f}_1(x_1,x_2)=\varnothing ,Ma{x}_S\bigcup _{x_2\in X_2}{f}_2(x_1,x_2)=\varnothing .$$

Remark 3.

In [13,14,15], these authors investigated the ideal Nash equilibria by the feasible criterion mapping with real-valued payoff functions. The following example illustrates that our results are different from existing results.

Example 3.

Let $N=\{1,2\}$, $X_1=[-\frac{1}{2},0],X_2=[0,1]$ and $S=\{(s_1,s_2)\in R^2:s_2\ge |s_1\left|\right\}$. Let $K_1\equiv X_1,K_2\equiv X_2$. Let $f_1,f_2:X_1\times X_2\to R^2$ be player’s vector payoffs functions:

$$f_1(x_1,x_2)=(f_1^1,f_1^2)=(x_1{x}_2,-2x_1{x}_2)$$

and

$$f_2(x_1,x_2)=(f_2^1,f_2^2)=(x_1{x}_2,x_1).$$

Clearly, all conditions of Theorem 2 hold, namely, Theorem 3.1 is applicable. Indeed, letting $x_1^{*}=-\frac{1}{2}$, $x_2^{*}=1$,

$$f_1(x_1^{*},x_2^{*})=Ma{x}_s\bigcup _{x_1\in K_1\left(x_2^{*}\right)}{f}_1(x_1,x_2^{*})$$

and

$$f_2(x_1^{*},x_2^{*})=Ma{x}_s\bigcup _{x_2\in K_2\left(x_1^{*}\right)}{f}_2(x_1^{*},x_2).$$

Namely, $(-\frac{1}{2},1)$ is a general strong Nash equilibrium point with vector payoff. However,

$$f_1^1(x_1^{*},x_2^{*})=-\frac{1}{2}<f_1^1(x_1^{*},0)=0.$$

Hence, our result are different from ones in [13,14,15].

Remark 4.

In [22], the authors investigated an existence theorem of general Ky Fan points with set-valued mappings F. The following example illustrates that when Theorem 2 in [22] is not applicable, Theorem 2 is applicable.

Example 4.

Let $N=\{1,2\}$, $X_1=[0,2],X_2=[0,1]$ and $S=R_{+}^2$. Let $K_1\equiv [0,2],K_2\equiv [0,1]$. Let $f_1,f_2:X_1\times X_2\to R^2$ be player’s vector payoffs functions:

$$f_1(x_1,x_2)=x_2(x_1,{(x_1-1)}^2)$$

and

$$f_2(x_1,x_2)=x_2(x_1,x_1^2).$$

Clearly, all conditions of Theorem 2 hold, namely, Theorem 2 is applicable. Indeed, letting $x_1^{*}=2$, $x_2^{*}=1$,

$$f_1(x_1^{*},x_2^{*})=(2,1)=Ma{x}_s\bigcup _{x_1\in K_1\left(x_2^{*}\right)}{f}_1(x_1,x_2^{*})$$

and

$$f_2(x_1^{*},x_2^{*})=(2,4)=Ma{x}_s\bigcup _{x_2\in K_2\left(x_1^{*}\right)}{f}_2(x_1^{*},x_2).$$

Namely, $(2,1)$ is a general strong Nash equilibrium point with vector payoff. Nevertheless, taking $x_2=1$, $x_1^1=0,x_1^2=1,l=\frac{1}{2}$, we have

$$f_1(l{x}_1^1+(1-l)x_1^2,1)=(\frac{1}{2},\frac{1}{4})\notin f_1(x_1^1,1)+R_{+}^2$$

and

$$f_1(l{x}_1^1+(1-l)x_1^2,1)=(\frac{1}{2},\frac{1}{4})\notin f_1(x_1^2,1)+R_{+}^2.$$

Thus, $f_1(·,x_2)$ is not $R_{+}^2$-properly quasiconcave. Namely, Theorem 2 in [22] is not applicable.

Let $N=\{1,2\},f_2=-f_1$. By Theorem 2, we can obtain the next existence result of general strong saddle point of vector-valued mapping soon.

Theorem 3.

