Existence of Weakly Pareto–Nash Equilibrium for Multiobjective Games with Infinitely Many Players

Abstract

Our work proves the existence of weakly Pareto–Nash equilibrium (PNE) in multiobjective games (MGs) with infinitely many players. First, we demonstrate the existence of weakly PNE under compactness assumptions by using the intersection theorem. Then, we extend the intersection theorem to the non-compact case and obtain a new intersection theorem. Finally, we prove the existence of weakly PNE in MGs with infinitely many players without compactness assumptions.

1. Introduction

In game theory, the existence of equilibrium points is often demonstrated using fixed point theorems. Nash [1] was the first to use this theorem to demonstrate non-cooperative equilibrium in n-person games. Subsequently, Shafer and Sonnenschein [2] extended Nash’s existence result [1] to generalized games. Researchers further explored equilibrium concepts in games involving an infinite number of participants. Aumann [3] introduced an economic model for a continuum of players. Based on Aumann’s [3] model, Khan [4] and Noguchi [5] demonstrated the existence of equilibrium for games defined over a measurable player space. In the case of games with countably infinite players, Yannelis and Prabhakar [6] proved that a Nash equilibrium exists. Yuan [7] further demonstrated that games with infinitely many participants admit an equilibrium. Recently, Yang and Song [8] proposed a unified framework to establish equilibria in settings involving infinitely many participants.

The above studies on Nash equilibrium are all single-objective games, but decision-makers often need to consider multiple objectives before making a decision. Therefore, compared to single-objective games, MGs are closer to real life and are of greater concern. In 1956, Blackwell [9] explored strategic interactions in zero-sum settings where payoffs were represented as vectors. In 1959, Shapley and Rigby [10] proposed the concept of equilibrium for a two-person MG. Then, the existence of PNE was proved by Wang [11] and Yu and Yuan [12]. The stability of weakly PNE has been studied in [13,14,15]. The work of MGs was extended to generalized MGs [16,17], multi-leader–follower MGs [18,19], fuzzy MGs [20,21,22,23], discontinuous MGs [24] and multiobjective population games [25,26,27,28,29].

In addition, Fan [30] first proved the convex intersection theorem. Then, Ma [31] extended Fan’s [30] intersection theorem to the case where the indexed set I is an infinite set. Furthermore, Abalo and Kostreva [32] extended Ma’s theorem [31] to the case of alliances and demonstrated the existence of Berge equilibrium. By extending Ma’s theorem [31], Yang and Pu [33] demonstrated that Nash equilibria can be guaranteed in large games. Shih and Tan [34] extended the intersection theorem to the non-compact case. For more information on the extension and application of the intersection theorem, refer to references [35,36,37,38].

In games involving infinitely many players, such as large-scale markets, public goods provision, or traffic networks, each individual’s action has an almost negligible impact on the overall outcome. Yet collectively, these actions shape the equilibrium behavior of the system. In such settings, the concept of weakly PNE has been proved to be useful. A weakly PNE is a Nash equilibrium that cannot be strictly Pareto dominated by any other equilibrium, ensuring that no other feasible outcome makes all players better off and at least one strictly better off. This relaxation of the Pareto efficiency condition provides a more practical and attainable notion of stability in large games where achieving global optima is often infeasible [39].

In this paper, we demonstrate that weakly PNE exists in MGs with infinitely many players under compactness assumption. Then, we further establish weakly PNE in MGs with infinitely many players under non-compactness assumption by extending the intersection theorem to the non-compact case. Our existence results generalize the findings of Wang [11], Yu and Yuan [12], Yuan and Tarafdar [40], and Yang and Song [8].

The remainder of the paper is structured as follows: Section 2 presents the necessary preliminaries and lemmas. In Section 3, we study the existence of weakly PNE in MGs with infinitely many participants under the compactness assumption. In Section 4, to prove the existence of weakly PNE under the non-compactness assumption, we extend the intersection theorem to the non-compact case. In Section 5, we make some brief and concise conclusions.

2. Preliminaries

First, we introduce several foundational definitions and auxiliary results.

Definition 1 

([41]). Let A and B be topological spaces, and consider a correspondence $\xi :A\to 2^B$. Then,

(i) 

ξ is called upper semicontinuous (usc) at $a\in A$ if for any open set $G\subset B$ with $\xi \left(a\right)\subset G$, there exists an open neighborhood U of a, such that for any $a^{\prime }\in U$, $\xi \left(a^{\prime }\right)\subset G$.

(ii) 

ξ is called lower semicontinuous (lsc) at $a\in A$ if for every open set $G\subset B$ satisfying $G\cap T\left(a\right)\ne \varnothing$, there exists an open neighborhood U of a, such that for any $a^{\prime }\in U$, $G\cap \xi \left(a^{\prime }\right)\ne \varnothing$.

(iii) 

ξ is continuous at $a\in A$ if ξ satisfies both usc and lsc at a.

For each positive integer k, define

$$\begin{array}{c}\hfill R_{+}^k=\{\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_k)\in R^k:{\xi }_{\mu }\ge 0,\mu =1,\cdots ,k\},\end{array}$$

and its interior by

$$\begin{array}{c}\hfill \mathrm{int}{R}_{+}^k=\{\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_k)\in R^k:{\xi }_{\mu }>0,\mu =1,\cdots ,k\}.\end{array}$$

Definition 2 

([42]). Let $\xi :B\to R_{+}^k$ be a vector-valued function defined on a nonempty convex set $B\subset R^n$. If for any $b_1,b_2\in B$ and for any $\theta \in (0,1)$, we have

$$\theta \xi \left(b_1\right)+(1-\theta )\xi \left(b_2\right)-\xi (\theta b_1+(1-\theta )b_2)\in R_{+}^k,$$

then ξ is called $R_{+}^k$-convex on B.

