# An existence theorem for single leader multi-follower games with direct preference maps

Yutthakan Chummongkhon, Poom Kumam, Parin Chaipunya

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Yutthakan Chummongkhon, Poom Kumam, Parin Chaipunya. An existence theorem for single leader multi-follower games with direct preference maps. 2025. hal-05020957

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# An existence theorem for single leader multi-follower games with direct preference maps

Yutthakan Chummongkhon¹, Poom Kumam¹,², and Parin Chaipunya*¹,²

¹ Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.

² Center of Excellence in Theoretical and Computational Science (TaCS-CoE), King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.
Email: yuttagan.24@mail.kmutt.ac.th, parin.cha@kmutt.ac.th and poom.kum@kmutt.ac.th;

## Abstract

This paper concerns with an existence of a solution for a single leader multi-follower game (SLMFG), where the followers jointly solve an abstract economy problem. Recall that an abstract economy problem is an extension of a generalized Nash equilibrium problem (GNEP) in the sense that the preference of each player can be described without numerical criteria. The results of this paper therefore extend the known literature concerning an existence of a solution of a SLMFG. Due to the lack of criterion functions in the lower-level, the technique we used is quite different from those that deal with GNEPs. In particular, we argue that the follower's abstract economy profiles, i.e., constraint-preference couples, belong to a particular metric space where the response map is proved to be upper semi-continuous.

**Keywords:** Single leader multi-follower game; Bilevel optimization; Nash equilibrium; Abstract economy.

## 1. Introduction

A bilevel programming, or bilevel optimization problem, refers to a class of optimization problems that involve two players, one is classified as the *leader* and the other one as the *follower*. When the leader makes her decision, the follower observes and makes an *optimal response* by solving the *lower-level* optimization problem, parametrized by the leader's action. The goal of the leader is then to make a decision so that, altogether with the follower's optimal response, gives her the *best* objective value. When the single follower is replaced with multiple followers, their response to the leader's action by solving a parametrized equilibrium-type problem. This approach is broadly referred to as a *single leader multi-follower game* (or briefly, *SLMFG*). These classes of bilevel problems have posed several interesting and important questions, theoretically and practically, in the literature. For instance, the issues regarding the KKT reformulation (or MPCC reformulation) were discussed in [5, 8, 9]. The existence results for several different settings of a SLMFG were studied in [1, 3, 4, 6, 7].

<footnote>* Corresponding author</footnote>
In this paper, we propose an existence theorem for a single leader multi-follower game (SLMFG) in which the followers' preferences are directly given by the preference maps without the need of objective functions. Thus, the lower-level problem here is presented as an abstract economy, à la Shafer and Sonnenschein [11], parametrized by the leader's action. Let us state precisely the problem of our study for this paper. Consider the $n$ followers that are represented by a set $\mathbf{F} := \{1, \dots, n\}$. The single leader, denoted by $\vartheta$, owns an action set $W$ and each follower $\nu \in \mathbf{F}$ owns an action set $X_\nu$. We write $X := \prod_{\nu \in \mathbf{F}} X_\nu$ and for each $\nu \in \mathbf{F}$ we write $X_{-\nu} := \prod_{\mu \in \mathbf{F} \setminus \{\nu\}} X_\mu$ whose generic element is denoted by $x_{-\nu}$. It is typical to represent $X$ as $X_\nu \times X_{-\nu}$ and to decompose $x \in X$ into $(x_\nu, x_{-\nu}) \in X_\nu \times X_{-\nu}$, and also to extract $x_{-\nu}$ from $x$ the other followers' component. The leader is equipped with an objective function $F: W \times X \to \mathbb{R}$, while each follower $\nu \in \mathbf{F}$ is equipped with a constraint map $A_\nu: W \times X_{-\nu} \rightleftharpoons X_\nu$ and a preference map $P_\nu: W \times X \rightleftharpoons X_\nu$. For each $w \in W$ and $\nu \in \mathbf{F}$, it is customary to write $A_\nu^w(\cdot) := A_\nu(w, \cdot)$ and $P_\nu^w(\cdot) := P_\nu(w, \cdot)$. The SLMFG that we focus on in this paper has the following form:

$$ 
\left\{ 
\begin{array}{ll}
\min_{w,x} & F(w,x) \tag{1.1a} \\
\text{s.t.} & w \in Y \subset W \tag{1.1b} \\
& \forall \nu \in \mathbf{F}: \tag{1.1c} \\
& \lfloor x_\nu \in A_\nu^w(x_{-\nu}), \ A_\nu^w(x_{-\nu}) \cap P_\nu^w(x) = \emptyset \tag{1.1d}
\end{array}
\right.
$$

In particular, the constraints (1.1c)–(1.1d) describe that the followers respond to the leader's signal $w$ with a Nash equilibrium of a $w$-parametrized abstract economy given by the pair of maps $(A_\nu^w, P_\nu^w)_{\nu \in \mathbf{F}}$. For convenience, at each $w \in W$ we write

$$ \mathrm{NE}(w) := \{x = (x_\nu)_{\nu \in \mathbf{F}} \in X \mid \forall \nu \in \mathbf{F}: x_\nu \in A_\nu^w(x_{-\nu}), \ A_\nu^w(x_{-\nu}) \cap P_\nu^w(x) = \emptyset\} $$

to denote the set of all such Nash equilibrium points.