Let $E_1,E_2$ be a real locally convex Hausdorff topological vector space and $X_1,X_2$ be nonempty compact convex subset of $E_1,E_2$, respectively. Let $f:X_1\times X_2\to V$ be a vector payoff function and $K_1:X_2\to 2^{X_1},K_2:X_1\to 2^{X_2}$ be set-valued mappings. The following conditions are satisfied:

(i) 

$\forall x_1\in X_1$, $f(x_1,·)$ is downward directed on $K_2\left(x_1\right)$; $\forall x_2\in X_2$, $-f(·,x_2)$ is downward directed on $K_1\left(x_2\right)$;

(ii) 

f is continuous on $X_1\times X_2$;

(iii) 

$K_1,K_2$ are continuous with nonempty convex compact valued;

(iv) 

$\forall x_1\in X_1$, $f(x_1,·)$ is S-quasiconvex; $\forall x_2\in X_2$, $f(·,x_2)$ is S-quasiconcave.

Then, there exist $x_1^{*}\in K_1\left(x_2^{*}\right)$ and $x_2^{*}\in K_2\left(x_1^{*}\right)$ such that

$$f(x_1^{*},x_2^{*})=Ma{x}_s\bigcup _{x_1\in K_1\left(x_2^{*}\right)}f(x_1,x_2^{*})=Mi{n}_s\bigcup _{x_2\in K_2\left(x_1^{*}\right)}f(x_1^{*},x_2).$$

Remark 5.

When $K_1\equiv X_1$ and $K_2\equiv X_2$, the above result reduces to Theorem 1 in [21].

In the following, we investigate the existence results for general strong noncooperative equilibrium point of vector-valued games with single-leader–multi-follower framework.

Next, we depict general strong noncooperative equilibria of vector-valued games with single-leader–multi-follower framework.

The leaders and the followers choose their strategies in sequence. After knowing the leaders’ strategy $y\in Y$, the followers play a parametric game. Let $N=\{1,2,\cdots ,n\}$ be a set of followers. Let $X_i$ be the pure strategy set of the ith follower and Y be the pure strategy set of the leader. Let $F_i:Y\times {\prod }_{i=1}^n{X}_i\to V$ be the parametric vector payoff function of the ith follower, ${\Theta }_i:Y\times X_{-i}\to 2^{X_i}$ be the parametric set-valued mapping of the ith follower and $\Gamma :Y\times {\prod }_{i=1}^n{X}_i\to V$ be the vector payoff function of the leader. We define $S\left(y\right)$ as the complete set of strong noncooperative equilibrium point of the followers:

$$S\left(y\right)=\{x^{*}\in \prod _{i=1}^n{X}_i:F_i(y,x_i^{*},x_{-i}^{*})=Ma{x}_s\bigcup _{x_i\in {\Theta }_i(y,x_{-i}^{*})}{F}_i(y,x_i,x_{-i}^{*}),\forall i\in N\},y\in Y.$$

A strategy $(y^{*},x^{*})$ is called a strong noncooperative equilibrium point of vector-valued games with single-leader–multi-follower framework if the following equality holds:

$$\Gamma (y^{*},x^{*})=Ma{x}_s\bigcup _{y\in Y,x\in S\left(y\right)}\Gamma (y,x).$$

First, we prove that $S\left(y\right)$ is u.s.c. and compact valued.

Lemma 3.

Assume that $S\left(y\right)\ne ,\forall y\in Y.$ Let $X_i$ be a compact set, $\forall i\in N$. For each $i\in N$, if $F_i:Y\times {\prod }_{i=1}^n{X}_i\to V$ is continuous and ${\Theta }_i:Y\times X_{-i}\to 2^{X_i}$ is continuous with nonempty compact valued, then S is u.s.c. and compact valued.

Proof. 

Since X is compact, it is sufficient to show that Graph(S) is closed. Let a net $\left\{(y_{\alpha },x_{\alpha })\right\}\subset Graph\left(S\right)$ and $(y_{\alpha },x_{\alpha })\to (y_0,x_0)$. By the definition of $S\left(y\right)$, for each $i\in N$,

$$F_i(y_{\alpha },x_{\alpha }^i,x_{\alpha }^{-i})=Ma{x}_s{F}_i(y_{\alpha },{\Theta }_i(y_{\alpha },x_{\alpha }^{-i}),x_{\alpha }^{-i}).$$

Since $F_i$ and ${\Theta }_i$ are continuous, by Lemma 2,

$$F_i(y_0,x_0^i,x_0^{-i})=Ma{x}_s{F}_i(y_0,{\Theta }_i(y_0,x_0^{-i}),x_0^{-i}).$$

Namely, $(y_0,x_0)\in Graph\left(S\right)$, i.e., S is u.s.c.

