# Extended Model on Structural Stability and Robustness to Bounded Rationality

Authors: Yi Liao; Lujiang Miao; Lei Wang; Fei Xu; Chi Zhang
Journal: Complexity
 (2021)

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## Abstract

In this article, we focus on an extended modelM¯ of bounded rationality. Based on a rationality function with lower semicontinuity, we analyze the relationship between structural stability and robustness of Ω¯. To further demonstrate the applicability of our theory, we introduce a model Ω¯0 containing an abstract rationality function and generalize abstract fuzzy economies. We demonstrate the structural stability of the extended model Ω¯0 at ξ¯,ɛ. That is to say, Ω¯0 is robust to the ξ¯,ɛ-equilibria.


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## Body

## 1. Introduction

Arrow and Debreu [1] studied the existence of equilibrium for a social competitive economy assuming that all participants are perfectly rational in the game in 1952. However, in reality, perfect rationality does not often occur. Hence, the assumption of perfect rationality restricted the application of the model. On the other hand, it is well known that the stability analysis is one of the significant topics in economic models. Anderlini and Canning [2] constructed an abstract framework Ω and derived necessary and sufficient conditions for the model being robust to ϵ-equilibria. Yu and Yu [3] extended the results in [2] under weaker conditions and proved the stability and robustness of the model. Wang et al. [4] generalized the model Ω [2] with abstract fuzzy economies. These authors studied the stability and showed its robustness. Miyazaki and Azuma [5] investigated the structural stability and robustness of the model introduced in [2]. They proved that if a system is ξ,ɛ-stable, then it is ξ,ɛ-robust. Loi and Matta [6] extended the model Ω introduced in [2] to a more general model Ω¯. They investigated a pure exchange economy using the abstract construction. Yu et al. [3] studied ξ,ɛ-stability and ξ,ɛ-robustness of model Ω introduced in [2]. In particular, they considered their relationship. Then, these results on stability and robustness in control and economy were further studied during the last decades [7–11].Zadeh [12] initiated fuzzy set theory to describe scenarios with imprecise parameters. Kim and Lee [13] studied a fuzzy game and obtained corresponding equilibrium existence theorem. Huang [14, 15] studied a generalized abstract fuzzy model of economics and considered the existence of its equilibrium. Patriche [16, 17] presented a Bayesian abstract fuzzy economic model with a measure space of agents and demonstrated the existence of equilibrium of the constructed model. More recently, Cui et al. [18] studied the loss aversion level of a bimatrix game with payoff function described by fuzzy variables. These authors obtained synchronization conditions for the fuzzy stochastic complex networks.Motivated by these existing studies, in this work, we study the extended modelΩ¯0 with generalized abstract fuzzy economies and an abstract rationality function. We are particularly interested in whether small deviations in the additional rationality of the extended model Ω¯0 will cause only minor changes in bounded rational equilibria. When we extend the model Ω to a complex model Ω¯, it is difficult to obtain its structural stability. It is essential to derive the relationship between the extended model Ω¯ and its structural stability. We mainly focus on the structure of Ω and how to extend Ω to Ω¯ in a natural way.The rest of the paper is organized as follows. In section2, we recall the notion of economic model Ω, structural stability, robustness to bounded rationality, and their connection. Moreover, we extend the model Ω to a complex model Ω¯. The relationship between ξ¯,ɛ-stability and ξ¯,ɛ-robustness of Ω¯ is discussed. In section 3, we prove a new theorem about the existence of equilibrium in LΓ-spaces for the generalized abstract fuzzy economic model. We further prove the structural stability of the extended model Ω¯0 with a category of generalized abstract fuzzy economies at ξ¯,ɛ. Finally, we present the conclusions of this article in section 4.


