On the existence of Nash equilibria in fuzzy generalized discontinuous games with infinite players

1 Introduction

In many economic game models, for example, the models in Dasgupta and Maskin [17] and Rothstein [31], the payoff functions of players are discontinuous. Motivated by economic problems modeled by games with discontinuous payoff functions, many authors have investigated suitable conditions guaranteeing the existence of Nash equilibria in discontinuous games; among the large literature, see Simon and Zame [32], Baye et al. [6], Reny [28], Bagh and Jofre [3], McLennan et al. [23], Carbonell-Nicolau [10], Barelli and Meneghel [5], Prokopovych [26], Nessah and Tian [24, 25], and references therein. Recently, focus has shifted to proving the existence of Nash equilibria in ordinal games with discontinuous preferences. Reny [29] proved several existence theorems of Nash equilibria in discontinuous games with the players’ preference relations over joint strategies. Carmona and Podczeck [13] extended the results of Reny [29] and provided conditions guaranteeing the existence of Nash equilibria in games in which players’ preferences can be arbitrary binary relations. Very recently, following the approach in Reny [29], Kukushkin [20] established the existence results of Nash equilibria in games with purely ordinal preferences, where payoff functions take values in arbitrary chains instead of the real line.

If a game possesses the property that the set of strategies for every player depends on the strategy choices of the other players, then we call it to be a generalized game. Debreu [16], Arrow and Debreu [1], and Rosen [30] first studied the existence of equilibria in generalized games. Since then, applications of generalized games have appeared in various fields (see Facchinei and Kanzow [19] and the references therein). Recently, Bagh [4] introduced the notion of variational continuity and obtained the existence results of equilibria in generalized discontinuous games.

Initially, Butnariu [7] used fuzzy sets in game theory. He represented the belief of each player for strategies of other players by using fuzzy sets. Since then, many authors have applied fuzzy game theory to a number of competitive decision-making situations; see, for example, Li et al. [22], Vidyottama et al. [35], Wilde [36], Chen and Larbani [14], Chakeri and Sheikholeslam [15], and the authors referenced by their works.

It should be pointed out that the authors mentioned above have investigated the game models with finite players whether in a crisp or fuzzy environments. On the other hand, due to many realistic applications (for example, Cournot competition and optimal taxation), a different line of literature studying the equilibrium problem of discontinuous games dropped the assumption that the number of players is finite. Integrating with the line of research on discontinuous games with finite players, Balder [8] discussed the existence problem of Cournot-Nash equilibria in discontinuous Cournot games with a measure space of players. Carmona and Podczeck [12] generalizes the concept of continuous security due to Barelli and Meneghel [5] for finite-players games to discontinuous games with a measure space of players and established the existence of Nash equilibria for such games. Recently, Tian [34] introduced the notion of recursive diagonal transfer continuity and established the existence of Nash equilibria in discontinuous games with arbitrary compact strategy spaces and countably infinite players.

However, to the best of our knowledge, there is no paper dealing with the existence of Nash equilibria in fuzzy generalized discontinuous games with infinite players. Inspired by the recent work on the existence of Nash equilibria in discontinuous games, in this paper, we introduce two new notions of transfer continuity for fuzzy generalized discontinuous games with infinite players, which are called fuzzy generalized quasi-weak transfer continuity and fuzzy generalized pseudo quasi-weak transfer continuity, respectively and next then, by means of a special case of a collectively maximal element theorem due to Lin and Ansari [21], we establish some existence results of Nash equilibria in fuzzy generalized discontinuous games with infinite players. As applications, we obtain some existence results of Nash equilibria in generalized discontinuous games in the setting of crisp senses.

The rest of this paper is organized as follows: Section 2 introduces some notation and definitions. Section 3 presents the main results of this paper on the existence of equilibria in fuzzy generalized discontinuous games with infinite players. Section 4 provides the existence results of Nash equilibria in generalized discontinuous games in crisp senses. Concluding remarks are given in Section 5.

2 Preliminaries

We recall some notation and definitions which will be used in the sequel. Let ℝ and ℕ denote the set of the real numbers and the set of the natural numbers, respectively. Let X be a set. We denote by 2 ^X the family of all subsets of X. Let X and Y be nonempty sets. A set-valued mapping T : X → 2 ^Y is a function from X into 2 ^Y , which implies that T assigns a unique value T (x) ∈2 ^Y for every x ∈ X. The inverse of T is a set-valued mapping T^-1 : Y → 2 ^X defined by T^-1 (y) = {x ∈ X : y ∈ T (x)} for every y ∈ Y.

Let X be a topological space and Y be a vector space. Then a function f : X → ℝ is called to be lower semicontinuous on X if the set {x ∈ X : f (x) > r} is open in X for every r ∈ ℝ and a function g : Y → ℝ is called to be quasiconcave on Y if the set {y ∈ Y : g (y) > r} is convex for every r ∈ ℝ .