By assumptions, it is easy to prove that S is closed valued. Since X is compact, S is compact valued. This completes the proof. □

Next, we give an existence result for general strong noncooperative equilibria of vector-valued games with single-leader–multi-follower framework.

Theorem 4.

For each $i\in N$, let $E_i$ be a real locally convex Hausdorff topological vector space and $X_i\subset E_i$ be a nonempty compact convex subset. Let Y be a compact set. The following conditions are satisfied:

(i) 

$\forall x_{-i}\in X_{-i}$, $-F_i(·,x_{-i})$ is downward directed on ${\Theta }_i\left(x_{-i}\right)$; $-\Gamma$ is downward directed on $Y\times {\prod }_{i=1}^n{X}_i$;

(ii) 

$F_i$ is continuous with nonempty compact values and ${\Theta }_i$ is continuous with nonempty convex compact values;

(iii) 

Γ is upper semicontinuous;

(iv) 

$\forall y\in Y,x_{-i}\in X_{-i}$, $F_i(y,·,x_{-i})$ is S-quasiconcave.

Then, there is at least a general strong Nash equilibrium point of of vector-valued games with single-leader–multi-follower framework.

Proof. 

Define a function $\Phi \left(y\right):Y\to V$,

$$\Phi \left(y\right):=Ma{x}_s\bigcup _{x\in S\left(y\right)}\Gamma (y,x).$$

Since $\Gamma$ is upper semicontinuous and $-\Gamma$ is downward directed, by Lemma 1, $\Phi \left(y\right)$ is well defined. By Theorem 2 and Lemma 3, for each $y\in Y$, $S\left(y\right)$ is u.s.c. nonempty compact valued. Thus, by the upper semicontinuity of $\Gamma$ and Lemma 2, the function $\Phi$ is upper semicontinuous. Since Y is compact and $-\Gamma$ is downward directed, by Lemma 1, there exists $y^{*}\in Y$ such that

$$\Phi \left(y^{*}\right)=Ma{x}_s\bigcup _{y\in Y}\Phi \left(y\right).$$

Take $x^{*}\in S\left(y^{*}\right)$ satisfying $\Gamma (y^{*},x^{*})=\Phi \left(y^{*}\right)$. Namely, $(y^{*},x^{*})$ is a general strong Nash equilibrium point of single-leader–multi-follower games with vector payoff. This completes the proof. □

Remark 6.

When $F_i,\Gamma$ is real-valued functions and $S=R_{+}$, condition (i) holds naturally. Hence, if $V=R$, $S=R_{+}$ and ${\Theta }_i\equiv X_i$, the above result reduces to Corollary 3.8 in [18].

4. Generic Stability

In this section, we investigate the stability of strong noncooperative equilibrium point with vector payoff when the vector payoff functions and constraint correspondence set-valued mappings are perturbed.

For any $i\in N$, let $E_i,V$ be Banach spaces. Let M be the space of $(f_1,\dots ,f_n,K_1,\dots ,K_n)$ with all assumptions of Theorem 2. For $m\in M$, let $\Omega \left(m\right)$ be the set of all strong noncooperation equilibria relative to ${\left\{(f_i,K_i)\right\}}_{i\in N}$.

We define the distance $\rho$ on M by

$$\rho (m,m^{\prime }):=\underset{x\in X}{sup}\sum _{i=1}^n\parallel f_i\left(x\right)-f_i^{\prime }\left(x\right)\parallel +\underset{x_{-i}\in X_{-i}}{sup}\sum _{i=1}^n{h}_i(K_i\left(x_{-i}\right),K_i^{\prime }\left(x_{-i}\right)),$$

where $m=(f_1,\dots ,f_n,K_1,\dots ,K_n),m^{\prime }=(f_1^{\prime },\dots ,f_n^{\prime },K_1^{\prime },\dots ,K_n^{\prime })\in M$ and $h_i$ is the Hausdorff distance by the norm on $E_i$. Obviously, $(M,\rho )$ is a complete metric space.

Definition 6

([5]). Let $m\in M$. $x\in \Omega \left(m\right)$ is said to be an essential point, if for any open neighborhood U of x, there exists $\delta >0$ such that

$$U\bigcap \Omega \left(m^{\prime }\right)\ne \varnothing ,$$

for any $m^{\prime }\in M$ and $\rho (m,m^{\prime })<\delta$. m is said to be essential, if for any $x\in \Omega \left(m\right)$, x is an essential point.