The SLMFG (1.1) indeed covers many particular cases of Nash equilibrium-like problems. For instance, if one fix $\varepsilon \ge 0$ and set

$$ P_\nu^w(x_\nu, x_{-\nu}) := \{y_\nu \in X_\nu \mid f_\nu(w, y_\nu, x_{-\nu}) < f_\nu(w, x_\nu, x_{-\nu}) - \varepsilon, \ (\forall y \in X_\nu)\}, \\ f_\nu: W \times X \to \mathbb{R}, $$

then (1.1) reduces to

$$ 
\left\{ 
\begin{array}{ll}
\min_{w,x} & F(w,x) \tag{1.2a} \\
\text{s.t.} & w \in Y \subset W \tag{1.2b} \\
& \forall \nu \in \mathbf{F}: \tag{1.2c} \\
& \lfloor x_\nu \in \varepsilon - \arg\min_{y_\nu} \{f_\nu(w, y_\nu, x_{-\nu}) \mid y_\nu \in A_\nu^w(x_{-\nu})\}, \tag{1.2d}
\end{array}
\right.
$$

which is a single leader multi-follower game where the followers solve for an $\varepsilon$-Nash equilibrium of a $w$-parametrized (generalized) Nash equilibrium problem. When $\varepsilon = 0$, this is a classical single leader multi-follower game.
Likewise, one could also retrieve the case where the followers play a set-valued Nash equilibrium problem by setting

$$ egin{cases} P_{\nu}^{w}(x_{\nu}, x_{-\nu}) := \{y_{\nu} \in X_{\nu} \mid (y_{\nu}, x_{-\nu}) \succeq_{\nu} (x_{\nu}, x_{-\nu})\}, \\ \succeq_{\nu} \subset X \times X. \end{cases} $$

Then $A_{\nu}^{w}(x_{-\nu}) \cap P_{\nu}^{w}(x_{\nu}, x_{-\nu}) = \emptyset$ describes that there is no feasible point $y_{\nu} \in A_{\nu}^{w}(x_{-\nu})$ in which $(y_{\nu}, x_{-\nu})$ is not preferred over $(x_{\nu}, x_{-\nu})$. In the language of vector optimization, this says that $(x_{\nu}, x_{-\nu})$ is weakly efficient.

The aim of this paper is to develop an existence theorem for a SLMFG of the form (1.1) where the followers' problem is an abstract economy problem parametrized by the leader's decision. We take the classical approach by showing that the solution map (or the response map) of the lower-level problem has closed graph. For this, we characterize the solution map with a parametric minimizer set of its associated gap function. We introduce a regularity condition and show that, under the uniform metric, the solution map is upper semicontinuous.

## 2. Preliminaries

In this section, we recall basic notions in set-valued analysis which includes distance between points and sets, convergence of sets, and continuity of set-valued maps.

Suppose that $E$ and $F$ are subsets of (possibly different) Banach spaces. A set-valued map $T: E \to F$ is said to be

* closed if its graph $\text{gr}(T) := \{(x, y) \in E \times F \mid y \in T(x)\}$ is closed in $E \times F$,

* compact if $\text{gr}(T)$ is compact in $E \times F$,

* closed-valued if $T(x)$ is closed for all $x \in E$,

* upper semicontinuous on $D \subset E$ if for every $x \in D$ and each open set $\emptyset$ with $T(x) \subset \emptyset$, there exists a neighborhood $\mathcal{V}$ of $x$ such that $T(x') \subset \emptyset$ for all $x' \in \mathcal{V}$,

* lower semicontinuous on $D \subset E$ if for every $x \in D$ and each open set $\emptyset$ with $T(x) \cap \emptyset \neq \emptyset$, there exists a neighborhood $\mathcal{V}$ of $x$ such that $T(x') \cap \emptyset \neq \emptyset$ for all $x' \in \mathcal{V}$,

* continuous on $D \subset E$ if it is both upper and lower semicontinuous on $D$.

The following results (see e.g. [2]) regarding $T: E \to F$ are widely known in set-valued analysis:

(i) If $T$ is closed-valued and upper semicontinuous, then it is closed.