## 2. The Extended ModelΩ¯

In this section, we introduce the following notation. Suppose thatX and Y are topological spaces. For a subset D of topological space X, we use 2D and D to denote, respectively, the set of all subsets of D and the family of all nonempty finite subset of D. If Gx is compact for all x∈X, we then say that G:X⟶2Y is a compact valued correspondence. Assume X and Y are metric spaces. If for any xn⟶x and yn∈Gxn,yn⟶y, we have y∈Gx, then G is upper semicontinuous at x∈X. If for any xn⟶x and y∈Gx, we have yn∈Gxn such that yn⟶y, then G is lower semicontinuous at x. If for any xn⟶x,hGxn,Gx⟶0, where h is the Hausdorff distance defined on Y, then G is continuous at x.Anderlini and Canning [2] studied an economic model of bounded rationality. The model Ω with a parameter space given by quadruple Ψ,X,F,R:Ψ,ϱ is a parameter space and X,d is an action space, in which ϱ and d are metrics. Here, F:Ψ×X⟶2X represents the feasibility correspondence inducing a further correspondence fξ=x∈X:x∈Fξ,x, for all ξ∈Ψ. The graph of f is denoted by Graphf=ξ,x∈Ψ×X:x∈fξ and R:Graphf⟶R+ is a rational function. When Rξ,x=0, we say that the full rationality is realized. Given a model Ω, for all ξ∈Ψ, we define the ɛ-equilibria set at ξ as follows:(1)Eξ,ɛ=x∈fξ:Rξ,x≤ɛ,∀ɛ≥0.We useEξ to denote all equilibria set at ξ as(2)Eξ=Eξ,0=x∈fξ:Rξ,x=0.Here, we extend the modelΩ proposed in [2] to a more complex model Ω¯. Using a rationality function with lower semicontinuity, we prove that Ω¯ is ξ¯,ɛ-stable, which implies that Ω¯ is robust to ξ¯,ɛ-equilibria.Loi and Matta [6] studied the extended model Ω¯ as follows.Definition 1.
(see [6]). For a model Ω=Ψ,X,F,R, its extended model Ω¯ is defined as a quadruple Ω¯=Ψ¯,X¯,F¯,R¯ satisfying the following:(1)
X¯,d¯=X,d;(2)
Ψ¯=KΨ,hϱ, where KΨ is to be the set of all compact subsets of Ψ with the Hausdorff distance hϱ related to the metric ϱ of Ψ;(3)
F¯:Ψ¯×X⟶2X is denoted by F¯=∪μ∈λ¯Fμ,x. We thus have f¯:Ψ¯⟶2X,f¯λ¯=x∈X|x∈F¯ξ¯,x;(4)
R¯:Graphf¯⟶R+ is an extended R. That is to say, for all ξ∈Ψ,R¯ξ,x=Rξ,x that satisfies(3)R¯ξ¯,x=0⟺Rξ,x=0,∀ξ∈ξ¯.Definition 2.
Given an extended modelΩ¯, for any ξ¯∈Ψ¯, the ϵ-equilibria set at ξ¯ is given by(4)E¯ξ¯,ɛ=x∈f¯ξ¯|R¯ξ¯,x⊂0,ɛ,∀ɛ≥0.
We useE¯ξ¯ to denote all equilibria set at ξ¯ as(5)E¯ξ¯=E¯ξ¯,0=x∈f¯ξ¯|R¯ξ¯,x=0.Definition 3.
The extended modelΩ¯ is structurally stable at ξ¯,ɛ if E¯:Ψ¯×R+⟶2X is continuous at ξ¯,ɛ.Definition 4.
The extended modelΩ¯ is robust to ξ¯,ɛ-equilibria if for all δ>0, we can find an ɛ¯>0 satisfying, for all ɛ′>0 with ɛ−ɛ′<ɛ¯ and for all ξ′∈Ψ¯ with hϱξ,ξ′<ɛ¯ and hdE¯ξ′¯,ɛ,E¯ξ′¯,ɛ′<δ, where hd is the Hausdorff distance defined on X.Theorem 1.