Zadeh [37] initiated the fuzzy sets theory which can be used to analyze and solve problems in the setting of fuzzy environments. Let X and Y be two nonempty sets. A function from X into [0, 1] is called a fuzzy set on X. We denote by F ( X ) the family of all fuzzy sets on X. A mapping from Y into F ( X ) is called a fuzzy mapping. If F : Y → F ( X ) is a fuzzy mapping, then for each y ∈ Y, F (y) (denoted it by F _y in the sequel) is a fuzzy set in F ( X ) and the number F _y  (x) is called the degree of membership of point x in F _y . If A ∈ F ( X ) is a fuzzy set, then the set (A)  _α  = {x ∈ X : A (x) > α}, α ∈ [0, 1), is called strong α-cut, and (A) ₀ is called the support of A. For more details, we refer the reader to Tripathy and Dutta [33] and the referencestherein.

We say that a family of ordered quadruple Θ = (X _i , H _i , f _i , φ _i ) _i∈I is a fuzzy generalized game if I is an infinite set of players, X _i is a nonempty topological space (a strategy set of player i), f i : X = Π i ∈ I X i → ℝ is player i’s payoff function which is bounded, φ _i  : X → [0, 1) is a function, and H i : X → F ( X i ) is a fuzzy constrained set-valued mapping of player i. For each i ∈ I, denote X - i = ∏ j ≠ i X j . If x = (x _i ) _i∈I ∈ X, then for every i ∈ I, x _i denotes the ith coordinate, x_-i ∈ X_-i, and we can rewrite x = (x _i , x_-i) ∈ X.

Definition 2.1. Given a fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I, a strategy profile x ¯ = ( x ¯ i , x ¯ - i ) ∈ X is said to be a Nash equilibrium of Θ if for each i ∈ I, x ¯ i ∈ ( H i x ¯ ) φ i ( x ¯ ) and f i ( x ¯ ) ≥ f i ( y i , x ¯ - i ) for every y i ∈ ( H i x ¯ ) φ i ( x ¯ ) .

Remark 2.1. Definition 2.1 generalizes the corresponding definitions due to Arrow and Debreu [1], Bagh [4], Debreu [16], Dasgupta and Maskin [18], Facchinei and Kanzow [19], and Rosen [30] from crisp senses and finite players settings to fuzzy senses and infinite players settings.

Now, we introduce the following definitions of fuzzy generalized quasi-weakly transfer continuity and fuzzy generalized pseudo quasi-weakly transfer continuity which are important for proving the existence theorems of Nash equilibria in fuzzy generalized discontinuous games with infinite players.

Definition 2.2. A fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I is called to be fuzzy generalized quasi-weakly transfer continuous if whenever x ∈ X is not an equilibrium, then there exist i ∈ I, y ¯ i ∈ ( H i x ) φ i ( x ) , γ > 0, and some neighborhood O of x such that for every z ∈ O and every neighborhood O z ⊆ O of z, we have f i ( y ¯ i , z - i ) > f i ( z i , z - i * ) + γ for some z * ∈ O z .

Remark 2.2. Fuzzy generalized quasi-weakly transfer continuity implies that if whenever x is not an equilibrium, then some player i can choose a strategy y ¯ i such that this strategy is contained in player i’s action domain as determined by H _i , φ _i , and x and yields a strictly large payoff at the local security level even if the others play slightly differently thanat x.

Definition 2.3. A fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I is called to be fuzzy generalized pseudo quasi-weakly transfer continuous if whenever x ∈ X is not an equilibrium, then there exist i ∈ I, y ¯ i ∈ ( H i x ) φ i ( x ) , γ > 0, and some neighborhood O of x such that for every z ∈ O , we have f i ( y ¯ i , z - i ) > f i ( z i , z - i * ) + γ for some z * ∈ O .

Remark 2.3. If every neighborhood O z in Definition 2.2 is equal to O , then Definition 2.2 coincides with Definition 2.3. Therefore, Definition 2.3 is a special case of Definition 2.2. Definitions 2.2 and 2.3 generalize the corresponding definitions given by Nessah and Tian [24].

Definition 2.4. Given a fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I, for every i ∈ I, the first approximation function Φ i : X × X i → ℝ of player i is defined by

Φ i ( x , y i ) = sup O ∈ Ω ( x ) inf z ∈ O [ f i ( y i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ] , ∀ ( x , y i ) ∈ X × X i ,

where Ω (x) is the set of all open neighborhoods of x and O z is a neighborhood of z.

Definition 2.5. Given a fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I, for every i ∈ I, the second approximation function ϒ i : X × X i → ℝ is defined by ϒ i ( x , y i ) = sup O ∈ Ω ( x ) inf z ∈ O [ f i ( y i , z - i ) - inf z ′ ∈ O f i ( z i , z - i ′ ) ] , ∀ ( x , y i ) ∈ X × X i , where Ω (x) is the set of all open neighborhoods of x and O z is a neighborhood of z.

Remark 2.4. For each i ∈ I, the above approximation functions Φ _i and ϒ _i are real-valued by the boundedness of player i’s payoff function. Definitions 2.4 and 2.5 generalizes the corresponding definitions in Nessah and Tian [24] from finite players to infinite players and from crisp settings to fuzzy settings.