(ii) If $T$ is closed and $F$ is compact, then it is upper semicontinuous.

(iii) If $F$ is compact, then $T$ is upper semicontinuous at $x \in E$ if and only if for any sequences $(x^n)$ in $E$ and $(y^n)$ in $F$ with $x^n \to x$ and $y^n \in T(x^n)$ for each $n \in \mathbb{N}$, it holds that $y \in T(x)$.

(iv) If $F$ is compact, then $T$ is lower semicontinuous at $x \in E$ if and only if for any sequence $(x^n)$ in $E$ with $x^n \to x$ and any $y \in T(x)$, there exists $y^n \in T(x^n)$ (for each $n \in \mathbb{N}$) such that $y^n \to y$.
Let us denote with $K(E)$ the family of compact subsets of $E$. Recall that the distance on $E$ induces a metric on $K(E)$ that preserves the completeness. For any $x \in E$ and $D \in K(E)$, we define the displacement from $D$ by

$$d(x, D) := \inf_{y \in D} d(x, y).$$

Observe that $x \in D \iff d(x, D) = 0$. For any $C, D \in K(E)$, the excess of $C$ from $D$ is defined by

$$e(C, D) := \sup_{x \in C} \inf_{y \in D} d(x, y).$$

It is important to notice that $e(C, D) \neq e(D, C)$ in general. Finally, the Hausdorff distance between $C, D \in K(E)$ is given by

$$h(C, D) := \max\{e(C, D), e(D, C)\}.$$

For any $D \subset E$ and $\varepsilon > 0$, we adopt the notation

$$\mathcal{U}_{\varepsilon}[D] := \{x \in E \mid d(x, D) < \varepsilon\}$$

to denote an open ball about a set $D$. When $C = \{x\}$ is singleton, we prefer to write $\mathcal{U}_{\varepsilon}[x]$ instead of $\mathcal{U}_{\varepsilon}[\{x\}]$. With this, one may also notice an equivalent formulation of the Hausdorff distance, which reads

$$h(C, D) = \inf\{\eta > 0 \mid \mathcal{U}_{\eta}[C] \supset D, \mathcal{U}_{\eta}[D] \supset C\}.$$

It is well-known that $(K(E), h)$ is a metric space which is complete if and only if $E$ is complete. Exploiting the Hausdorff metric, we have a further characterization of continuity of a set-valued map. That is, if $F$ is compact, then $T: E \rightrightarrows F$ is continuous at $x \in E$ if and only if $\lim_{n \to \infty} h(T(x^n), T(x)) = 0$ for any sequence $(x^n)$ in $E$ with $x^n \to x$.

Next, consider a sequence $(C^n)$ of subsets of $E$. Its outer and inner limits could be defined, respectively, by

$$\limsup_{n \to \infty} C^n := \left\{ x \in E \mid \begin{array}{l} x \text{ is an accumulation point of some sequence } (x^n) \\ \text{such that } x^n \in C^n \text{ for all } n \in \mathbb{N}. \end{array} \right\}$$

and

$$\liminf_{n \to \infty} C^n := \left\{ x \in E \mid \begin{array}{l} x \text{ is the limit of some convergent sequence } (x^n) \\ \text{such that } x^n \in C^n \text{ for all } n \in \mathbb{N}. \end{array} \right\}.$$

We say that $C \subset E$ is the Painlevé-Kuratowski limit of $(C^n)$, or symbolically that $C^n \xrightarrow{\text{PK}} C$, if

$$C = \limsup_{n \to \infty} C^n = \liminf_{n \to \infty} C^n.$$

If $C$ and all $C^n$'s are compact, then $C^n \xrightarrow{\text{PK}} C$ if and only if $\lim_{n \to \infty} h(C^n, C) = 0$. That is, the Painlevé-Kuratowski and the Hausdorff limits coincide for compact sequences.
### 3. Stability analysis for abstract economy problems

This section is dedicated to the stability analysis of the lower-level abstract economy. The idea is that a decision $w$ of the leader *selects* a tuple $(A_\nu^w, P_\nu^w)_{\nu \in F}$ representing an abstract economy profile, from some particular class $\Lambda$, that defines the followers's abstract economy problem.

#### 3.1. An existence result

For the moment, we neglect the leader's ingredient and focus on the solvability of an abstract economy problem.