Given a modelΩ=Ψ,X,F,R and Ω¯=Ψ¯,X¯,F¯,R¯ is its corresponding extended model. If the model Ω satisfies the following assumptions:(1)
Ψ,ϱ is a complete metric space and X is a compact metric space,(2)
f:Ψ⟶2X is nonempty compact valued and upper semicontinuous,(3)
R:Graghf⟶R+ is lower semicontinuous,(4)
for allξ∈Ψ, Eξ=x∈fξ:Rξ,x≠∅,then Ω¯ being ξ¯,ɛ-stable guarantees that Ω¯ is robust to ξ¯,ɛ-equilibria.Proof.
Obviously, ifΨ,ϱ is a complete metric space, then it follows from Theorem 3.3 in Henrikson [19] that Ψ¯,hϱ is a complete metric space.
Suppose modelΩ¯ is not robust to ξ¯,ɛ-equilibria. Consequently, there exist ξ¯⟶ξ,ɛn⟶ɛ, and hdE¯ξ¯n,ɛn,E¯ξ¯n,ɛ≥δ0, where δ0>0, ɛn⊂R+, and ξ¯ is a sequence of compact subsets Ψ. Take the subsequence ɛnk of ɛn such that ɛnk≥ɛnk+1, ɛnk≥ɛ or ɛnk≤ɛnk+1, ɛnk≤ɛ.
First, consider the subsequenceɛnk satisfying ɛnk≥ɛnk+1 and ɛnk≥ɛ. Then, we have E¯ξ¯nk,ɛ⊂E¯ξ¯nk,ɛnk. Thus, we can choose xnk∈E¯ξ¯nk,ɛnk such that(6)miny∈E¯ξ¯nk,ɛdxnk,y>δ02.
Sincexnk∈E¯ξ¯nk,ɛnk, then xnk∈f¯ξ¯nk, and it follows from Definition 1 that R¯ξ¯nk,ɛnk⊂0,ɛnk. Moreover, since f:Ψ⟶2X is nonempty compact valued and upper semicontinuous, it follows from Lemma 3.7 in [6] that the correspondence f¯:Ψ¯⟶2X is nonempty compact valued and upper semicontinuous. Since ξnk⟶ξ, without loss of generality, by Lemma 2.1 in [3], we can assume xnk⟶x∈f¯ξ¯.
Because of the lower semicontinuity ofR¯, we obtain that R¯ξ¯nk,ɛnk⊂0,ɛnk,ɛnk⟶ɛ which implies that R¯ξ¯,ɛ⊂0,ɛ. Therefore, x∈E¯ξ¯,ɛ. Since the extended model Ω¯ is ξ¯,ɛ-stable, we have that hdE¯ξ¯n,ɛn,E¯ξ¯,ɛ⟶0. It follows from Lemma 2.5 in [20] and (6) that(7)miny∈E¯ξ¯nk,ɛdxnk,y≥δ02.
However, this is in contradiction withx∈E¯ξ¯,ɛ.
Next, consider the subsequenceɛnk satisfying ɛnk≤ɛnk+1 and ɛnk≤ɛ. We then have E¯ξ¯nk,ɛnk⊂E¯ξ¯nk,ɛ. Therefore, we can choose xnk∈E¯ξ¯nk,ɛ such that(8)miny∈E¯ξ¯nk,ɛnkdxnk,y>δ02.
Sincef¯:Ψ¯⟶2X is nonempty compact valued and upper semicontinuous and ξnk⟶ξ, without loss of generality, it follows from Lemma 2.1 in [3] that we can assume xnk⟶x∈f¯ξ¯.
Because of the lower semicontinuity ofR¯, we have that R¯ξ¯nk,ɛnk⊂0,ɛnk,ɛnk⟶ɛ. That is to say, R¯ξ¯,ɛ⊂0,ɛ. Therefore, x∈E¯ξ¯,ɛ. Since the extended model Ω¯ is ξ¯,ɛ-stable, we have that hdE¯ξ¯n,ɛn,E¯ξ¯,ɛ⟶0. It follows from Lemma 2.5 in [20] and (8) that(9)miny∈E¯ξ¯nk,ɛdxnk,y≥δ02.
However, this is in contradiction withx∈E¯ξ¯,ɛ, which completes the proof.Remark 1.
Theorem1 improves Theorem 4.6 in [5] and Theorem 3.4 in [3], where the extended model Ω¯ was extended. We also generalize Theorem 3.10 in [6] where the structurally stability at ξ and robustness to ϵ-equilibria are extended to the structurally stability at ξ¯,ɛ and the robustness to ξ¯,ɛ-equilibria, respectively.