For each $m\in M$, m is essential if and only if the set-valued mapping $\Omega :M\to 2^X$ is l.s.c. (see Theorem 5 in [5]).

Definition 7

([32]). Let Q be a subset of M.

(i) 

Q is said to be dense if $clQ=M$;

(ii) 

Q is residual if $Q={\bigcap }_{i=1}^{\infty }{Q}_i$, where $Q_i$ is an open dense subset of M, $\forall i$.

Theorem 5.

The set-valued mapping $\Omega :M\to 2^X$ is u.s.c. and compact valued.

Proof. 

Since $X_i$ is compact, it is sufficient to show that Graph($\Omega$) is closed. Let the net $\left\{(f_{\alpha }^1,\dots ,f_{\alpha }^n,K_{\alpha }^1,\dots ,K_{\alpha }^n,x_{\alpha }^1,\dots ,x_{\alpha }^n)\right\}\subseteq Graph(\Omega )$ and $(f_{\alpha }^1,\dots ,f_{\alpha }^n,K_{\alpha }^1,\dots ,K_{\alpha }^n,x_{\alpha }^1,\dots ,x_{\alpha }^n)\to (f_0^1,\dots ,f_0^n,K_0^1,\dots ,K_0^n,x_0^1,\dots ,x_0^n).$ Hence, for each $i\in N$,

$$f_{\alpha }^i(x_{\alpha }^i,x_{\alpha }^{-i})=Ma{x}_s{f}_{\alpha }^i(K_{\alpha }^i\left(x_{\alpha }^{-i}\right),x_{\alpha }^{-i}).$$

First, we show that for each $i\in N$, $x_0^i\in K_0^i\left(x_0^{-i}\right)$. Let d be the distance on $X_i\times X_i$.

$$\begin{array}{ccc}\hfill d(x_0^i,K_0^i\left(x_0^{-i}\right))& \le & d(x_0^i,x_{\alpha }^i)+d(x_{\alpha }^i,K_{\alpha }^i\left(x_{\alpha }^{-i}\right))+h(K_{\alpha }^i\left(x_{\alpha }^{-i}\right),K_0^i\left(x_{\alpha }^{-i}\right))+h(K_0^i\left(x_{\alpha }^{-i}\right),K_0^i\left(x_0^{-i}\right))\hfill \\ & \le & d(x_0^i,x_{\alpha }^i)+\rho (m_{\alpha },m_0)+h(K_0^i\left(x_{\alpha }^{-i}\right),K_0^i\left(x_0^{-i}\right))\to 0.\hfill \end{array}$$

Namely, $x_0^i\in K_0^i\left(x_0^{-i}\right)$.

Next, we show that for each $i\in N$,

$$f_0^i(x_0^i,x_0^{-i})=Ma{x}_s{f}_0^i(K_0^i\left(x_0^{-i}\right),x_0^{-i}).$$

Suppose that this is not true. There exist $i\in N$ and $u_0^i\in K_0^i\left(x_0^{-i}\right)$ such that

$$f_0^i(u_0^i,x_0^{-i})\notin f_0^i(x_0^i,x_0^{-i})-S.$$

By the closedness of the cone S, there exists an open neighborhood U of $\theta$,

$$f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,x_0^{-i})+U\subset V\backslash (-S).$$

By the l.s.c. of $K_0^i$, for $u_0^i\in K_0^i\left(x_0^{-i}\right)$, there exists $u_{\alpha }^i\in K_0^i\left(x_{\alpha }^{-i}\right)$ such that $u_{\alpha }^i\to u_0^i$. Since $h(K_{\alpha }^i\left(x_{\alpha }^{-i}\right),K_0^i\left(x_{\alpha }^{-i}\right))\to 0$, we can choose subnets $\left\{K_{{\beta }_k}^i\right\},\left\{x_{{\beta }_k}^i\right\}$ such that

$$h(K_{{\beta }_k}^i\left(x_{{\beta }_k}^{-i}\right),K_0^i\left(x_{{\beta }_k}^{-i}\right))<\frac{1}{k},$$

which implies that there exists $\overline{u_{{\beta }_k}^i}\in K_{{\beta }_k}^i\left(x_{{\beta }_k}^{-i}\right)$ such that $\parallel \overline{u_{{\beta }_k}^i}-u_{{\beta }_k}^i\parallel <\frac{1}{k}$. Then,