3 Existence of Nash equilibria in fuzzy environments

From now on, we assume that X _i is a Hausdorff topological vector space for every i ∈ I. In order to prove the existence theorems of Nash equilibria in fuzzy generalized discontinuous games with infinite players, we need the following lemma which can be deduced from Corollary 4.4 and Remark 4.6 due to Lin and Ansari [21].

Lemma 3.1. Let I be an arbitrary index set and K be a nonempty compact subset of X = ∏ i ∈ I X i . For each i ∈ I, let G _i  : X → 2 ^{X _i} be a set-valued mapping such that

for each x ∈ X, G _i  (x) is convex;

Then there exists x ¯ ∈ K such that G i ( x ¯ ) = ∅ for every i ∈ I.

The following theorem provides full characterizations for the existence of Nash equilibria in fuzzy generalized discontinuous games with infinite players. Before proving this theorem, we briefly explain why fuzzy quasi-weakly transfer continuity can guarantee the existence of Nash equilibra in fuzzy generalized discontinuous games with quasiconcave payoff functions. If a fuzzy generalized game Θ = (X _i , H _i , f _i , φ _i ) _i∈I fails to possess a Nash equilibrium, then it follows from the definition of fuzzy generalized quasi-weakly transfer continuity that there exists i ∈ I such that player i can choose a strategy y ¯ i ∈ ( H i x ) φ i ( x ) which yields a strictly large payoff at the local security level as long as the others play slightly differently than at x. So, the difference between the payoff at deviation strategy profile and the payoff at the disequilibrium strategy x is uniformly positive. On the other hand, for each i ∈ I, we can construct a set-valued mapping G _i  : X → 2 ^{X _i} by using H _i , φ _i , and the first approximation function Φ _i . Subsequently, by showing the lower semicontinuity and the quasiconcavity of Φ _i and combining the other conditions of Theorem 3.1, we can check that G _i satisfies all the conditions of Lemma 3.1. Therefore, by Lemma 3.1, there exists a strategy profile x ¯ such that x ¯ i ∈ ( H i x ¯ ) φ i ( x ¯ ) for every i ∈ I and the value of the first approximation function Φ _i is nonpositive at every deviation strategy y i ∈ ( H i x ¯ ) φ i ( x ¯ ) , which is impossible.

Theorem 3.1. Let Θ = (X _i , H _i , f _i , φ _i ) _i∈I be a fuzzy generalized game satisfying the fuzzy generalized quasi-weakly transfer continuity and K be a nonempty compact subset of X. For each i ∈ I, assume that:

for each x ∈ X, (H _{i x} ) _{φ i (x)} is nonempty convex;

Then Θ has at least a Nash equilibrium in K.

Proof. For each i ∈ I, define a set-valued mapping L _i  : X → 2 ^{X _i} by L _i  (x) = {y _i  ∈ X _i  : Φ _i  (x, y _i ) >0} for every x ∈ X. Further, for every i ∈ I, let us define a set-valued mapping G _i  : X → 2 ^{X _i} by setting, for every x ∈ X, G i ( x ) = { ( H i x ) φ i ( x ) ⋂ L i ( x ) , if x ∈ F i , ( H i x ) φ i ( x ) , if x ∉ F i . We prove Theorem 3.1 in the following five steps.

Step 1. Show that the set L i - 1 ( y i ) is open in X for every i ∈ I and every y _i  ∈ X _i .