If $\xi$ is $R_{+}^k$-convex on B, then $-\xi$ is said to be $R_{+}^k$-concave on B.

Definition 3 

([42]). Let $\xi :B\to R_{+}^k$ be a vector-valued function defined on a nonempty convex set $B\subset R^n$. For any $b_1,b_2\in B$, for any $\theta \in (0,1)$ and for any $w\in R^k$, if $\xi \left(b_1\right)\in w+R_{+}^k,\xi \left(b_2\right)\in w+R_{+}^k$, we have

$$\xi (\theta b_1+(1-\theta )b_2)\in w+R_{+}^k,$$

then ξ is called $R_{+}^k$-quasiconcave on B.

If $\xi$ is $R_{+}^k$-quasiconcave on B, then $-\xi$ is said to be $R_{+}^k$-quasiconvex on B.

Lemma 1 

(Fan [30]). Let $\{S_{\mu }:\mu \in \mathcal{M}\}$ be a nonempty compact convex set in a Hausdorff topological vector space (TVS) $E_{\mu }$, where $\mathcal{M}=\{1,2,\cdots ,n\}$. Let $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }$ and $S_{-\mu }={\prod }_{\nu \in \mathcal{M},\nu \ne \mu }{S}_{\nu }$. Let $\left\{A_{\mu }:\mu \in \mathcal{M}\right\}$ be a subset of S, and meet the following conditions:

(i) 

for any $\mu \in \mathcal{M}$ and $s_{\mu }\in S_{\mu }$, the section $A_{\mu }\left(s_{\mu }\right)=\left\{s_{-\mu }\in S_{-\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu }\right\}$ is open in $S_{-\mu }$;

(ii) 

for any $\mu \in \mathcal{M}$ and $s_{-\mu }\in S_{-\mu }$, the section $A_{\mu }\left(s_{-\mu }\right)=\left\{s_{\mu }\in S_{\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu }\right\}$ is nonempty and convex.

Then ${\bigcap }_{\mu \in \mathcal{M}}{A}_{\mu }\ne \varnothing$.

Then, Ma [31] extended Lemma 1 to the case where the indexed set $\mathcal{M}$ is an infinite set and obtained the following result.

To avoid ambiguity in the subsequent exposition, we uniformly assume that $\{S_{\mu }:\mu \in \mathcal{M}\}$ is a nonempty compact convex set in a Hausdorff TVS $E_{\mu }$, and $\{{\tilde{S}}_{\mu }:\mu \in \mathcal{M}\}$ is a nonempty convex set in a Hausdorff locally convex TVS ${\tilde{E}}_{\mu }$, where $\mathcal{M}$ is a finite or infinite indexed set.

Lemma 2 

(Ma [31]). Let $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }$ and $S_{-\mu }={\prod }_{\nu \in \mathcal{M},\nu \ne \mu }{S}_{\nu }$. Let ${\left\{A_{\mu }\right\}}_{\mu \in \mathcal{M}}\subset S$ satisfy the following conditions:

(i) 

for any $\mu \in \mathcal{M}$ and $s_{\mu }\in S_{\mu }$, the section $A_{\mu }\left(s_{\mu }\right)=\left\{s_{-\mu }\in S_{-\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu }\right\}$ is open in $S_{-\mu }$;

(ii) 

for any $\mu \in \mathcal{M}$ and $s_{-\mu }\in S_{-\mu }$, the section $A_{\mu }\left(s_{-\mu }\right)=\left\{s_{\mu }\in S_{\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu }\right\}$ is nonempty and convex.

Then ${\bigcap }_{\mu \in \mathcal{M}}{A}_{\mu }\ne \varnothing$.

The following is an extended version of Lemma 2 in Yang and Pu [33].

Lemma 3 

([33]). Let $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }$ and $S_{-\mu }={\prod }_{\nu \in \mathcal{M},\nu \ne \mu }{S}_{\nu }$. Let $\left\{A_{\mu ,\epsilon }:\mu \in \mathcal{M},\epsilon >0\right\}\subset S$ and meet the following conditions:

(i) 

for any $\mu \in \mathcal{M},\epsilon >0$ and $s_{\mu }\in S_{\mu }$, the section $A_{\mu ,\epsilon }\left(s_{\mu }\right)=\left\{s_{-\mu }\in S_{-\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu ,\epsilon }\right\}$ is open in $S_{-\mu }$;

(ii) 

for any $\mu \in \mathcal{M},\epsilon >0$ and $s_{-\mu }\in S_{-\mu }$, the section $A_{\mu ,\epsilon }\left(s_{-\mu }\right)=\left\{s_{\mu }\in S_{\mu }\mid \left(s_{\mu },s_{-\mu }\right)\in A_{\mu ,\epsilon }\right\}$ is nonempty and convex;

(iii) 

for any $\mu \in \mathcal{M}$ and ${\epsilon }_2\ge {\epsilon }_1>0$, $A_{\mu ,{\epsilon }_1}\subset A_{\mu ,{\epsilon }_2}$.

Then ${\displaystyle \bigcap _{\mu \in \mathcal{M},\epsilon >0}}{\overline{A}}_{\mu ,\epsilon }\ne \varnothing$, where $\overline{A}$ represents the closure of the set A.