**Theorem 3.1.** Consider a finite set $F$ of players. For each $\nu \in F$, let $\mathcal{X}_\nu$ be a Banach space and $X_\nu \subset \mathcal{X}_\nu$ be nonempty, closed and convex. Suppose, for each $\nu \in F$, that the following assumptions hold

(A1) the maps $A_\nu : X_{-\nu} \rightrightarrows X_\nu$ and $P_\nu : X \rightrightarrows X_\nu$ are upper semicontinuous with closed convex values,

(A2) the map $A_\nu$ has nonempty values,

(A3) the set $E_\nu := \{x \in X \mid A_\nu(x_{-\nu}) \cap P_\nu(x) = \emptyset\}$ is open,

(A4) for each $x \in X$, $x_\nu \in A_\nu(x_{-\nu})$ implies $x_\nu \notin P_\nu(x)$.

Then there is a point $\bar{x} \in X$ such that

$$ \bar{x}_\nu \in A_\nu(\bar{x}_{-\nu}), \quad A_\nu(\bar{x}_{-\nu}) \cap P_\nu(\bar{x}) = \emptyset \quad (3.1) $$

for all $\nu \in F$.

*Proof.* Let $\nu \in F$. Define $G_\nu : X \rightrightarrows X_\nu$ by

$$ G_\nu(x) := \begin{cases} A_\nu(x_{-\nu}) \cap P_\nu(x) & \text{if } x \in E_\nu, \\ A_\nu(x_{-\nu}) & \text{otherwise.} \end{cases} \quad (3.2) $$

Let $G := \prod_{\nu \in F} G_\nu$. It is clear that if $x \in X$ satisfies (3.1) for all $\nu \in F$, then $x \in G(x)$. The converse holds with the use of (A4). This means (3.1) is characterized with $x \in X$ being a fixed point of the map $G$. From [2, Theorem 3.1.8], $G$ is upper semicontinuous and compact on $X$ with nonempty closed convex values. The existence of such a fixed point then follows from [10, Theorem 1 and Lemma 3] ■

#### 3.2. A stability result

In this subsection, we consider a class of abstract economy profiles, which are described by tuples $(A_\nu, P_\nu)_{\nu \in F}$ of constraint-preference maps, where the continuity in the perturbation implies the continuity of the corresponding Nash equilibrium points.

Always consider $F$ a finite set of players and for each $\nu \in F$, $X_\nu$ is a closed convex subset of a Banach space $\mathcal{X}_\nu$. Consider $\Lambda$ the class containing $\lambda = (A_\nu, P_\nu)_{\nu \in F}$, where $A_\nu : X_{-\nu} \rightrightarrows X_\nu$ and $P_\nu : X \rightrightarrows X_\nu$, satisfying the following two conditions
(Λ₁) the assumptions (A1)–(A3) hold,

(Λ₂) there exists τ > 0 such that for all ν ∈ F,

$$x_{\nu} \in A_{\nu}(x_{-\nu}) \implies \mathcal{U}_{\tau}[x_{\nu}] \cap P_{\nu}(x) = \emptyset.$$

One could directly observe that (Λ₂) implies (A4). Therefore any λ = (A_ν, P_ν)_{ν∈F} ∈ Λ induces a nonempty solution set NE(λ), where

$$\mathrm{NE}(\lambda) := \{x \in X \mid x_{\nu} \in A_{\nu}(x_{-\nu}), \ A_{\nu}(x_{-\nu}) \cap P_{\nu}(x) = \emptyset, \ \forall \nu \in F\}.$$

Given E := (E_ν)_{ν∈F}, where for each ν ∈ F, E_ν ⊂ X is open relative to X, we define a subclass Λ(E) ⊂ Λ by

$$\Lambda(\mathcal{E}) := \{\lambda \in \Lambda \mid E_{\nu}(\lambda) = \mathcal{E}_{\nu}, \ \forall \nu \in F\},$$

where E_ν(λ) := {x ∈ X | A_ν(x_{-ν}) ∩ P_ν(x) ≠ ∅}. On this subset Λ(E), we impose a distance function ρ : Λ(E) × Λ(E) → R_+ given by

$$\rho(\lambda, \hat{\lambda}) = \sum_{\nu \in F} \left[ \sup_{x \in X} h(A_{\nu}(x_{-\nu}), \hat{A}_{\nu}(x_{-\nu})) + \sup_{x \in X} h(P_{\nu}(x), \hat{P}_{\nu}(x)) \right],$$

where λ = (A_ν, P_ν)_{ν∈F} and λ = (A_ν, P_ν)_{ν∈F}. The pair (Λ(E), ρ) is clearly a metric space, but its completeness is not so easy to see. In fact, to prove its completeness, we need to assume a regularity condition below.