We prove that for every i ∈ I and every y _i  ∈ X _i , the function X ∋ x ↦ Φ _i  (x, y _i ) is lower semicontinuous on X. In fact, for every i ∈ I and every y _i  ∈ X _i , we define the function X ∋ x ↦ ψ O i ( x , y i ) as follows: ψ O i ( x , y i ) = { inf z ∈ O [ f i ( y i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ] , if x ∈ O , - ∞ , if x ∉ O , where O is an open neighborhood and O z is a neighborhood of z. We show that the function X ∋ x ↦ ψ O i ( x , y i ) is lower semicontinuous on X. Indeed, it suffices to prove that the set { x ∈ X : ψ O i ( x , y i ) > r } is open in X for every i ∈ I, every y _i  ∈ X _i , and every r ∈ ℝ . In order to do so, let x ∈ { x ∈ X : ψ O i ( x , y i ) ≤ r } ¯ , where { x ∈ X : ψ O i ( x , y i ) ≤ r } ¯ denotes the closure of { x ∈ X : ψ O i ( x , y i ) ≤ r } . Then there exists a net { x ρ } ρ ∈ Λ ⊆ { x ∈ X : ψ O i ( x , y i ) ≤ r } such that x _ρ  → x. We prove that x ∈ { x ∈ X : ψ O i ( x , y i ) ≤ r } by contradiction. Suppose that x ∉ { x ∈ X : ψ O i ( x , y i ) ≤ r } . Then we get ψ O i ( x , y i ) > r . Further, if x ∉ O , then by the definition of ψ O i ( x , y i ) , we can see that - ∞ > r, which is a contradiction. Thus, we have x ∈ O and ψ O i ( x , y i ) = inf z ∈ O [ f i ( y i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ] > r . Since { x ρ } ρ ∈ Λ ⊆ { x ∈ X : ψ O i ( x , y i ) ≤ r } , we have φ O i ( x ρ , y i ) ≤ r for every ρ ∈ Λ. Suppose that there exists ρ₀ ∈ Λ such that x ρ 0 ∈ O . Then we have inf z ∈ O [ f i ( y i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ] ≤ r , which contradicts the fact that inf z ∈ O [ f i ( y i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ] > r , so we conclude that x ρ ∉ O for every ρ ∈ Λ. Since x ρ → x ∈ O , we have x ρ ∈ O for sufficiently large ρ ∈ Λ, which contradicts the fact that x ρ ∉ O for every ρ ∈ Λ. Therefore, x ∈ { x ∈ X : ψ O i ( x , y i ) ≤ r } . This shows that the set { x ∈ X : ψ O i ( x , y i ) ≤ r } is closed in X and thus, the set { x ∈ X : ψ O i ( x , y i ) > r } is open in X, which implies that the function X ∋ x ↦ ψ O i ( x , y i ) is lower semicontinuous on X. Since the function X ∋ x ↦ Φ _i  (x, y _i ) is the pointwise supremum of a set of lower semicontinuous functions on X, it follows from Proposition 1.5 in Aubin [2] that the function X ∋ x ↦ Φ _i  (x, y _i ) is lower semicontinuous on X and thus, the set L i - 1 ( y i ) = { x ∈ X : Φ i ( x , y i ) > 0 } is open in X for every i ∈ I and every y _i  ∈ X _i .

Step 2. Show that the set L _i  (x) is convex for every i ∈ I and every x ∈ X.

In order to do so, we first prove that the set {y _i ∈X _i  : Φ _i  (x, y _i ) ≥ t} is convex for every i ∈ I, every x ∈ X, and every t ∈ ℝ . In fact, let β ∈ [0, 1]and y i 1 , y i 2 ∈ { y i ∈ X i : Φ i ( x , y i ) ≥ t } . Then by the definition of Φ _i , there exists O 1 ∈ Ω ( x ) such that inf z ∈ O 1 [ f i ( y i 1 , z - i ) - sup O z ⊆ O 1 inf z ′ ∈ O z f i ( z i , z - i ′ ) ] ≥ t . Therefore, we have the following:

f i ( y i 1 , z - i ) - sup O z ⊆ O 1 inf z ′ ∈ O z f i ( z i , z - i ′ ) ≥ t , ∀ z ∈ O 1 .(1) Similarly, there exists O 2 ∈ Ω ( x ) such that

f i ( y i 2 , z - i ) - sup O z ⊆ O 2 inf z ′ ∈ O z f i ( z i , z - i ′ ) ≥ t , ∀ z ∈ O 2 .(2) Let O * = O 1 ⋂ O 2 . Since O * ⊆ O j , j ∈ { 1 , 2 } , the following inequality holds:

sup O z ⊆ O * inf z ′ ∈ O z f i ( z i , z - i ′ ) ≤ sup O z ⊆ O j inf z ′ ∈ O z f i ( z i , z - i ′ ) , j ∈ { 1 , 2 } .(3) By (1), (2), and (3), for every z ∈ O * , we have

f i ( y i 1 , z - i ) ≥ sup O z ⊆ O * inf z ′ ∈ O z f i ( z i , z - i ′ ) + t ,(4)

f i ( y i 2 , z - i ) ≥ sup O z ⊆ O * inf z ′ ∈ O z f i ( z i , z - i ′ ) + t .(5) Now, we can conclude that the set {y _i  ∈ X _i  : f _i  (y _i , z_-i) ≥ r} is convex for every i ∈ I, every z_-i ∈ X_-i, and every r ∈ ℝ in view of (iv) and the following fact:

{ y i ∈ X i : f i ( y i , z - i ) ≥ r } = ⋂ ɛ > 0 { y i ∈ X i : f i ( y i , z - i ) > r - ɛ } .(6) Thus, by (4), (5), and the fact that {y _i  ∈ X _i  : f _i  (y _i , z_-i) ≥ r} is convex for every i ∈ I, every z_-i ∈ X_-i, and every r ∈ ℝ , we have

f i ( β y i 1 + ( 1 - β ) y i 2 , z - i ) ≥ sup O z ⊆ O * inf z ′ ∈ O z f i ( z i , z - i ′ ) + t , ∀ z ∈ O * .(7) By (7) and the definition of Φ _i , we have Φ i ( x , β y i 1 + ( 1 - β ) y i 2 ) ≥ t and thus, β y i 1 + ( 1 - β ) y i 2 ∈ { y i ∈ X i : Φ i ( x , y i ) ≥ t } , which implies that the set {y _i  ∈ X _i  : Φ _i  (x, y _i ) ≥ t} is convex for every i ∈ I, every x ∈ X, and every t ∈ ℝ .