## 3. Application

Here, we propose a category of new generalized abstract fuzzy economies in anLΓ-space.We then study the equilibrium existence theorem for the models. Furthermore, we establish an extended modelΩ¯0 with an abstract rationality function and a category of generalized abstract fuzzy economies. We demonstrate that Ω¯0 is structurally stable at ξ¯,ɛ. In other words, Ω¯0 is robust to the ξ¯,ɛ-equilibria.


### 3.1. The Equilibria of Generalized Abstract Fuzzy Economic Model in anLΓ-Space

The notions of abstract convex space andLΓ-space were introduced by Park [21, 22].Definition 5.
(see [21]). An abstract convex space E,D;Γ consists of a topological space E, a nonempty set D, and a map Γ:D⟶E with nonempty values.
For anyA∈D, we denote ΓA=ΓA. Let coΓD′=∪ΓA|A∈D⊂E be the Γ-convex hull of D′, where D′⊂D. If for any N∈D, we have ΓN⊂X, i.e., coΓD′⊂X, then X is said to be a Γ-convex subset of E,D;Γ related to D′. If D⊂E, the space is defined as E⊃D;Γ. Here, if coΓX∩D⊂X, then X is called Γ-convex. That is to say, X is Γ-convex related to D′=X∩D. When E=D, we let E;Γ=E,E;Γ.Definition 6.
(see [22]). If an abstract convex space E,D;Γ has a basis ℬ of a uniformity of E, then it is said to be an abstract convex uniform space E,D;Γ;ℬ. If D is dense in E and for any U∈ℬ and for any Γ-convex subset A⊂E, the set x∈E:A∩Ux≠∅ is Γ-convex, then an abstract convex uniform space E⊃D;Γ;ℬ is said to be an LΓ-space.
Based on Proposition 5.1 and Theorem 8.5 in Park [22], we can obtain Lemma 1.Lemma 1.
LetX,D;Γ be a LΓ-space. Suppose that G:X⟶2X is nonempty closed Γi-convex valued and compact upper semicontinuous. Then, we can find an x^∈X satisfying x^∈Gx^.Lemma 2.
LetXi,Di;Γii∈I be a family of LΓ-spaces. For each i∈I, suppose that Gi:X=∏Xi⟶2Xi is nonempty closed Γi-convex valued and compact upper semicontinuous. Thus, for each i∈I, we can find an x^=x^ii∈I∈X satisfying x^i∈Gix^.Proof.
LetX=∏i∈IXi and D=∏i∈IDi. For each i∈I, let πi:X⟶Xi be the projection of X onto Xi. Then, for any x∈X, G:X⟶2X is defined by Gx=∏i∈IGix. We denote Γ:D⟶2X\∅ as ΓA=∏i∈IΓiπiA,∀A∈D. It follows from Lemma 2 in [23] that X,D,Γ is a LΓ-space. Since each Gi is nonempty closed Γi-convex valued and compact upper semicontinuous, it follows from Lemma 3 of Fan [24] that G is also nonempty closed Γi-convex valued and compact upper semicontinuous. Next, we prove that, for any x∈X, Gx is Γ-convex. For every A∈Gx and A⊂D, we derive that for every i∈I,πA⊂Gix∩Di. Since every Gix is Γi-convex, we have that ΓiπiA⊂Gix for each i∈I. Therefore, ΓA=∏i∈IΓiπiA⊂∏i∈IGix=Gx. Thus, we obtain that Gx is Γ-convex and G:X⟶2X is nonempty closed Γi-convex valued and compact upper semicontinuous. It follows from Lemma 1, for each i∈I, we can find an x^=x^ii∈I∈X satisfying x^∈Gx^=∏i∈IGix^, i.e., x^i∈Gix^.
LetXandY be two nonempty convex subsets of a Hausdorff topological vector space. In this article, we use FX to denote all the fuzzy sets on X. F:X⟶FY is said to be a fuzzy mapping. Thus, we have that for each x∈X,Fx(denote by Fx in the sequel) is a fuzzy set in FY. Fxy is defined as the degree of membership of point y in Fx.
For everyq∈0,1, we use Aq=x∈X:Ax≥q to denote the q-cut set of A∈FX. If I is a set of agents (finite or infinite), Xi is a nonempty topological space (a choice set), Ai,Bi:X=∏i∈IXi⟶FXi are fuzzy constraint mappings (fuzzy constraint correspondences), and Pi:X⟶FXi is a fuzzy preference mapping (fuzzy preference correspondence), then ϒ=Xi,Ai,Bi,Pii∈I is said to be a generalized abstract fuzzy economy. If, for every i∈I,x^i∈Bix^bix^ and Aix^aix^∩Pix^pix^=∅, where ai,bi,pi:X⟶0,1, then the point x^∈X is said to be an equilibrium of ϒ.Theorem 2.
Suppose thatϒ=Xi,Ai,Bi,Pii∈I is an abstract fuzzy model of economics, X=∏i∈IXi, and ai,bi,pi:X⟶0,1. Suppose for every i∈I, ϒ=Xi,Ai,Bi,Pii∈I satisfies the following conditions:(1)
Xi,Di;Γi is a compact LΓ-space(2)
for everyx∈X,x↦Aixaix:X⟶2Xi and x↦Bixbix:X⟶2Xi are compact and nonempty closed Γi-convex valued(3)
for everyx∈X,x↦Pixpix:X⟶2Xi is closed Γi-convex valued(4)
Ei=x∈X:Aixaix∩Pixpix≠∅ is open in X(5)
for everyx∈X,xi∉Aixaix∩Pixpix
Then, for eachi∈I, we can find an x^∈X satisfying x^i∈Bix^bix^ and Aix^aix^∩Pix^pix^=∅.Proof.
LetX=∏i∈IXi,D=∏i∈IDi,andΓA=∏i∈IΓiπiA, where πi is a projection of X onto Xi. By Lemma 2 in [23], we have that X,U,φN is an LΓ-space. For each i∈I, Gi:X⟶2Xi is defined as(10)Gix=Aixaix∩Pixpix,ifx∈Ei,Bixbix,ifx∉Ei.
From conditions (2) and (3) and Theorem 3.18 in [25], we have that x⟶Aixaix∩Pixpix is nonempty closed valued and compact upper semicontinuous. On the other hand, from conditions (2)–(4), Lemma 3 in [24], and Lemma 1 in [26], we have that Gi is nonempty closed Γi-convex valued and compact upper semicontinuous. From Lemma 2, for each i∈I, we can find an x^∈X satisfying x^i∈Gix^. If, for some j∈I,x^∈Ej, we obtain that x^j∈Ajx^ajx^∩Pjx^pjx^. However, this is in contradiction with condition (5). Therefore, for each i∈I, x^∉Ei. By the definition of Gi, we have that for each i∈I,x^i∈Bix^bix^, and Aix^aix^∩Pix^pix^=∅. The proof is complete.Remark 2.
Theorem2 improves Theorem 2 in [14] and Theorem 2 in [15]. We also generalize the related results in [4] from generalized convex space to LΓ-space. Theorem 2 can be considered as an extended variant of Theorem 2 in [27] with respect to LΓ-spaces. We note that LΓ-spaces include a variety of topological spaces such as the LC-spaces and the LG-spaces (see [21, 22] and references therein).