**Definition 3.2.** A closed subset Ψ ⊂ Λ(E) is said to be regular w.r.t. E if

(Ψ₁) the condition (Λ₂) holds with the same τ > 0 for any λ ∈ Ψ,

(Ψ₂) there exists α > 0 such that for any ε > 0, any λ = (A_ν, P_ν)_{ν∈F} ∈ Ψ, and any x ∈ E_ν = E_ν, it holds

$$\mathcal{U}_{\varepsilon}[A_{\nu}(x_{-\nu})] \cap \mathcal{U}_{\varepsilon}[P_{\nu}(x)] \subset \mathcal{U}_{\alpha\varepsilon}[A_{\nu}(x_{-\nu}) \cap P_{\nu}(x)].$$

Following directly from the definition, one could observe that the above regularity always holds in finite dimensional spaces when the intersections A_ν(x_{-ν}) ∩ P_ν(x) are bounded.

**Lemma 3.3.** If Ψ ⊂ Λ(E) is regular w.r.t. E, then (Ψ, ρ) is complete.

*Proof.* Let (λ^n) = (A_ν^n, P_ν^n)_{ν∈F} be a Cauchy sequence in Ψ. For every x ∈ X and ν ∈ F, the sequence (A_ν^n(x_{-ν}))^n∈F of compact sets is Cauchy w.r.t. to the Hausdorff metric on K(X_ν), which in turn converges, in the Hausdorff metric, to a compact set A_ν(x_{-ν}) ∈ K(X_ν). In the same way, the sequence (P_ν^n(x))^n∈F converges to a set P_ν(x) ∈ K(X_ν). Through these limits, we have defined the maps A_ν : X_{-ν} ⇒ X_ν and P_ν : X ⇒ X_ν which produce compact convex values. The goal is to prove that λ := (A_ν, P_ν)_{ν∈F belongs to Λ, through the three claims below.

**Claim 1:** The maps A_ν and P_ν are upper semicontinuous.
Take any sequences $(x^n)$ and $(y^n)$ in $X$ that are convergent, respectively, to $\hat{x} \in X$ and $\hat{y} \in X$ with the property that $y^n \in \hat{A}_\nu(x^n)$ for all $n \in \mathbb{N}$. It suffices to show that $\hat{y} \in \hat{A}_\nu(\hat{x})$. Now, for any $\nu \in \mathbb{F}$ and $r, n \in \mathbb{N}$, we get

$$d(y^n, \hat{A}_\nu(\hat{x})) \leq d(y^n, \hat{A}_\nu(x^n)) + h(\hat{A}_\nu(x^n), A_\nu^r(x^n)) + e(A_\nu^r(x^n), \hat{A}_\nu(\hat{v})) \quad (3.3)$$

$$= h(\hat{A}_\nu(x^n), A_\nu^r(x^n)) + e(A_\nu^r(x^n), \hat{A}_\nu(\hat{x})). \quad (3.4)$$

Let $\varepsilon > 0$ be arbitrary. Since $\lim_{r \to \infty} h(A_\nu^r(x), \hat{A}_\nu(x))$ uniformly on $x \in X$, there exists $R > 0$ in which

$$h(\hat{A}_\nu(x), A_\nu^r(x)) < \frac{\varepsilon}{2} \quad (3.5)$$

for any $r > R$ and $x \in X$. Take any $r > R$, the above inequality gives

$$A_\nu^r(\hat{x}) \subset U_{\varepsilon/\alpha}[\hat{A}_\nu(\hat{x})].$$

Together with the facts that each map $A_\nu^r$ is upper semicontinuous and $x^n \to \hat{x}$, there exists $N > 0$ in which

$$A_\nu^r(x^n) \subset U_{\varepsilon/2}[\hat{A}_\nu(\hat{x})]$$

for all $n > N$. This means, for any $r, n > \max\{R, N\}$, it holds that

$$e(A_\nu^r(x^n), \hat{A}_\nu(\hat{x})) < \frac{\varepsilon}{2}. \quad (3.6)$$

Combining (3.3)–(3.4) with (3.5) and (3.6), we obtain that

$$d(y^n, \hat{A}_\nu(\hat{x})) < \varepsilon$$

for each $n > \max\{R, N\}$. It implies that $\hat{y} \in \hat{A}_\nu(\hat{x})$, and we conclude that $\hat{A}_\nu$ is upper semicontinuous.

With a similar technique, we could show that $\hat{P}_\nu$ is upper semicontinuous, so that Claim 1 is proved.

**Claim 2:** $E_\nu(\hat{\lambda}) = \varepsilon_\nu$ for all $\nu \in \mathbb{F}$.

This claim follows directly from the regularity of $\Psi$ w.r.t $\varepsilon$.

**Claim 3:** For $x \in X$, $x_\nu \in \hat{A}_\nu(x_{-\nu})$ implies $U_\tau[x_\nu] \cap \hat{P}_\nu(x) = \emptyset$ for all $x \in X$.