$$\parallel \overline{u_{{\beta }_k}^i}-u_0^i\parallel \le \parallel \overline{u_{{\beta }_k}^i}-u_{{\beta }_k}^i\parallel +\parallel u_{{\beta }_k}^i-u_0^i\parallel <\frac{1}{k}+\parallel u_{{\beta }_k}^i-u_0^i\parallel ,$$

and hence $\overline{u_{{\beta }_k}^i}\to u_0^i.$ Since $(f_{\alpha }^1,\dots ,f_{\alpha }^n)\to (f_0^1,\dots ,f_0^n)$ by $\rho$, for the above open neighborhood U, for any sufficiently large $\beta$,

$$(f_{\beta }^i(\overline{u_{\beta }^i},x_{\beta }^{-i})-f_{\beta }^i(x_{\beta }^i,x_{\beta }^{-i}))-(f_0^i(\overline{u_{\beta }^i},x_{\beta }^{-i})-f_0^i(x_{\beta }^i,x_{\beta }^{-i}))\in \frac{1}{2}U.$$

By the continuity of $f_0$,

$$f_0^i(\overline{u_{\beta }^i},x_{\beta }^{-i})-f_0^i(x_{\beta }^i,x_{\beta }^{-i})\to f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,x_0^{-i}).$$

For the open neighborhood $(f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,x_0^{-i})+\frac{1}{2}U+S)$ of $f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,$$x_0^{-i})$, we have that for any sufficiently large $\beta$,

$$f_0^i(\overline{u_{\beta }^i},x_{\beta }^{-i})-f_0^i(x_{\beta }^i,x_{\beta }^{-i})\in f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,x_0^{-i})+\frac{1}{2}U+S.$$

Thus, we have that for sufficiently large k,

$$\begin{array}{ccc}& & f_{{\beta }_k}^i(\overline{u_{{\beta }_k}^i},x_{{\beta }_k}^{-i})-f_{{\beta }_k}^i(x_{{\beta }_k}^i,x_{{\beta }_k}^{-i})\hfill \\ & =& (f_{{\beta }_k}^i(\overline{u_{{\beta }_k}^i},x_{{\beta }_k}^{-i})-f_{{\beta }_k}^i(x_{{\beta }_k}^i,x_{{\beta }_k}^{-i}))-(f_0^i(\overline{u_{{\beta }_k}^i},x_{{\beta }_k}^{-i})-f_0^i(x_{{\beta }_k}^i,x_{{\beta }_k}^{-i}))+(f_0^i(\overline{u_{{\beta }_k}^i},x_{{\beta }_k}^{-i})-f_0^i(x_{{\beta }_k}^i,x_{{\beta }_k}^{-i}))\hfill \\ & \in & f_0^i(u_0^i,x_0^{-i})-f_0^i(x_0^i,x_0^{-i})+U+S\hfill \\ & \subseteq & V\backslash (-S)+S=V\backslash (-S),\hfill \end{array}$$

which contradicts the fact that $\left\{(f_{\alpha }^1,\dots ,f_{\alpha }^n,K_{\alpha }^1,\dots ,K_{\alpha }^n,x_{\alpha }^1,\dots ,x_{\alpha }^n)\right\}\subseteq Graph(\Omega )$. Hence, Graph($\Omega$) is closed. It is easy concerning that $\Omega$ is compact valued. This completes the proof. □

Lemma 4

([33]). Let M be a complete metric space and X be a topological space. Suppose that $\Omega :M\to 2^X$ is compact valued and upper semicontinuous with nonempty values. Then, a dense residual subset Q of M exists such that Ω is lower semicontinuous on Q.

By Definition 6, Lemma 4 and Theorem 5, we obtain the following result immediately.

Theorem 6.

There exists a dense residual set $Q\subseteq M$ such that m is essential for any $m\in Q$.

Remark 7.

The results in [17,18] require that the ordering cone has a nonempty interior, owing to the concept of weakly Pareto solution. Hence, our results are different from ones in the literatures.

5. Concluding Remark

In this paper, by virtue of the Kakutani–Fan–Glicksberg fixed point theorem, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals. Moreover, we establish the upper semicontinuous of parametric strong noncooperative equilibrium point set and the existence of strong equilibrium point of vector-valued games with the single-leader–multi-follower framework. Finally, we obtain some results on the stability of general strong noncooperative equilibrium point vector-valued games when the vector payoff functions and constraint correspondence set-valued mappings are perturbed.