### 3.2. Structural Stability and Robustness to Generalized Abstract Fuzzy Economies

Letϒ=Xi,Ai,Bi,Pii∈I be a generalized abstract fuzzy economic model satisfying all the conditions of Theorem 2. We introduce the definition Ψ=ξ=Gii∈I, where(11)Gix=Aixaix∩Pixpix,ifx∈Ei,Bixbix,ifx∉Ei.By the proof of Theorem2, we have that Gi:X=∏i∈IXi⟶2Xi is compact upper semicontinuous on X and there exists a point x∈X such that xi∈Gix.LetXi be a metric space induced by a metric di. For each ξ1=G1ii∈I∈Ψ and ξ2=G2ii∈I∈Ψ, we define(12)hϱξ1,ξ2=∑i∈Isupx∈XhiG1ix,G2ix,where hi is the Hausdorff metric induced by di on Xi, for each i∈I.Next, we consider the modelΩ0=Ψ,X,F,R, where Fξ,x=Gx=∏i∈IGix, for all ξ=Gii∈I∈Ψ and for any x∈X. Here, we denote f:Ψ⟶2X by fξ=x∈X:x∈Fξ,x=x∈X:xi∈Gix and R:Graghf⟶R+ is a rationality function, which is defined as(13)Rξ,x=maxi∈Idixi,Gix=maxi∈Iminyi∈Gixdixi,yi,for any x∈fξ. Here, for each i∈I, di is the distance on Xi.For everyξ∈Ψ, the ϵ-equilibria set of the generalized abstract fuzzy model of economics ϒ at ξ is defined as follows:(14)Eξ,ɛ=x∈fξ:Rξ,x≤ɛ.We useEξ to denote all equilibria set of the generalized abstract fuzzy model of economics ϒ at ξ as(15)Eξ=Eξ,0=x∈fξ:Rξ,x=0.We notice that for allξ,ɛ∈Ψ×R+, Eξ,ɛ is compact if R is lower semicontinuous. It follows from Theorem 1 that Eξ≠∅.Obviously, if and only if for eachi∈I, xi∈Gix, and x∈Eξ, Rξ,x≥0 and Rξ,x=0.Theorem 3.
LetΩ¯0=Ψ¯,X¯,F¯,R¯ be the extended model of Ω0; if Ω¯0 is structurally stable at ξ¯,ɛ, then Ω¯0 is robust to ξ¯,ɛ-equilibria.Proof.
First, we prove thatΨ,ϱ is a complete metric space. Let ξm be any Cauchy sequence in Ψ. Consequently, for any ɛ>0, we can find a positive integer Pɛ such that(16)ϱξm,ξp=∑i∈Isupx∈XhiGmix,Gpix<ɛ,∀m,p≥Pɛ.
SinceXi is compact, Xi is complete. As a result, X=∏i∈IXi is compact. Obviously, under the Hausdorff distance, the family of compact subsets of Xi is a complete metric space. We notice that Gmi:X⟶2Xi is compact upper semicontinuous. Based on Proposition 3.1.11 in [25], for each i∈I, we can always find a compact set Gix⊂Xi satisfying limp⟶∞Gpix=Gix. From (16), we obtain that(17)∑i∈Isupx∈XhiGmix,Gix≤ɛ,∀m≥Pɛ.
Sinceξm=Gmii∈I∈Ψ, we can always find an xm∈X, such that xmi∈Gmixm, for each i∈I. Because of the compactness of X, we can let xm⟶x. Since Gi is upper semicontinuous on X, we can find a positive integer m0≥Pɛ satisfying(18)Gixm⊂Uɛ,Gix,∀m≥m0.
Therefore, we get(19)xmi∈Gmixm⊂U2ɛ,Gixm⊂U3ɛ,Gix,∀m≥m0.
Letm⟶∞, we thus have that dixi,Gix≤3ɛ. Because of the arbitrariness of ϵ, we obtain that dixi,Gix=0 and as such xi∈Gix. That is to say, ξ=Gii∈I∈Ψ. Thus, Ψ,ϱ is a complete metric space.
In what follows, we aim to prove thatRξ,x is lower semicontinuous at ξ,x.
For everyξn=Gnii∈I∈Ψ with ϱξn,ξ⟶0, in which ξ=Gii∈I∈Ψ, and for any xn∈X with xn⟶x, we obtain that(20)Rξn,xn>Rξ,x−ε,∀ε>0.
SinceGi:X⟶2Xi is compact upper semicontinuous, it follows from Proposition 3.1.19 in [25] that dixi,Gix=minyi∈Gixdixi,yi is a lower semicontinuous mapping at xi. Hence, we can find a positive integer N1 satisfying, for all n≥N1,(21)minyi∈Gixndixni,yi>minyi∈Gixdixi,yi−ε2,∀i∈I.
For anyn=1,2,…, since Gni is compact, then we can find a yni∈Gnixn satisfying(22)dixni,yni=minyi∈Gnixndixni,yi,∀i∈I.
FromhiGnixn,Gixn⟶0 and yni∈Gnixn, there is always a positive integer N2 such that, for all n≥N2, there exists yni′∈Gixn with dyni′,yni<ε/2 for every i∈I. Let N=maxN1,N2. Then, for all n∈N, we have(23)Rξn,xn=maxi∈Iminyi∈Gnixndixni,yi=maxi∈Idixni,yni≥maxi∈Idixni,yni′−diyni′,yni≥maxi∈Idixni,yni′−maxi∈Idiyni′,yni≥maxi∈Iminyi∈Gixndixni,yi−ε2>maxi∈Iminyi∈Gixdixi,yi−ε=Rξ,x−ε.
Now, we prove thatRξ,x is lower semicontinuous at ξ,x.
Consider the correspondence(24)fξ=x∈X:x∈Fξ,x=x∈X:xi∈Gix.
It follows from Lemma 6 in [28] that f:Ψ⟶2X is an upper semicontinuous mapping. Therefore, according to Theorem 2, we know that if Ω¯0 is structurally stable at ξ¯,ε, then Ω¯0 is robust to ξ¯,ε-equilibria. Thus, the proof is complete.Remark 3.
Theorem3 improves Theorem 2 of [2]. First, the compactness of Ψ is dropped. Second, f is upper semicontinuous weakened compared with the continuity. Third, F is lower semicontinuity weakened than the continuity. Compared with Theorem 2 in [4], we introduce a more general model in Theorem 3. The structural stability at ξ and robustness to ε-equilibria are extended to the structural stability at ξ¯,ε and the robustness to ξ¯,ε-equilibria, respectively.