Let $\nu \in \mathbb{F}$ and take any $\hat{x} \in X$ with $\hat{x}_\nu \in \hat{A}_\nu(\hat{x}_{-\nu})$. Using the PK convergence of $(A_\nu^n)$ to its limit $\hat{A}_\nu$, there exists a sequence $(x^n)$ in $X$ such that $x^n \to \hat{x}$ and $x_\nu^n \in A_\nu^n(\hat{x}_{-\nu})$ for all $n \in \mathbb{N}$. Applying $(\Psi_1)$ to the point $(x_\nu^n, \hat{x}_{-\nu})$ with $A_\nu^n(\hat{x}_{-\nu})$, we obtain

$$\tau \leq d(x_\nu^n, P_\nu^n(\hat{x})) \leq d(x_\nu^n, \hat{P}_\nu(\hat{x})) + h(P_\nu^n(\hat{x}), P_\nu(\hat{x}))$$

for large $n \in \mathbb{N}$. Letting $n \to \infty$, we have $\tau \leq d(\hat{x}_\nu, \hat{P}_\nu(\hat{x}))$, which is equivalent to saying $U_\tau[\hat{x}_\nu] \cap P_\nu(\hat{x}) = \emptyset$ The claim is thus proved.

Accumulating the Claims 1–3, one conclude that $(\Lambda_1)$ and $(\Lambda_2)$ are satisfied at the limit $\hat{\lambda}$. This proves $\hat{\lambda} \in \Lambda$ and the closedness of $\Psi$ guarantees that it is complete. ■
Following the proof of Theorem 3.1, one have a fixed point characterization of elements in NE(·). We define a function $\mathcal{R}: \Lambda \times X \to \mathbb{R}_+$ by

$$ \mathcal{R}(\lambda; x) := \max_{\nu \in \mathbb{F}} d(x_\nu, G_\nu(\lambda; x)), $$ 
where $$ G_\nu(\lambda; x) := \begin{cases} A_\nu(x_{-\nu}) \cap P_\nu(x) & \text{if } x \in E_\nu(\lambda), \\ A_\nu(x_{-\nu}) & \text{otherwise,} \end{cases} $$

for any $\lambda = (A_\nu, P_\nu)_{\nu \in \mathbb{F}} \in \Lambda$. One may observe that $\mathcal{R}(\lambda; \cdot)$ is a gap function for the game $\lambda$, i.e. it is nonnegative and $\mathcal{R}(\lambda; x) = 0$ if and only if $x \in \text{NE}(\lambda)$.

**Lemma 3.4.** *The gap function $\mathcal{R}: \Lambda \times X \to \mathbb{R}_+$ is lower semicontinuous on $\Psi \times X$, provided that $\Psi \subset \Lambda(\mathcal{E})$ is regular w.r.t. $\mathcal{E}$.*

*Proof.* Let $(\lambda^n)$, $\lambda^n = (A_\nu^n, P_\nu^n)_{\nu \in \mathbb{F}}$, be a sequence in $\Psi$ that is convergent to $\hat{\lambda} = (\hat{A}_\nu, \hat{P}_\nu) \in \Psi$.

**Claim 1:** For any $\nu \in \mathbb{F}$, $G_\nu(\lambda^n; x) \to G_\nu(\hat{\lambda}; x)$ uniformly over $x \in X$.

It is sufficient to show that $A_\nu^n(x_{-\nu}) \cap P_\nu^n(x) \to \hat{A}_\nu(x_{-\nu}) \cap \hat{P}_\nu(x)$ uniformly on $x \in E_\nu$. It is clear that we have

$$ A_{\nu}^{n}(x) \to \hat{A}_{\nu}(x) \quad \text{and} \quad P_{\nu}^{n}(x) \to \hat{P}_{\nu}(x) \qquad (3.7) $$

uniformly on $x \in X$. Let us fix $\nu \in \mathbb{F}$ and $x \in E_\nu$. For any $\varepsilon > 0$, and $n \in \mathbb{N}$ sufficiently large, we have

$$ \begin{aligned} \hat{A}_{\nu}(x_{-\nu}) &\subset \mathcal{U}_{\varepsilon/\alpha}[A_{\nu}^{n}(x_{-\nu})], \quad A_{\nu}^{n}(x_{-\nu}) &\subset \mathcal{U}_{\varepsilon/\alpha}[\hat{A}_{\nu}(x_{-\nu})] \\ \hat{P}_{\nu}(x) &\subset \mathcal{U}_{\varepsilon/\alpha}[P_{\nu}^{n}(x)], \quad P_{\nu}^{n}(x) &\subset \mathcal{U}_{\varepsilon/\alpha}[\hat{P}_{\nu}(x)]. \end{aligned} $$