Lemma 4 

(Fan [43]). Let $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }$. Assume C is a subset of $S\times S$ and that it meets the following conditions:

(i) 

for any $s\in S,\{b\in S|(s,b)\in C\}$ is an open set of S;

(ii) 

for any $b\in S,\{s\in S|(s,b)\in C\}$ is a convex set;

(iii) 

for any $s\in S,(s,s)\notin C$.

Then there exists at least one point $b^{*}\in S$ such that for any $s\in S$, $(s,b^{*})\notin C$.

Lemma 5. 

[44] For any $\mu \in \mathcal{M}$, let $D_{\mu }$ be a nonempty compact and convex subset of $S_{\mu }$. Assume that $C_{\mu },B_{\mu }:{\prod }_{\mu \in \mathcal{M}}{\tilde{S}}_{\mu }\to 2^{D_{\mu }}$ satisfy the following conditions:

(i) 

for any $\tilde{s}\in \tilde{S}$, co $C_{\mu }\left(\tilde{s}\right)\subset B_{\mu }\left(\tilde{s}\right)$ and $B_{\mu }\left(\tilde{s}\right)\ne \varnothing$;

(ii) 

for any ${\xi }_{\mu }\in D_{\mu }$, $C_{\mu }^{-1}\left({\xi }_{\mu }\right)$ is open in $\tilde{S}$.

Then there exists at least one point $\tilde{s}\in D$ such that for any $\mu \in \mathcal{M},{\tilde{s}}_{\mu }\in B_{\mu }\left({\tilde{s}}_{\mu }\right)$.

3. The Existence of Weakly PNE with Compactness Assumption

First, we give the model of MGs.

Let $\mathcal{M}$ be a infinite indexed set. Assume that a MG model is denoted as $\Gamma ={\left\{(S_{\mu },F_{\mu })\right\}}_{\mu \in \mathcal{M}}$, where

(i)

$S_{\mu }$ is the strategy set of player $\mathcal{M}$;

(ii)

$F_{\mu }=\{f_{\mu }^1,\cdots ,f_{\mu }^k\}:S={\displaystyle \prod _{\mu =1}^n}{S}_{\mu }\to R^k$ is the vector-valued payoff function of player $\mathcal{M}$, where $K=\{1,2\cdots ,k\}$ is the target set.

Definition 4. 

For any $\mu \in \mathcal{M}$ and $y_{\mu }\in S_{\mu }$, if

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\notin int{R}_{+}^k,$$

then $s^{*}$ is called a weakly PNE of MG.

Remark 1. 

If $K=1$, then the game model Γ is consistent with the game model of Yang and Song [8]. If $\mathcal{M}$ is a finite indexed set, then the game model Γ is consistent with the game models of Wang [11] and Yu and Yuan [12].

Theorem 1. 

Given that the following conditions hold:

(i) 

for any $\mu \in \mathcal{M}$, $F_{\mu }$ is $R_{+}^k$-continuous;

(ii) 

for any $\mu \in \mathcal{M}$ and $s_{-\mu }\in S_{-\mu }$, $y_{\mu }\to F_{\mu }(y_{\mu },s_{-\mu })$ is $R_{+}^k$-quasi-concave.

Then, there exists $s^{*}=(s_{\mu }^{*},s_{-\mu }^{*})\in S$ such that for any $\mu \in \mathcal{M}$ and $y_{\mu }\in S_{\mu }$,

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\notin int{R}_{+}^k.$$

Proof. 

For any $\epsilon >0$ and $\mu \in \mathcal{M}$, we define the following sets:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }=\{s=(s_{\mu },s_{-\mu })\in S\mid & F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall y_{\mu }\in S_{\mu }\},\hfill \end{array}\end{array}$$

(1)

where $\overline{\epsilon }=(\epsilon ,\cdots ,\epsilon )\in R_{+}^k$. Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }\left(s_{\mu }\right)& =\{s_{-\mu }\in S_{-\mu }\mid (s_{\mu },s_{-\mu })\in A_{\mu ,\epsilon }\}\hfill \\ & =\{s_{-\mu }\in S_{-\mu }\mid F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall y_{\mu }\in S_{\mu }\}.\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }\left(s_{-\mu }\right)& =\{s_{\mu }\in S_{\mu }\mid (s_{\mu },s_{-\mu })\in A_{\mu ,\epsilon }\}\hfill \\ & =\{s_{\mu }\in S_{\mu }\mid F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall y_{\mu }\in S_{\mu }\}.\hfill \end{array}\end{array}$$

(1) First, we prove that $A_{\mu ,\epsilon }\left(s_{\mu }\right)$ is an open set in $S_{-\mu }$.

For any $\mu \in \mathcal{M}$, $y_{\mu }\in S_{\mu }$ and $s_{\mu }\in S_{\mu }$, if $s_{-\mu }\in A_{\mu ,\epsilon }\left(s_{\mu }\right)$, then there exists an open neighborhood V of zero in $R^k$ such that

$$(F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }+V)\cap R_{+}^k=\varnothing .$$

By condition (i), since $F_{\mu }$ is $R_{+}^k$-continuous, then $s_{-\mu }\to F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })$ is continuous on $S_{-\mu }$. Thus, there exists an open neighborhood U of $s_{-\mu }$ such that for any $s_{-\mu }^{\prime }\in U$,

$$F_{\mu }(y_{\mu },s_{-\mu }^{\prime })-F_{\mu }(s_{\mu },s_{-\mu }^{\prime })-\overline{\epsilon }\in F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }+V.$$

Then

$$F_{\mu }(y_{\mu },s_{-\mu }^{\prime })-F_{\mu }(s_{\mu },s_{-\mu }^{\prime })-\overline{\epsilon }\notin R_{+}^k.$$

Therefore, $A_{\mu ,\epsilon }\left(s_{\mu }\right)$ is open in $S_{-\mu }$.