## 3.1. The Equilibria of Generalized Abstract Fuzzy Economic Model in anLΓ-Space

The notions of abstract convex space andLΓ-space were introduced by Park [21, 22].Definition 5.
(see [21]). An abstract convex space E,D;Γ consists of a topological space E, a nonempty set D, and a map Γ:D⟶E with nonempty values.
For anyA∈D, we denote ΓA=ΓA. Let coΓD′=∪ΓA|A∈D⊂E be the Γ-convex hull of D′, where D′⊂D. If for any N∈D, we have ΓN⊂X, i.e., coΓD′⊂X, then X is said to be a Γ-convex subset of E,D;Γ related to D′. If D⊂E, the space is defined as E⊃D;Γ. Here, if coΓX∩D⊂X, then X is called Γ-convex. That is to say, X is Γ-convex related to D′=X∩D. When E=D, we let E;Γ=E,E;Γ.Definition 6.
(see [22]). If an abstract convex space E,D;Γ has a basis ℬ of a uniformity of E, then it is said to be an abstract convex uniform space E,D;Γ;ℬ. If D is dense in E and for any U∈ℬ and for any Γ-convex subset A⊂E, the set x∈E:A∩Ux≠∅ is Γ-convex, then an abstract convex uniform space E⊃D;Γ;ℬ is said to be an LΓ-space.
Based on Proposition 5.1 and Theorem 8.5 in Park [22], we can obtain Lemma 1.Lemma 1.
LetX,D;Γ be a LΓ-space. Suppose that G:X⟶2X is nonempty closed Γi-convex valued and compact upper semicontinuous. Then, we can find an x^∈X satisfying x^∈Gx^.Lemma 2.
LetXi,Di;Γii∈I be a family of LΓ-spaces. For each i∈I, suppose that Gi:X=∏Xi⟶2Xi is nonempty closed Γi-convex valued and compact upper semicontinuous. Thus, for each i∈I, we can find an x^=x^ii∈I∈X satisfying x^i∈Gix^.Proof.
LetX=∏i∈IXi and D=∏i∈IDi. For each i∈I, let πi:X⟶Xi be the projection of X onto Xi. Then, for any x∈X, G:X⟶2X is defined by Gx=∏i∈IGix. We denote Γ:D⟶2X\∅ as ΓA=∏i∈IΓiπiA,∀A∈D. It follows from Lemma 2 in [23] that X,D,Γ is a LΓ-space. Since each Gi is nonempty closed Γi-convex valued and compact upper semicontinuous, it follows from Lemma 3 of Fan [24] that G is also nonempty closed Γi-convex valued and compact upper semicontinuous. Next, we prove that, for any x∈X, Gx is Γ-convex. For every A∈Gx and A⊂D, we derive that for every i∈I,πA⊂Gix∩Di. Since every Gix is Γi-convex, we have that ΓiπiA⊂Gix for each i∈I. Therefore, ΓA=∏i∈IΓiπiA⊂∏i∈IGix=Gx. Thus, we obtain that Gx is Γ-convex and G:X⟶2X is nonempty closed Γi-convex valued and compact upper semicontinuous. It follows from Lemma 1, for each i∈I, we can find an x^=x^ii∈I∈X satisfying x^∈Gx^=∏i∈IGix^, i.e., x^i∈Gix^.
LetXandY be two nonempty convex subsets of a Hausdorff topological vector space. In this article, we use FX to denote all the fuzzy sets on X. F:X⟶FY is said to be a fuzzy mapping. Thus, we have that for each x∈X,Fx(denote by Fx in the sequel) is a fuzzy set in FY. Fxy is defined as the degree of membership of point y in Fx.
For everyq∈0,1, we use Aq=x∈X:Ax≥q to denote the q-cut set of A∈FX. If I is a set of agents (finite or infinite), Xi is a nonempty topological space (a choice set), Ai,Bi:X=∏i∈IXi⟶FXi are fuzzy constraint mappings (fuzzy constraint correspondences), and Pi:X⟶FXi is a fuzzy preference mapping (fuzzy preference correspondence), then ϒ=Xi,Ai,Bi,Pii∈I is said to be a generalized abstract fuzzy economy. If, for every i∈I,x^i∈Bix^bix^ and Aix^aix^∩Pix^pix^=∅, where ai,bi,pi:X⟶0,1, then the point x^∈X is said to be an equilibrium of ϒ.Theorem 2.
Suppose thatϒ=Xi,Ai,Bi,Pii∈I is an abstract fuzzy model of economics, X=∏i∈IXi, and ai,bi,pi:X⟶0,1. Suppose for every i∈I, ϒ=Xi,Ai,Bi,Pii∈I satisfies the following conditions:(1)
Xi,Di;Γi is a compact LΓ-space(2)
for everyx∈X,x↦Aixaix:X⟶2Xi and x↦Bixbix:X⟶2Xi are compact and nonempty closed Γi-convex valued(3)
for everyx∈X,x↦Pixpix:X⟶2Xi is closed Γi-convex valued(4)
Ei=x∈X:Aixaix∩Pixpix≠∅ is open in X(5)
for everyx∈X,xi∉Aixaix∩Pixpix
Then, for eachi∈I, we can find an x^∈X satisfying x^i∈Bix^bix^ and Aix^aix^∩Pix^pix^=∅.Proof.
LetX=∏i∈IXi,D=∏i∈IDi,andΓA=∏i∈IΓiπiA, where πi is a projection of X onto Xi. By Lemma 2 in [23], we have that X,U,φN is an LΓ-space. For each i∈I, Gi:X⟶2Xi is defined as(10)Gix=Aixaix∩Pixpix,ifx∈Ei,Bixbix,ifx∉Ei.
From conditions (2) and (3) and Theorem 3.18 in [25], we have that x⟶Aixaix∩Pixpix is nonempty closed valued and compact upper semicontinuous. On the other hand, from conditions (2)–(4), Lemma 3 in [24], and Lemma 1 in [26], we have that Gi is nonempty closed Γi-convex valued and compact upper semicontinuous. From Lemma 2, for each i∈I, we can find an x^∈X satisfying x^i∈Gix^. If, for some j∈I,x^∈Ej, we obtain that x^j∈Ajx^ajx^∩Pjx^pjx^. However, this is in contradiction with condition (5). Therefore, for each i∈I, x^∉Ei. By the definition of Gi, we have that for each i∈I,x^i∈Bix^bix^, and Aix^aix^∩Pix^pix^=∅. The proof is complete.Remark 2.
Theorem2 improves Theorem 2 in [14] and Theorem 2 in [15]. We also generalize the related results in [4] from generalized convex space to LΓ-space. Theorem 2 can be considered as an extended variant of Theorem 2 in [27] with respect to LΓ-spaces. We note that LΓ-spaces include a variety of topological spaces such as the LC-spaces and the LG-spaces (see [21, 22] and references therein).