For large $n \in \mathbb{N}$, the regularity of $\Psi$ yields

$$ \begin{aligned} \hat{A}_{\nu}(x_{-\nu}) \cap \hat{P}_{\nu}(x) &\subset \mathcal{U}_{\varepsilon/\alpha}[A_{\nu}^{n}(x_{-\nu})] \cap \mathcal{U}_{\varepsilon/\alpha}[P_{\nu}^{n}(x)] \subset \mathcal{U}_{\varepsilon}[A_{\nu}^{n}(x_{-\nu}) \cap P_{\nu}^{n}(x)] \\ A_{\nu}^{n}(x_{-\nu}) \cap P_{\nu}^{n}(x) &\subset \mathcal{U}_{\varepsilon/\alpha}[\hat{A}_{\nu}(x_{-\nu})] \cap \mathcal{U}_{\varepsilon/\alpha}[\hat{P}_{\nu}(x)] \subset \mathcal{U}_{\varepsilon}[\hat{A}_{\nu}(x_{-\nu}) \cap \hat{P}_{\nu}(x)]. \end{aligned} $$

This means we have

$$ h(\hat{A}_{\nu}(x_{-\nu}) \cap \hat{P}_{\nu}(x), A_{\nu}^{n}(x_{-\nu}) \cap P_{\nu}^{n}(x)) < \varepsilon $$

whenever $n \in \mathbb{N}$ is sufficiently large. Since this is uniform over $x \in E_\nu$, we conclude that

$$ \sup_{x \in E_\nu} h(\hat{A}_{\nu}(x_{-\nu}) \cap \hat{P}_{\nu}(x), A_{\nu}^{n}(x_{-\nu}) \cap P_{\nu}^{n}(x)) \to 0. $$

Together with the uniform convergence of $(A_\nu^n(x))$ over $x \in X \setminus E_\nu$, which could be observed in (3.7). This proves the claim.

Now, note that $G_\nu(\lambda^n; \cdot)$ and $G_\nu(\hat{\lambda}; \cdot)$ are upper semicontinuous, and hence the functions $x \mapsto d(x, G_\nu(\lambda^n; x))$ and $x \mapsto d(x, G_\nu(\hat{\lambda}; \cdot))$ are both lower semicontinuous (by [2, Corollary 1.4.17]). Take any convergent sequence $(x^n)$ in $X$ with the limit $\hat{x} \in X$. Let $\varepsilon > 0$. For $n \in \mathbb{N}$ sufficiently large, we have

$$ d(x^n, G_\nu(\hat{\lambda}; x^n)) > d(\hat{x}, G_\nu(\hat{\lambda}; \hat{x})) - \frac{\varepsilon}{2}. $$
Using the compactness, for any $n \in \mathbb{N}$, there exists $u^n \in G_\nu(\lambda^n; x^n)$ such that

$$d(x^n, u^n) = d(x^n, G_\nu(\lambda^n; x^n)).$$

From the convergence proved in Claim 1, for large enough $n \in \mathbb{N}$, we could pick $v^n \in G_\nu(\hat{\lambda}; x^n)$ such that

$$d(u^n, v^n) = d(u^n, G_\nu(\hat{\lambda}; x^n)) \le h(G_\nu(\lambda^n; x^n), G_\nu(\hat{\lambda}; x^n)) < \frac{\varepsilon}{2}.$$

Therefore, we have

$$d(x^n, G_\nu(\lambda^n; x^n)) = d(x^n, u^n) \\ \ge d(x^n, v^n) - d(v^n, u^n) \\ \ge d(x^n, G_\nu(\hat{\lambda}; x^n)) - d(v^n, u^n) \\ > d(\hat{x}, G_\nu(\hat{\lambda}; \hat{x})) - \varepsilon$$

for large $n \in \mathbb{N}$. Repeat the process for all $\nu \in \mathcal{F}$, we get

$$\mathcal{R}(\lambda^n; x^n) > \mathcal{R}(\hat{\lambda}; \hat{x}) - \varepsilon$$

for large $n \in \mathbb{N}$. Thus, the function $\mathcal{R}$ is lower semicontinuous on $\Psi \times X$.

We finally have all the tools to prove our main stability theorem. We now state and prove the continuity of $NE(\cdot)$ based on Lemmas 3.3 and 3.4.

**Theorem 3.5.** The map $\lambda \mapsto NE(\lambda)$ is upper semicontinuous on $\Psi \subset \Lambda(\mathcal{E})$, provided that $\Psi$ is regular w.r.t. $\mathcal{E}$.