(2) Next, we prove that $A_{\mu ,\epsilon }\left(s_{-\mu }\right)$ is convex.

Since $y_{\mu }\to F_{\mu }(y_{\mu },s_{-\mu })$ is $R_{+}^k$-quasi-concave, then for any $s_{-\mu }\in S_{-\mu }$, $\lambda \in [0,1]$, $w\in R^k$ and for any $s_{\mu }^1,s_{\mu }^2\in A_{\mu ,\epsilon }\left(s_{-\mu }\right)$,

$$F_{\mu }(\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2,s_{-\mu })\in w+R_{+}^k.$$

Let $w=F_{\mu }(s_{\mu }^1,s_{-\mu })$ then

$$-F_{\mu }(s_{\mu }^1,s_{-\mu })\in -F_{\mu }(\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2,s_{-\mu })+R_{+}^k.$$

If there exists $y_{\mu }\in S_{\mu }$ such that

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2,s_{-\mu })-\overline{\epsilon }\in R_{+}^k,$$

then

$$\begin{array}{c}\hfill \begin{array}{cc}& F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu }^1,s_{-\mu })-\overline{\epsilon }\hfill \\ & \in F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2,s_{-\mu })-\overline{\epsilon }+R_{+}^k\hfill \\ & \subset R_{+}^k+R_{+}^k=R_{+}^k.\hfill \end{array}\end{array}$$

It conflicts with $s_{\mu }^1\in A_{\mu ,\epsilon }\left(s_{-\mu }\right)$, then

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2,s_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall y_{\mu }\in S_{\mu },$$

i.e., $\lambda s_{\mu }^1+(1-\lambda )s_{\mu }^2\in A_{\mu ,\epsilon }\left(s_{-\mu }\right)$.

Thus,

$$A_{\mu ,\epsilon }\left(s_{-\mu }\right)=\{s_{\mu }\in S_{\mu }\mid (s_{\mu },s_{-\mu })\in A_{\mu ,\epsilon }\}$$

is convex.

(3) Next, we prove that $A_{\mu ,\epsilon }\left(s_{-\mu }\right)$ is nonempty.

For any $\mu \in \mathcal{M}$ and $s_{-\mu }\in S_{-\mu }$, we define

$$C=\{s=(y_{\mu },s_{-\mu })\in S_{\mu }\times S_{-\mu }\mid F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })\in int{R}_{+}^k\}.$$

Similarly, according to condition (i), for any $y_{\mu }\in S_{\mu }$, the set $\{s_{\mu }\in S_{\mu }\mid (y_{\mu },s_{\mu })\in C\}$ is easily proved to be open in $S_{\mu }$. By condition (ii), the set $s_{\mu }\in S_{\mu }$, $\{s_{\mu }\in S_{\mu }\mid (y_{\mu },s_{\mu })\in C\}$ is easily proved to be convex in $S_{\mu }$. Obviously, for any $s\in S$, $(s,s)\notin C$. Therefore, by Lemma 4, for any $y_{\mu }\in S_{\mu }$, there exists $s_{\mu }\in S_{\mu }$ such that $(y_{\mu },s_{\mu })\notin B$, i.e.,

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })\notin int{R}_{+}^k,\forall y_{\mu }\in S_{\mu }.$$

It also satisfies

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall y_{\mu }\in S_{\mu }.$$

Thus, $A_{\mu ,\epsilon }\left(s_{-\mu }\right)$ is nonempty.

(4) Finally, we prove for all $\mu \in \mathcal{M}$, $A_{\mu ,{\epsilon }_1}\subset A_{\mu ,{\epsilon }_2}$, where $0<{\epsilon }_1⩽{\epsilon }_2$.

For any $\mu \in \mathcal{M}$ and $s\in A_{\mu ,{\epsilon }_1}$,

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })\notin {\overline{\epsilon }}_1+R_{+}^k,\forall y_{\mu }\in S_{\mu }.$$

Since ${\epsilon }_1⩽{\epsilon }_2$, then ${\overline{\epsilon }}_2+R_{+}^k\subset {\overline{\epsilon }}_1+R_{+}^k$. We have

$$F_{\mu }(y_{\mu },s_{-\mu })-F_{\mu }(s_{\mu },s_{-\mu })-{\overline{\epsilon }}_2\notin R_{+}^k,\forall y_{\mu }\in S_{\mu }.$$

Thus, $s\in A_{\mu ,{\epsilon }_1}\subset A_{\mu ,{\epsilon }_2}$.

Therefore, according to Lemma 3, there exists $s^{*}\in {\bigcap }_{\mu \in \mathcal{M}}{\overline{A}}_{\mu ,\epsilon }$, i.e., for any $\mu \in \mathcal{M}$ and $\epsilon >0$,there exists a net $\left\{s^{\gamma }\right\}$ convergence to $s^{*}$, and $s^{\gamma }\in A_{\mu ,\epsilon }$ such that

$$\begin{array}{c}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{\gamma })-F_{\mu }(s_{\mu }^{\gamma },s_{-\mu }^{\gamma })-{\overline{\epsilon }}_1\notin R_{+}^k,\forall y_{\mu }\in S_{\mu },\end{array}$$

(2)

then

$$\begin{array}{c}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{\gamma })-F_{\mu }(s_{\mu }^{\gamma },s_{-\mu }^{\gamma })\in R^k\setminus (\overline{\epsilon }+R_{+}^k),\forall y_{\mu }\in S_{\mu }.\end{array}$$

(3)

Since $F_{\mu }$ is continuous and $\epsilon$ is arbitrary, then

$$\begin{array}{c}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\in \overline{R^k\setminus R_{+}^k},\forall y_{\mu }\in S_{\mu },\end{array}$$

(4)

where $\overline{R^k\setminus R_{+}^k}$ is the closure of $R^k\setminus R_{+}^k$. We have

$$\begin{array}{c}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\notin int{R}_{+}^k,\forall y_{\mu }\in S_{\mu }.\end{array}$$

(5)

Therefore, $s^{*}$ is a weakly PNE for MG. □

First, we present an example of a MG with finitely many players as follows:

Example 1. 