## 3.2. Structural Stability and Robustness to Generalized Abstract Fuzzy Economies

Letϒ=Xi,Ai,Bi,Pii∈I be a generalized abstract fuzzy economic model satisfying all the conditions of Theorem 2. We introduce the definition Ψ=ξ=Gii∈I, where(11)Gix=Aixaix∩Pixpix,ifx∈Ei,Bixbix,ifx∉Ei.By the proof of Theorem2, we have that Gi:X=∏i∈IXi⟶2Xi is compact upper semicontinuous on X and there exists a point x∈X such that xi∈Gix.LetXi be a metric space induced by a metric di. For each ξ1=G1ii∈I∈Ψ and ξ2=G2ii∈I∈Ψ, we define(12)hϱξ1,ξ2=∑i∈Isupx∈XhiG1ix,G2ix,where hi is the Hausdorff metric induced by di on Xi, for each i∈I.Next, we consider the modelΩ0=Ψ,X,F,R, where Fξ,x=Gx=∏i∈IGix, for all ξ=Gii∈I∈Ψ and for any x∈X. Here, we denote f:Ψ⟶2X by fξ=x∈X:x∈Fξ,x=x∈X:xi∈Gix and R:Graghf⟶R+ is a rationality function, which is defined as(13)Rξ,x=maxi∈Idixi,Gix=maxi∈Iminyi∈Gixdixi,yi,for any x∈fξ. Here, for each i∈I, di is the distance on Xi.For everyξ∈Ψ, the ϵ-equilibria set of the generalized abstract fuzzy model of economics ϒ at ξ is defined as follows:(14)Eξ,ɛ=x∈fξ:Rξ,x≤ɛ.We useEξ to denote all equilibria set of the generalized abstract fuzzy model of economics ϒ at ξ as(15)Eξ=Eξ,0=x∈fξ:Rξ,x=0.We notice that for allξ,ɛ∈Ψ×R+, Eξ,ɛ is compact if R is lower semicontinuous. It follows from Theorem 1 that Eξ≠∅.Obviously, if and only if for eachi∈I, xi∈Gix, and x∈Eξ, Rξ,x≥0 and Rξ,x=0.Theorem 3.
LetΩ¯0=Ψ¯,X¯,F¯,R¯ be the extended model of Ω0; if Ω¯0 is structurally stable at ξ¯,ɛ, then Ω¯0 is robust to ξ¯,ɛ-equilibria.Proof.
First, we prove thatΨ,ϱ is a complete metric space. Let ξm be any Cauchy sequence in Ψ. Consequently, for any ɛ>0, we can find a positive integer Pɛ such that(16)ϱξm,ξp=∑i∈Isupx∈XhiGmix,Gpix<ɛ,∀m,p≥Pɛ.
SinceXi is compact, Xi is complete. As a result, X=∏i∈IXi is compact. Obviously, under the Hausdorff distance, the family of compact subsets of Xi is a complete metric space. We notice that Gmi:X⟶2Xi is compact upper semicontinuous. Based on Proposition 3.1.11 in [25], for each i∈I, we can always find a compact set Gix⊂Xi satisfying limp⟶∞Gpix=Gix. From (16), we obtain that(17)∑i∈Isupx∈XhiGmix,Gix≤ɛ,∀m≥Pɛ.
Sinceξm=Gmii∈I∈Ψ, we can always find an xm∈X, such that xmi∈Gmixm, for each i∈I. Because of the compactness of X, we can let xm⟶x. Since Gi is upper semicontinuous on X, we can find a positive integer m0≥Pɛ satisfying(18)Gixm⊂Uɛ,Gix,∀m≥m0.
Therefore, we get(19)xmi∈Gmixm⊂U2ɛ,Gixm⊂U3ɛ,Gix,∀m≥m0.
Letm⟶∞, we thus have that dixi,Gix≤3ɛ. Because of the arbitrariness of ϵ, we obtain that dixi,Gix=0 and as such xi∈Gix. That is to say, ξ=Gii∈I∈Ψ. Thus, Ψ,ϱ is a complete metric space.
In what follows, we aim to prove thatRξ,x is lower semicontinuous at ξ,x.
For everyξn=Gnii∈I∈Ψ with ϱξn,ξ⟶0, in which ξ=Gii∈I∈Ψ, and for any xn∈X with xn⟶x, we obtain that(20)Rξn,xn>Rξ,x−ε,∀ε>0.