Secondly, we show that Φ _i is quasiconcave on X _i , that is, the set {y _i  ∈ X _i  : Φ _i  (x, y _i ) > s} is convex for every i ∈ I, every x ∈ X, and every s ∈ ℝ . In fact, let λ ∈ [0, 1], y i 1 , y i 2 ∈ { y i ∈ X i : Φ i ( x , y i ) > s } , andlet t = min { Φ i ( x , y i 1 ) , Φ i ( x , y i 2 ) } > s . Then y i 1 , y i 2 ∈ {y _i  ∈ X _i  : Φ _i  (x, y _i ) ≥ t}. Since {y _i  ∈ X _i  : Φ _i  (x, y _i ) ≥ t} is convex for every i ∈ I, every x ∈ X, and every t ∈ ℝ , it follows that λ y i 1 + ( 1 - λ ) y i 2 ∈ { y i ∈ X i : Φ i ( x , y i ) ≥ t > s } , which implies that the set {y _i  ∈ X _i  : Φ _i  (x, y _i ) > s} is convex for every i ∈ I, every x ∈ X, and every s ∈ ℝ . Therefore, the set L _i  (x) = {y _i  ∈ X _i  : Φ _i  (x, y _i ) >0} is convex for every i ∈ I and every x ∈ X.

Step 3. Show that G _i  (x) is convex and x _i  ∉ G _i  (x) for every i ∈ I and every x ∈ X.

By (i) and the fact that L _i  (x) is convex for every i ∈ I, we can see that G _i  (x) is convex for every i ∈ I and every x ∈ X. Next, we show that x _i  ∉ G _i  (x) for every x ∈ X. In light of the proof of Theorem 2.1 in Nessah and Tian [24], we have Φ _i  (x, x _i ) ≤0 for every i ∈ I and every x ∈ X. For the sake of completeness, we give the proof as follows. Suppose the contrary. Then it follows that there exist i ∈ I and x ∈ X such that Φ _i  (x, x _i ) >0 and thus, we can choose a real number ξ > 0 such that Φ _i  (x, x _i ) > ξ > 0. By the definition of Φ _i  (x, x _i ), there exists a neighborhood O of x such that f i ( x i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( x i , z - i ′ ) > ξ for every z = ( x i , z - i ) ∈ O . Taking O z = O , we obtain f i ( x i , z - i ) - inf z ′ ∈ O f i ( x i , z - i ′ ) > ξ for every z = ( x i , z - i ) ∈ O . Further, by the definition of infimum, for the given ξ > 0, there exists z ∈ O such that f i ( x i , z - i ) - inf z ′ ∈ O f i ( x i , z - i ′ ) < ξ , which contradicts f i ( x i , z - i ) - inf z ′ ∈ O f i ( x i , z - i ′ ) > ξ . Therefore, we have Φ _i  (x, x _i ) ≤0 for every i ∈ I and every x ∈ X and thus, x _i  ∉ L _i  (x) for every i ∈ I and every x ∈ X. By the definitions of G _i and F i , we can easily deduce that x _i  ∉ G _i  (x) for every i ∈ I and every x ∈ X.

Step 4. Show that ⋃ y i ∈ X i G i - 1 ( y i ) = ⋃ y i ∈ X i int G i - 1 ( y i ) for every i ∈ I.

For each i ∈ I and each y _i  ∈ X _i , we have G i - 1 ( y i ) = { x ∈ F i : y i ∈ ( H i x ) φ i ( x ) ⋂ L i ( x ) } ⋃ { x ∉ F i : y i ∈ ( H i x ) φ i ( x ) } = ( ( X \ F i ) ⋂ { x ∈ X : H i x ( y i ) > φ i ( x ) } ) ⋃ ( F i ⋂ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ) = ( ( X \ F i ) ⋃ ( F i ⋂ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ) ) ⋂ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋃ ( F i ⋂ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ) ) = ( X ⋂ ( ( X \ F i ) ⋃ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ) ) ⋂ ( ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋃ F i ) ⋂ { x ∈ X : H i x ( y i ) > φ i ( x ) } ) = ( ( X \ F i ) ⋃ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ) ⋂ ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ) = ( { x ∈ X : H i x ( y i ) > φ i ( x ) } ⋂ L i - 1 ( y i ) ) ⋃ ( { x ∈ X \ F i : H i x ( y i ) > φ i ( x ) } ) . We are going to show that ⋃ y i ∈ X i G i - 1 ( y i ) = ⋃ y i ∈ X i int G i - 1 ( y i ) for every i ∈ I. In fact, it is obvious that ⋃ y i ∈ X i int G i - 1 ( y i ) ⊆ ⋃ y i ∈ X i G i - 1 ( y i ) . In order to prove that ⋃ y i ∈ X i G i - 1 ( y i ) ⊆ ⋃ y i ∈ X i int G i - 1 ( y i ) , we take x 0 ∈ ⋃ y i ∈ X i G i - 1 ( y i ) arbitrarily. Then there exists y i * ∈ X i such that

x 0 ∈ G i - 1 ( y i * ) = ( { x ∈ X : H i x ( y i * ) > φ i ( x ) } ⋂ L i - 1 ( y i * ) ) ⋃ ( { x ∈ X \ F i : H i x ( y i * ) > φ i ( x ) } ) .