Let $M{G}_1$ be a game with finitely many players and two objectives. Let $\mathcal{M}=\{1,2,\dots ,n\}$, for any $\mu \in \mathcal{M}$, the strategy set is $S_{\mu }=[0,1]$ and the strategy space is $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }={\prod }_{\mu =1}^n[0,1]$. Let the target set $K=\{1,2\}$. Define the payoff function of player μ as:

$$\begin{array}{c}\hfill F_{\mu }\left(s\right)=(f_{\mu }^1\left(s\right),f_{\mu }^2\left(s\right))=(s_{\mu }-s_{\mu }^2,-\sum _{\nu =1}^n{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }),\end{array}$$

(6)

where $s=(s_1,s_2,\dots s_n)$ denotes the strategy profile of all players.

Next, we verify that $F_{\mu }\left(s\right)$ satisfies all the conditions of Theorem 1.

(i) For any $\mu \in \mathcal{M}$, we verify that $F_{\mu }$ is $R_{+}^2$-continuous.

Since $f_{\mu }^1\left(s\right)=s_{\mu }-s_{\mu }^2$, $f_{\mu }^1\left(s\right)$ is continuous.

Since ${\sum }_{\nu =1}^n{\displaystyle \frac{1}{2^{\nu }}}=1$ and a linear combination of finitely many continuous functions is still continuous, $f_{\mu }^2\left(s\right)=-{\sum }_{\nu =1}^n{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }$ is continuous.

Therefore, for any $\mu \in \mathcal{M}$, $F_{\mu }\left(s\right)$ is $R_{+}^2$-continuous.

(ii) For any $\mu \in \mathcal{M}$ and $s_{-\mu }\in S_{-\mu }$, we verify that $y_{\mu }\to F_{\mu }(y_{\mu },s_{-\mu })$ is $R_{+}^2$-quasi-concave.

By Formula (6),

$$f_{\mu }^1(y_{\mu },s_{-\mu })=y_{\mu }-y_{\mu }^2.$$

Since

$${\displaystyle \frac{\partial f_{\mu }^1\left(y_{\mu }\right)}{\partial y_{\mu }}}=1-2y_{\mu },{\displaystyle \frac{{\partial }^2{f}_{\mu }^1}{\partial y_{\mu }^2}}=-2<0,$$

$f_{\mu }^1\left(s\right)$ is a strictly concave function. Therefore, $y_{\mu }\to f_{\mu }^1(y_{\mu },s_{-\mu })$ is quasi-concave.

Since

$$f_{\mu }^2(y_{\mu },s_{-\mu })=-\sum _{\nu =1}^n{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }=-{\displaystyle \frac{y_{\mu }}{2^{\mu }}}-\sum _{\nu \ne \mu }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }$$

is a linear function, $y_{\mu }\to f_{\mu }^2(y_{\mu },s_{-\mu })$ is quasi-concave.

According to Theorem 1, there exists at least one weakly PNE $s^{*}=(s_1^{*},s_2^{*},\dots )\in S$ such that for all $\mu \in \mathcal{M}$ and $y_{\mu }\in [0,1]$,

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\notin int{R}_{+}^2.$$

By Formula (6),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})& =(f_{\mu }^1(y_{\mu },s_{-\mu }^{*})-f_{\mu }^1(s_{\mu }^{*},s_{-\mu }^{*}),f_{\mu }^2(y_{\mu },s_{-\mu }^{*})-f_{\mu }^2(s_{\mu }^{*},s_{-\mu }^{*}))\hfill \\ & =((y_{\mu }-s_{\mu }^{*})(1-s_{\mu }^{*}-y_{\mu }),{\displaystyle \frac{1}{2^{\mu }}}(s_{\mu }^{*}-y_{\mu }))\notin int{R}_{+}^2.\hfill \end{array}\end{array}$$

Take

$$s^{*}=(0,0,\dots ,0).$$

Then for any $\mu \in \mathcal{M}$ and any $y_{\mu }\in [0,1]$,

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})=\left(y_{\mu }(1-y_{\mu }),-{\displaystyle \frac{y_{\mu }}{2^{\mu }}}\right)\notin \mathrm{int}{R}_{+}^2,$$

so $s^{*}=(0,0,\dots ,0)$ is a weakly PNE of $M{G}_1$.

Next, we verify that as $n\to \infty$, a weakly PNE still exists in the MG.

Example 2. 