SinceGi:X⟶2Xi is compact upper semicontinuous, it follows from Proposition 3.1.19 in [25] that dixi,Gix=minyi∈Gixdixi,yi is a lower semicontinuous mapping at xi. Hence, we can find a positive integer N1 satisfying, for all n≥N1,(21)minyi∈Gixndixni,yi>minyi∈Gixdixi,yi−ε2,∀i∈I.
For anyn=1,2,…, since Gni is compact, then we can find a yni∈Gnixn satisfying(22)dixni,yni=minyi∈Gnixndixni,yi,∀i∈I.
FromhiGnixn,Gixn⟶0 and yni∈Gnixn, there is always a positive integer N2 such that, for all n≥N2, there exists yni′∈Gixn with dyni′,yni<ε/2 for every i∈I. Let N=maxN1,N2. Then, for all n∈N, we have(23)Rξn,xn=maxi∈Iminyi∈Gnixndixni,yi=maxi∈Idixni,yni≥maxi∈Idixni,yni′−diyni′,yni≥maxi∈Idixni,yni′−maxi∈Idiyni′,yni≥maxi∈Iminyi∈Gixndixni,yi−ε2>maxi∈Iminyi∈Gixdixi,yi−ε=Rξ,x−ε.
Now, we prove thatRξ,x is lower semicontinuous at ξ,x.
Consider the correspondence(24)fξ=x∈X:x∈Fξ,x=x∈X:xi∈Gix.
It follows from Lemma 6 in [28] that f:Ψ⟶2X is an upper semicontinuous mapping. Therefore, according to Theorem 2, we know that if Ω¯0 is structurally stable at ξ¯,ε, then Ω¯0 is robust to ξ¯,ε-equilibria. Thus, the proof is complete.Remark 3.
Theorem3 improves Theorem 2 of [2]. First, the compactness of Ψ is dropped. Second, f is upper semicontinuous weakened compared with the continuity. Third, F is lower semicontinuity weakened than the continuity. Compared with Theorem 2 in [4], we introduce a more general model in Theorem 3. The structural stability at ξ and robustness to ε-equilibria are extended to the structural stability at ξ¯,ε and the robustness to ξ¯,ε-equilibria, respectively.


## 4. Conclusions

In this paper, we discuss the relationship betweenξ¯,ε-stability and ξ¯,ε-robustness of Ω¯ using a lower semicontinuous rationality function. As an application, we introduce the model Ω¯0 consisting of a category of the parameterized generalized abstract fuzzy economic model with related abstract rationality function. We prove that the model Ω¯0 is structurally stable at ξ¯,ε. Therefore, we can conclude the robustness of Ω¯0 to ξ¯,ε-equilibria. In short, our work generalizes the results in the extant literature.For economic models, stability analysis is one of the most significant issues. Our assumptions and methods employed in this research can be adopted to solve other economic models. In summary, the results derived in this paper can help solve a variety of economic and financial problems. The conclusions drawn in this article are also helpful in solving optimal control problems, financial optimization problems, and Nash equilibrium problems. In our future research, we will use the results obtained in this article to study various economic models.



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*Source: 4902786-2021-05-17.xml*