If x 0 ∈ { x ∈ X : H i x ( y i * ) > φ i ( x ) } ⋂ L i - 1 ( y i * ) , then by (ii) and the fact that L i - 1 ( y i * ) is open in X, we have x 0 ∈ { x ∈ X : H i x ( y i * ) > φ i ( x ) } ⋂ L i - 1 ( y i * ) = int ( { x ∈ X : H i x ( y i * ) > φ i ( x ) } ⋂ L i - 1 ( y i * ) ) ⊆ int G i - 1 ( y i * ) . If x 0 ∈ { x ∈ X \ F i : H i x ( y i * ) > φ i ( x ) } , then according to (iii), there exists y i ˜ ∈ X i such that

x 0 ∈ int { x ∈ X \ F i : H i x ( y i ˜ ) > φ i ( x ) } ⊆ int G i - 1 ( y i ˜ ) . Combining these two cases, we can conclude that ⋃ y i ∈ X i G i - 1 ( y i ) ⊆ ⋃ y i ∈ X i int G i - 1 ( y i ) . Therefore, we have ⋃ y i ∈ X i G i - 1 ( y i ) = ⋃ y i ∈ X i int G i - 1 ( y i ) for every i ∈ I.

Step 5. Show that Θ has at least a Nash equilibrium in K.

We check (iv) of Lemma 3.1. By (v) and the definition of L _i , there exists a nonempty compact convex subset C _i of X _i for every i ∈ I such that for all x ∈ X \ K, there exists i ∈ I such that (H _{i x} ) _{φ i (x)}⋂ L _i  (x) ⋂ C _i  ≠ ∅. If x ∈ F i , then by the definition of G _i , we have G i ( x ) ⋂ C i = ( H i x ) φ i ( x ) ⋂ L i ( x ) ⋂ C i ≠ ∅ . If x ∉ F i , then by the definition of G _i again, we obtain G i ( x ) ⋂ C i = ( H i x ) φ i ( x ) ⋂ C i ⊇ ( H i x ) φ i ( x ) ⋂ L i ( x ) ⋂ C i ≠ ∅ . Therefore, in both cases, we have G _i  (x)⋂ C _i  ≠ ∅. All the requirements of Lemma 3.1 are fulfilled. So, by Lemma 3.1, there exists x ¯ ∈ K such that G i ( x ¯ ) = ∅ for every i ∈ I. If x ¯ ∉ F i for some i ∈ I, then ( H i x ¯ ) φ i ( x ¯ ) = ∅ which contradicts (i). Therefore, x ¯ ∈ F i for every i ∈ I and thus, we have x ¯ i ∈ ( H i x ¯ ) φ i ( x ¯ ) and ( H i x ¯ ) φ i ( x ¯ ) ⋂ L i ( x ¯ ) = ∅ for every i ∈ I, which implies the following:

Φ i ( x ¯ , y i ) ≤ 0 , ∀ i ∈ I , ∀ y i ∈ ( H i x ¯ ) φ i ( x ¯ ) .(8) Now, we show that x ¯ is a Nash equilibrium of the fuzzy generalized game Θ. In fact, suppose to the contrary that x ¯ is not a Nash equilibrium. Then by the definition of fuzzy generalized quasi-weak transfer continuity, there exist i ∈ I, y ¯ i ∈ ( H i x ¯ ) φ i ( x ¯ ) , γ > 0, and some neighborhood O of x ¯ such that for every z ∈ O and every neighborhood O z ⊆ O of z, f i ( y ¯ i , z - i ) > f i ( z i , z - i * ) + γ for some z * ∈ O z . Thus, for every z ∈ O , we have

f i ( y ¯ i , z - i ) - sup O z ⊆ O inf z ′ ∈ O z f i ( z i , z - i ′ ) ≥ f i ( y ¯ i , z - i ) - f i ( z i , z - i * ) > γ .(9) By (9), we have Φ i ( x ¯ , y ¯ i ) ≥ γ > 0 , which contradicts (8). Therefore, x ¯ is a Nash equilibrium of the fuzzy generalized game Θ. This completes the proof. □

Remark 3.1. (1) (ii) of Theorem 3.1 can be replaced by the following conditions.

(ii)^′ For each y _i  ∈ X _i , the function X ∋ x ↦ H _{i x}  (y _i ) is lower semicontinuous on X;

(ii)^″ φ _i  : X → [0, 1) is an upper semicontinuous function.

In fact, by (ii)^′ and (ii)^″, we know that the function X ∋ x ↦ H _{i x}  (y _i ) - φ _i  (x) is lower semicontinuous on X for every y _i  ∈ X _i . Therefore, the set {x ∈ X : H _{i x}  (y _i ) > φ _i  (x)} is open in X for every y _i  ∈ X _i .

(2) (iii) of Theorem 3.1 can be replaced by the following equivalent condition.