Let $M{G}_2$ be a game with infinitely many players and two objectives. Let $\mathcal{M}=\{1,2,\dots \}$, for any $\mu \in \mathcal{M}$, the strategy set is $S_{\mu }=[0,1]$ and the strategy space is $S={\prod }_{\mu \in \mathcal{M}}{S}_{\mu }={\prod }_{\mu =1}^{\infty }[0,1]$. Let the target set $K=\{1,2\}$. Define the payoff function of player μ as:

$$\begin{array}{c}\hfill F_{\mu }\left(s\right)=(f_{\mu }^1\left(s\right),f_{\mu }^2\left(s\right))=(s_{\mu }-s_{\mu }^2,-\sum _{\nu =1}^{\infty }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }),\end{array}$$

(7)

where $s=(s_1,s_2,\dots s_{\mu },\dots )$ denotes the strategy profile of all players.

Since $f_{\mu }^1\left(s\right)=s_{\mu }-s_{\mu }^2$ and for any $\mu \in \mathcal{M}$, $s_{\mu }\in [0,1]$, then $f_{\mu }^1\left(s\right)$ is continuous.

Since ${\sum }_{\nu =1}^{\infty }{\displaystyle \frac{1}{2^{\nu }}}=1$ and $s_{\nu }\in [0,1]$, by the Weierstrass M-test, the series

$$\sum _{\nu =1}^{\infty }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }$$

converges uniformly under the product topology. Therefore, $f_{\mu }^2\left(s\right)=-{\sum }_{\nu =1}^{\infty }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }$ is continuous.

Therefore, $F_{\mu }\left(s\right)$ is $R_{+}^2$-continuous.

By Formula (7),

$$f_{\mu }^1(y_{\mu },s_{-\mu })=y_{\mu }-y_{\mu }^2.$$

Since

$${\displaystyle \frac{\partial f_{\mu }^1\left(y_{\mu }\right)}{\partial y_{\mu }}}=1-2y_{\mu },{\displaystyle \frac{{\partial }^2{f}_{\mu }^1}{\partial y_{\mu }^2}}=-2<0,$$

$f_{\mu }^1\left(s\right)$ is a strictly concave function. Therefore, $y_{\mu }\to f_{\mu }^1(y_{\mu },s_{-\mu })$ is quasi-concave.

Since

$$f_{\mu }^2(y_{\mu },s_{-\mu })=-\sum _{\nu =1}^{\infty }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }=-{\displaystyle \frac{y_{\mu }}{2^{\mu }}}-\sum _{\nu \ne \mu }{\displaystyle \frac{1}{2^{\nu }}}{s}_{\nu }$$

is a linear function, $y_{\mu }\to f_{\mu }^2(y_{\mu },s_{-\mu })$ is quasi-concave.

According to Theorem 1,

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})\notin int{R}_{+}^2.$$

By Formula (7),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})& =(f_{\mu }^1(y_{\mu },s_{-\mu }^{*})-f_{\mu }^1(s_{\mu }^{*},s_{-\mu }^{*}),f_{\mu }^2(y_{\mu },s_{-\mu }^{*})-f_{\mu }^2(s_{\mu }^{*},s_{-\mu }^{*}))\hfill \\ & =((y_{\mu }-s_{\mu }^{*})(1-s_{\mu }^{*}-y_{\mu }),{\displaystyle \frac{1}{2^{\mu }}}(s_{\mu }^{*}-y_{\mu }))\notin int{R}_{+}^2.\hfill \end{array}\end{array}$$

Again choose

$$s^{*}=(0,0,0,\dots ).$$

Then for any $\mu \in \mathcal{M}$ and $y_{\mu }\in [0,1]$,

$$F_{\mu }(y_{\mu },s_{-\mu }^{*})-F_{\mu }(s_{\mu }^{*},s_{-\mu }^{*})=\left(y_{\mu }(1-y_{\mu }),-{\displaystyle \frac{y_{\mu }}{2^{\mu }}}\right)\notin \mathrm{int}{R}_{+}^2,$$

so $s^{*}=(0,0,0,\dots )$ is a weakly PNE of $M{G}_2$.

Therefore, there exists at least one Nash equilibrium in the MG with infinitely many players.

4. The Existence of Weakly PNE Without Compactness Assumption

Theorem 2. 

For any $\mu \in \mathcal{M}$, let $D_{\mu }$  be a nonempty compact and convex subset of ${\tilde{S}}_{\mu }$. Let $\tilde{S}={\prod }_{\mu \in \mathcal{M}}{\tilde{S}}_{\mu }$ and ${\tilde{S}}_{-\mu }={\prod }_{\nu \in \mathcal{M},\nu \ne \mu }{\tilde{S}}_{\nu }$. Suppose $\left\{A_{\mu ,\epsilon }:\mu \in \mathcal{M},\epsilon >0\right\}\subset \tilde{S}$ is a collection that satisfies the following assumptions:

(i) 

For any $\mu \in \mathcal{M},\epsilon >0$ and ${\tilde{s}}_{\mu }\in D_{\mu }$, the section $A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)=\left\{{\xi }_{-\mu }\in {\tilde{S}}_{-\mu }\mid \left({\tilde{s}}_{\mu },{\xi }_{-\mu }\right)\in A_{\mu ,\epsilon }\right\}$ is open in ${\tilde{S}}_{-\mu }$;

(ii) 

For any $\mu \in \mathcal{M},\epsilon >0$ and ${\xi }_{-\mu }\in {\tilde{S}}_{-\mu }$, the section $A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu }=\left\{{\tilde{s}}_{\mu }\in D_{\mu }\mid \left({\tilde{s}}_{\mu },{\xi }_{-\mu }\right)\in A_{\mu ,\epsilon }\right\}$ is nonempty and convex;

(ii) 

For any $\mu \in \mathcal{M}$, ${\epsilon }_2\ge {\epsilon }_1>0,A_{\mu ,{\epsilon }_1}\subset A_{\mu ,{\epsilon }_2}.$

Then ${\displaystyle \bigcap _{\mu \in \mathcal{M},\epsilon >0}}{\overline{A}}_{\mu ,\epsilon }\ne \varnothing$.