*Proof.* Observe that, for each $\lambda = (A_\nu, P_\nu)_{\nu \in \mathcal{F}} \in \Psi$, one may now deduce that

$$NE(\lambda) = S(\lambda) := \{x \in X \mid \mathcal{R}(\lambda; x) = 0\} = \arg \min_{x \in X} \mathcal{R}(\lambda; x).$$

This set above is nonempty for all $\lambda \in \Psi$ in view of Theorem 3.1. To show that $NE = S$ is upper semicontinuous, we take a convergent sequence $(\lambda^n)$ in $\Psi$ with the limit $\hat{\lambda} \in \Psi$. Let $(x^n)$ be a sequence in $X$ such that $x^n \in S(\lambda^n)$, for all $n \in \mathbb{N}$, and is convergent to $\hat{x} \in X$. This means $\mathcal{R}(\lambda^n; x^n) = 0$ for all $n \in \mathbb{N}$. The lower semicontinuity of $\mathcal{R}$ implies that $\mathcal{R}(\hat{\lambda}; \hat{x}) = 0$ and that $\hat{x} \in S(\hat{\lambda})$. Therefore, the map $NE = S$ is upper semicontinuous on $\Psi$.

## 4. An existence result for SLMFG with direct preference maps

This final section presents an existence result for the SLMFG with direct preference maps of the form (1.1), with a more specific structure of the followers' problem. The theorem is, in fact, just an application of the Weierstraß theorem combined with the stability result from Theorem 3.5.

**Theorem 4.1.** Suppose that

(a) $W$ is a topological space and $Y \subset W$ is closed,
(b) $F: W \times X \to \mathbb{R}$ is lower semicontinuous,

(c) $\sigma: W \to \Psi$ is continuous, where $\Psi \subset \Lambda(\mathcal{E})$ is regular w.r.t. $\mathcal{E}$.

Then the following SLMFG has a solution:

$$ \min_{w,x} F(w,x) \qquad (4.1a) $$

$$ \text{s.t.} \quad w \in Y \qquad (4.1b) $$

$$ \sigma(w) = (A_{\nu}^{w}, P_{\nu}^{w})_{\nu \in \mathbf{F}} \qquad (4.1c) $$

$$ \forall \nu \in \mathbf{F}: \qquad (4.1d) $$

$$ \lfloor x_{\nu} \in A_{\nu}^{w}(x_{-\nu}), \quad A_{\nu}^{w}(x_{-\nu}) \cap P_{\nu}^{w}(x) = \emptyset. \qquad (4.1e) $$

*Proof.* One could observe that the constraints (4.1c)-(4.1e) reduce simply to $x \in \mathbf{NE}(\psi(w))$, or equivalently, to $(w, x) \in \text{gr}(\mathbf{NE} \circ \sigma)$. This means the SLMFG (4.1) reduces to

$$ \min_{w,x} F(w,x) \qquad (4.2a) $$

$$ \text{s.t.} \quad w \in Y \qquad (4.2b) $$

$$ (w, x) \in \text{gr}(\mathbf{NE} \circ \sigma). \qquad (4.2c) $$

From Theorem 3.5, the response map NE is upper semicontinuous. Since $\sigma$ is continuous, the composition $NE \circ \sigma: W \Rightarrow X$ is again upper semicontinuous. As a consequent, the graph of $NE \circ \sigma$ is closed due to the compactness of $X$. The Weierstraß theorem then guarantees that (4.2), and hence (4.1), has a solution. ■

In the formulation of SLMFG (4.1), the leader's action $w$ is passed to the followers through the signal map $\psi$, which explicitly selects an abstract economy profile $\lambda \in \Psi$. One could also present Theorem 4.1 (and the problem (4.1) herein) more transparently by adopting the parametric approach, i.e., each follower $\nu \in \mathbf{F}$ has the parametrized constraint map $A_{\nu}: W \times X_{-\nu} \Rightarrow X_{\nu}$ and the parametrized preference map $P_{\nu}: W \times X \Rightarrow X_{\nu}$. An alternative presentation of Theorem 4.1 now reads: If the conditions (a) and (b) hold, and the tuple $(A_{\nu}(w, \cdot), P_{\nu}(w, \cdot))_{\nu \in \mathbf{F}}$ belongs to $\Psi$, then the corresponding SLMFG of the form (1.1) has a solution.

## Acknowledgments

The authors are grateful to the peer reviewers for their valuable comments, which have had a great impact on the presentation of this paper. The first author was supported by the Petchra Pra Jom Klao M.Sc. Research Scholarship from King Mongkut's University of Technology Thonburi (Grant No.19/2564). This research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund under project number FRB670073/0164.

## Author Contributions

Supervision: Poom Kumam, Parin Chaipunya; Investigation: Yutthakan Chummongkhon; Writing – Original draft: Yutthakan Chummongkhon, Parin Chaipunya; Writing – Review & editing, Poom Kumam, Parin Chaipunya.
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