(iii)^′ For each y _i  ∈ X _i and each x ∈ { x ∈ X \ F i : H i x ( y i ) > φ i ( x ) } , there exists a y ˆ i ∈ X i such that x ∈ int { x ∈ X \ F i : H i x ( y ˆ i ) > φ i ( x ) } , where F i = { x ∈ X : x i ∈ ( H i x ) φ i ( x ) } .

(3) Theorem 3.1 is different from Theorem 1 of Carbonell-Nicolau and McLean [11] in the following ways: (a) Theorem 3.1 deals with the existence of Nash equilibria in fuzzy generalized discontinuous games, while Theorem 1 of Carbonell-Nicolau and McLean [11] discusses the existence of Nash Equilibra in Bayesian games; (b) the game model in Theorems 3.1 permits infinite players, while the game model in Theorem 1 of Carbonell-Nicolau and McLean [11] is only suitable for finite players setting because this theorem needs the upper semicontinuity of the sum of all players’ payoff functions; (c) every strategy space X _i in Theorem 3.1 is a noncompact Hausdorff topological vector space, while every strategy space X _i in Theorem 1 of Carbonell-Nicolau and McLean [11] needs to be a compact metric space; (d) the game model in Theorem 3.1 has fuzzy constraint set-valued mappings, which allow for the possibility that a player’s set of strategies might depend on the strategy choices of other players in the setting of fuzzy environments (for details, see Dasgupta and Maskin [16] and the references therein), while the game model in Theorem 1 of Carbonell-Nicolau and McLean [11] has no constraint set-valued mappings; (e) Theorem 3.1 is proved based on a collectively maximal element theorem due to Lin and Ansari [21], while Carbonell-Nicolau and McLean [11] proved Theorem 1 by transforming Bayesian games into normal games and using Theorem 3.1 of Renny [28].

Also Theorem 3.1 can be compared with Theorem 4.2 of Renny [29], Theorem 2 of Prokopovych and Yannelis [27], Theorem 1 of Nessah and Tian [25], and Theorem 3.3 of Bich and Laraki [9] in several aspects.

Example 3.1. Let Θ = ( X i , H i , f i , φ i ) i ∈ ℕ be a fuzzy generalized game with a countable set of players such that for every i ∈ ℕ , X _i  = (0, 1] is a non-empty convex strategy set and X = Π_i∈IX _i is equipped with box topology. For each i ∈ ℕ , let the fuzzy constraint set-valued mapping H _i  : X → 2 ^{X _i} be defined by

H i x ( y i ) = { ( x i 2 + 1 8 ) y i i , if x i ∈ ( 0 , 1 2 ] , x - i ∈ X - i , y i ∈ ( 0 , 1 ] , ( 1 3 x i 2 + 1 8 ) y i i , if x i ∈ ( 1 2 , 1 ] , x - i ∈ X - i , y i ∈ ( 0 , 1 ] . The function φ _i  : X → [0, 1) is defined by φ i ( x ) = { ( 1 9 x i + 1 36 ) 1 i , if x i ∈ ( 0 , 1 2 ] , x - i ∈ X - i , ( 2 27 x i 2 + 7 108 ) 1 i , if x i ∈ ( 1 2 , 1 ] , x - i ∈ X - i . Then, for each i ∈ ℕ , we have ( H i x ) φ i ( x ) = { ( 2 9 , 1 ] , if x i ∈ ( 0 , 1 2 ] , x - i ∈ X - i , ( 16 x i 2 + 14 72 x i 2 + 27 , 1 ] , if x i ∈ ( 1 2 , 1 ] , x - i ∈ X - i . It is obvious that for each i ∈ ℕ and each x ∈ X, the set (H _{i x} ) _{φ i (x)} is nonempty convex. For each i ∈ ℕ and each y _i  ∈ (0, 1], by the continuity of the functions x ↦ H _{i x}  (y _i ) and φ _i on segmentation intervals, we can see that the set {x ∈ X : H _{i x}  (y _i ) > φ _i  (x)} is (possible empty) open in X. By calculation, we know that the set X \ F i = { x ∈ X : x i ∉ ( H i x ) φ i ( x ) } = ( 0 , 2 9 ] × X - i is open in X. Thus, (iii) of Theorem 3.1 holds.

Now, we define the payoff functions by setting, for each i ∈ ℕ and each (x _i , x_-i) ∈ X, f i ( x ) = f i ( x i , x - i ) = { x i + 1 , if x - i ≠ ( 1 2 , 1 2 , … , 1 2 , … ) , x i - 1 , if x - i = ( 1 2 , 1 2 , … , 1 2 , … ) . It is easy to see that f _i is quasiconcave on X _i . Next, we prove that the game Θ is quasi-weakly transfer continuous in the following two steps.