Proof. 

For any $\epsilon >0$, define $C_{\mu ,\epsilon },B_{\mu ,\epsilon }:S\to 2^{D_{\mu }}$ as follows:

$$C_{\mu ,\epsilon }\left(\xi \right)=B_{\mu ,\epsilon }\left(\xi \right)=A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu },\forall \xi \in \tilde{S}.$$

Then according to condition (ii), for any $\mu \in \mathcal{M}$ and $\xi \in \tilde{S}$, co $C_{\mu ,\epsilon }\left(\xi \right)=$ co $(A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu })\subset (A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu })=B_{\mu ,\epsilon }\left(\xi \right)$, and $C_{\mu ,\epsilon }\left(\xi \right)=A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu }\ne \varnothing$. By condition (i), for any $\mu \in \mathcal{M},{\tilde{s}}_{\mu }\in D_{\mu }$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{\mu ,\epsilon }^{-1}\left({\tilde{s}}_{\mu }\right)& =\{\xi \in \tilde{S}:{\tilde{s}}_{\mu }\in C_{\mu ,\epsilon }\left(\xi \right)\}\hfill \\ & =\{\xi \in \tilde{S}:{\tilde{s}}_{\mu }\in A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\cap D_{\mu }\}\hfill \\ & =\{\xi \in \tilde{S}:{\tilde{s}}_{\mu }\in A_{\mu ,\epsilon }\left({\xi }_{-\mu }\right)\}\hfill \\ & =\{\xi \in \tilde{S}:({\tilde{s}}_{\mu },{\xi }_{-\mu })\in A_{\mu ,\epsilon }\}\hfill \\ & ={\tilde{S}}_{\mu }\otimes A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)\hfill \end{array}\end{array}$$

is open in $\tilde{S}$, where ${\tilde{S}}_{\mu }\otimes A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)$ denotes the set $\{({\tilde{s}}_{\mu },{\xi }_{-\mu })\in \tilde{S}:{\tilde{s}}_{\mu }\in {\tilde{S}}_{\mu },{\xi }_{-\mu }\in A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)\}$.

According to Lemma 5, there exists a point $\tilde{s}\in D={\Pi }_{\mu \in \mathcal{M}}{D}_{\mu }$ such that $\tilde{s}\in B\left(\tilde{s}\right)={\Pi }_{\mu \in \mathcal{M}}{B}_{\mu }\left(\tilde{s}\right)$, that is, ${\tilde{s}}_{\mu }\in A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)$. Then $\tilde{s}\in {\bigcap }_{\mu \in \mathcal{M}}{A}_{\mu ,\epsilon }.$ Let ${\left\{{\tilde{s}}_{\epsilon }\right\}}_{\epsilon >0}$ be a net of $A_{\mu ,\epsilon }$ and ${\tilde{s}}_{\epsilon }\to \tilde{s}$. Since $D_{\mu }$ is compact, by Tychonoff theorem, $D={\Pi }_{\mu \in \mathcal{M}}{D}_{\mu }$ is also compact. Consider the net $\left\{{\tilde{s}}_{\epsilon }\right\}$ indexed by the directed set $(0,+\infty )$ with the order ${\epsilon }_1⪯{\epsilon }_2\Rightarrow {\epsilon }_2\le {\epsilon }_1$. By compactness, this net has a subnet ${\left\{{\tilde{s}}_{{\epsilon }_{\delta }}\right\}}_{\delta \in \Lambda }$ converges to a point $\tilde{s}\in D$, where $\Lambda$ is a directed set and ${\epsilon }_{\delta }\to 0^{+}$, i.e., for every $\epsilon >0$, there exists ${\delta }_0\in \Lambda$ such that $\delta ⪰{\delta }_0\Rightarrow {\epsilon }_{\delta }\le \epsilon$. By condition (iii), ${\tilde{s}}_{{\epsilon }_{\delta }}\in A_{\mu ,{\epsilon }_{\delta }}\subset A_{\mu ,\epsilon }$. Since ${\tilde{s}}_{{\epsilon }_{\delta }}\to \tilde{s}$, and ${\tilde{s}}_{{\epsilon }_{\delta }}\in A_{\mu ,\epsilon }$, then $\tilde{s}\in {\overline{A}}_{\mu ,\epsilon }$. Therefore, ${\displaystyle \bigcap _{\mu \in \mathcal{M},\epsilon >0}}{\overline{A}}_{\mu ,\epsilon }\ne \varnothing$. □

Remark 2. 

Fierro’s non-compact intersection theorem [37] is a general intersection result that does not rely on the sectional structure. It is based on the finite intersection property and a global non-compactness condition (C), which is analogous to the Kuratowski measure of non-compactness [45], and applies to families of closed sets in complete spaces to ensure a nonempty compact intersection. In contrast, our theorem specifically addresses sectional structures, using an ε-parameterization and locally compact convex sets $D_{\mu }$ instead of global compactness or condition (C).

Shih and Tan’s non-compact intersection theorem [34] generalizes Ky Fan’s intersection theorem [30] to a non-compact setting but without any parameterization. Their result relies on a global compact convex set K and an auxiliary family of sets $B_{\mu }$ to ensure convexity. By comparison, our result defines convex sections directly over locally compact sets $D_{\mu }$, and introduces ε-parameterization to control the evolution of the sets. Moreover, our conclusion concerns the non-emptiness of the intersection of closures.