Step 1. Let i ∈ ℕ , ɛ > 0, and x ∈ X be given arbitrarily. We show that the game Θ has the property that there exist a strategy y _i  ∈ (H _{i x} ) _{φ i (x)} and a neighborhood V of x_-i such that f i ( y i , z - i ) ≥ f i ( x ) - ɛ 2 for every z - i ∈ V . In fact, if x - i ≠ ( 1 2 , 1 2 , … , 1 2 , … ) , then it is clear that there exist y _i  = 1 ∈ (H _{i x} ) _{φ i (x)} and some neighborhood V of x_-i such that f i ( y i , z - i ) ≥ f i ( x ) - ɛ 2 for every z - i ∈ V . If x - i = ( 1 2 , 1 2 , … , 1 2 , … ) , then there also exist y _i  = 1 ∈ (H _{i x} ) _{φ i (x)} and some neighborhood V of x_-i such that f i ( x ) - ɛ 2 = x i - 1 - ɛ 2 ≤ 0 ≤ f i ( y i , z - i ) for every z - i ∈ V .

Step 2. Let x = (x _i , x_-i) be a nonequilibrium point of Θ. Then there must exist a player i such that 0 < x _i  < 1. Otherwise, for each i ∈ ℕ , x _i  = 1 and it is easy to check that (1, 1, …, 1, …) is a Nash equilibrium of Θ, which is a contradiction. Thus, there exists a player i such that 0 < x _i  < 1. Then we can choose some γ > 0 such that x _i  + 2γ < 1. If x - i ≠ ( 1 2 , 1 2 , … , 1 2 , … ) , then there exist a strategy x ¯ i = 1 and some neighborhood O 1 ⊆ ( x i - γ , x i + γ ) × X - i of x such that for each z ∈ O 1 and each neighborhood O z 1 of z, we have f i ( x ¯ i , x - i ) = 2 > x i + 1 + 2 γ > z i + 1 + γ = f i ( z i , z - i ′ ) + γ for some z ′ ∈ O z 1 with z - i ′ ≠ ( 1 2 , 1 2 , … , 1 2 , … ) . If x - i = ( 1 2 , 1 2 , … , 1 2 , … ) , then there exist a strategy x ¯ i = 1 and some neighborhood O 1 ⊆ ( x i - γ , x i + γ ) × X - i of x such that for each z ∈ O 1 and each neighborhood O z 1 of z, there exits z ′ ∈ O z 1 with z - i ′ = ( 1 2 , 1 2 , … , 1 2 , … ) , so we have f i ( z i , z - i ′ ) + γ = z i - 1 + γ ≤ x i + 2 γ - 1 < f i ( x ¯ i , x - i ) = 0 . In summary, if whenever x is not a nonequilibrium point of Θ, then there exist i ∈ ℕ , a strategy x ¯ i ∈ X i , γ > 0, and some neighborhood O 1 of x such that for each z ∈ O 1 and each neighborhood O z 1 of z, we have f i ( x ¯ i , x - i ) > f i ( z i , z - i ′ ) + γ for some z ′ ∈ O z 1 . By Step 1, for the ( x ¯ i , x - i ) ∈ X , there exist y _i  ∈ (H _{i x} ) _{φ i (x)} and a neighborhood O 2 = X i × V of x such that f i ( y i , z - i ) ≥ f i ( x ¯ i , x - i ) - γ 2 for every z ∈ O 2 . Thus, for each z ∈ O = O 1 ⋂ O 2 and each neighborhood O z ⊆ O of z, there exists z ′ ∈ O z such that f i ( y i , z - i ) ≥ f i ( x ¯ i , x - i ) - γ 2 > f i ( z i , z - i ′ ) + γ 2 , which implies that the game Θ is quasi-weakly transfer continuous.

Now, we check that (v) of Theorem 3.1 holds. For each i ∈ ℕ , let C i = [ i i + 1 , 1 ] be a compact convex subset of X _i . Let K = [ 1 2 , 1 ] × [ 1 2 , 1 ] × … × [ 1 2 , 1 ] ×… be a nonempty compact sunset of X. Then for each x ∈ X \ K, it is easy to verify that x is is not a nonequilibrium point of Θ. Thus, it follows from the quasi-weakly transfer continuity of Θ that there exist i ∈ ℕ , y _i  = 1 ∈ (H _{i x} ) _{φ i (x)}, γ > 0, and some neighborhood O of x such that for every z ∈ O and every neighborhood O z ⊆ O of z, we have f i ( y i , z - i ) > f i ( z i , z - i ′ ) + γ 2 for some z ′ ∈ O z . Therefore, for every z ∈ O , we have Φ i ( x , y i ) ≥ f i ( y i , z - i ) - f i ( z i , z - i ′ ) > 0 . Hence, (H _{i x} ) _{φ i (x)}⋂ {y _i  ∈ X _i  : Φ _i  (x, y _i ) >0} ⋂ C _i  ≠ ∅. Therefore, all the conditions of Theorem 3.1 are satisfied. Thus, we can find a Nash equilibrium point of Θ in K. In fact, there exists x ¯ = ( 1 , 1 , … , 1 , … ) ∈ K such that for each i ∈ I, x ¯ i ∈ ( H i x ¯ ) φ i ( x ¯ ) and f i ( x ¯ ) ≥ f i ( y i , x ¯ - i ) for every y i ∈ ( H i x ¯ ) φ i ( x ¯ ) .