Theorem 3. 

Given that the following conditions hold:

(i) 

For any $\mu \in \mathcal{M}$, $F_{\mu }$ is continuous on $\tilde{S}$;

(ii) 

For any $\mu \in \mathcal{M}$ and ${\tilde{s}}_{-\mu }\in {\tilde{S}}_{-\mu }$, ${\xi }_{\mu }\to F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })$ is $R_{+}^k-$quasi-concave on ${\tilde{S}}_{\mu }$;

(iii) 

For any $\mu \in \mathcal{M}$, there exists a nonempty compact and convex subset $D_{\mu }$ of ${\tilde{S}}_{\mu }$ and a point ${\tilde{s}}^{\#}\in D$, such that for any $\tilde{s}\in S\setminus D$, where $D={\prod }_{\mu \in \mathcal{M}}{D}_{\mu }$,

$$F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })-F_{\mu }({\tilde{s}}_{\mu }^{\#},{\tilde{s}}_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall {\xi }_{\mu }\in {\tilde{S}}_{\mu },$$

where $\overline{\epsilon }=(\epsilon ,\dots ,\epsilon )\in R_{+}^k$.

Then there exists at least one weakly PNE for MG.

Proof. 

For any $\epsilon >0$ and $\mu \in \mathcal{M}$, we define the following sets:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }=\{\tilde{s}=({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })\in \tilde{S}\mid & F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })-F_{\mu }({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall {\xi }_{\mu }\in {\tilde{S}}_{\mu }\},\hfill \end{array}\end{array}$$

(8)

where $\overline{\epsilon }=(\epsilon ,\dots ,\epsilon )\in R_{+}^k$. Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)& =\{{\tilde{s}}_{-\mu }\in {\tilde{S}}_{-\mu }\mid ({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })\in A_{\mu ,\epsilon }\}\hfill \\ & =\{{\tilde{s}}_{-\mu }\in {\tilde{S}}_{-\mu }\mid F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })-F_{\mu }({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall {\xi }_{\mu }\in {\tilde{S}}_{\mu }\}.\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)& =\{{\tilde{s}}_{\mu }\in {\tilde{S}}_{\mu }\mid ({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })\in A_{\mu ,\epsilon }\}\hfill \\ & =\{{\tilde{s}}_{\mu }\in {\tilde{S}}_{\mu }\mid F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })-F_{\mu }({\tilde{s}}_{\mu },{\tilde{s}}_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall {\xi }_{\mu }\in {\tilde{S}}_{\mu }\}.\hfill \end{array}\end{array}$$

Similar to the proof of Theorem 1, we easily prove that $A_{\mu ,\epsilon }$ meets the following conditions: (1) For any $\mu \in \mathcal{M}$ and ${\tilde{s}}_{\mu }\in D_{\mu }$, $A_{\mu ,\epsilon }\left({\tilde{s}}_{\mu }\right)$ is open in ${\tilde{S}}_{-\mu }$; (2) $A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)\cap D_{\mu }$ is convex in ${\tilde{S}}_{\mu }$; and (3) for any $\mu \in \mathcal{M}$, $0<{\epsilon }_1⩽{\epsilon }_2$, $A_{\mu ,{\epsilon }_1}\subset A_{\mu ,{\epsilon }_2}$. Thus, we only need to prove that $A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)\cap D_{\mu }$ is nonempty. When $\tilde{s}\in D$, by Theorem 1, for any $\mu \in \mathcal{M}$, there exists ${\tilde{s}}_{\mu }\in D_{\mu }$ and ${\tilde{s}}_{-\mu }\in D_{-\mu }$ such that $A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)\ne \varnothing$, then $A_{\mu ,\epsilon }\left({\tilde{s}}_{-\mu }\right)\cap D_{\mu }\ne \varnothing$. When $\tilde{s}\in \tilde{S}\setminus D$, by condition (iii), there exists ${\tilde{s}}^{\#}\in D$ such that

$$F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu })-F_{\mu }({\tilde{s}}_{\mu }^{\#},{\tilde{s}}_{-\mu })-\overline{\epsilon }\notin R_{+}^k,\forall {\xi }_{\mu }\in S_{\mu }.$$

Therefore, $A_{\mu ,\epsilon }\cap D\ne \varnothing$. By Theorem 2, ${\bigcap }_{\mu \in \mathcal{M},\epsilon >0}{\overline{A}}_{\mu ,\epsilon }\ne \varnothing$. Similar to Formulas (2)–(5), we can prove that for any $\mu \in \mathcal{M}$, there exists ${\tilde{s}}^{*}\in \tilde{S}$ such that

$$F_{\mu }({\xi }_{\mu },{\tilde{s}}_{-\mu }^{*})-F_{\mu }({\tilde{s}}_{\mu }^{*},{\tilde{s}}_{-\mu }^{*})\notin int{R}_{+}^k,\forall {\xi }_{\mu }\in {\tilde{S}}_{\mu }.$$

Thus there exists at least one weakly PNE for MG. This completes the proof. □

5. Conclusions

The primary goal of this work is to demonstrate the existence of weakly PNE in MGs with infinitely many players. Then, we extend Lemma 3 to the non-compact case and obtain a new intersection theorem. Finally, we apply the new intersection theorem to obtain the existence of weakly PNE under non-compactness assumption in games with infinitely many players. Significantly, if our game model considers only the single-objective case, then Theorem 1 is the same result as Yang and Song [8]. If the index set $\mathcal{M}$ in our game model is finite, then Theorem 1 is consistent with the results of Wang [11].