Control and Stability Analysis of Complex Dynamical Systems involved in Finance, Ecology and EngineeringView this Special Issue
Extended Model on Structural Stability and Robustness to Bounded Rationality
In this article, we focus on an extended model of bounded rationality. Based on a rationality function with lower semicontinuity, we analyze the relationship between structural stability and robustness of . To further demonstrate the applicability of our theory, we introduce a model containing an abstract rationality function and generalize abstract fuzzy economies. We demonstrate the structural stability of the extended model at . That is to say, is robust to the -equilibria.
Arrow and Debreu  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  constructed an abstract framework and derived necessary and sufficient conditions for the model being robust to -equilibria. Yu and Yu  extended the results in  under weaker conditions and proved the stability and robustness of the model. Wang et al.  generalized the model  with abstract fuzzy economies. These authors studied the stability and showed its robustness. Miyazaki and Azuma  investigated the structural stability and robustness of the model introduced in . They proved that if a system is -stable, then it is -robust. Loi and Matta  extended the model introduced in  to a more general model . They investigated a pure exchange economy using the abstract construction. Yu et al.  studied -stability and -robustness of model introduced in . 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  initiated fuzzy set theory to describe scenarios with imprecise parameters. Kim and Lee  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.  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 model 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 will cause only minor changes in bounded rational equilibria. When we extend the model to a complex model , it is difficult to obtain its structural stability. It is essential to derive the relationship between the extended model and its structural stability. We mainly focus on the structure of and how to extend to in a natural way.
The rest of the paper is organized as follows. In section 2, we recall the notion of economic model , structural stability, robustness to bounded rationality, and their connection. Moreover, we extend the model to a complex model . The relationship between -stability and -robustness of is discussed. In section 3, we prove a new theorem about the existence of equilibrium in -spaces for the generalized abstract fuzzy economic model. We further prove the structural stability of the extended model with a category of generalized abstract fuzzy economies at . Finally, we present the conclusions of this article in section 4.
2. The Extended Model
In this section, we introduce the following notation. Suppose that and are topological spaces. For a subset of topological space , we use and to denote, respectively, the set of all subsets of and the family of all nonempty finite subset of . If is compact for all , we then say that is a compact valued correspondence. Assume and are metric spaces. If for any and , we have , then is upper semicontinuous at . If for any and , we have such that , then is lower semicontinuous at . If for any , where is the Hausdorff distance defined on , then is continuous at .
Anderlini and Canning  studied an economic model of bounded rationality. The model with a parameter space given by quadruple is a parameter space and is an action space, in which and are metrics. Here, represents the feasibility correspondence inducing a further correspondence , for all . The graph of is denoted by and is a rational function. When , we say that the full rationality is realized. Given a model , for all , we define the -equilibria set at as follows:
We use to denote all equilibria set at as
Here, we extend the model proposed in  to a more complex model . Using a rationality function with lower semicontinuity, we prove that is -stable, which implies that is robust to -equilibria.
Loi and Matta  studied the extended model as follows.
Definition 1. (see ). For a model , its extended model is defined as a quadruple satisfying the following:(1);(2), where is to be the set of all compact subsets of with the Hausdorff distance related to the metric of ;(3) is denoted by . We thus have ;(4) is an extended . That is to say, for all that satisfies
Definition 2. Given an extended model , for any , the -equilibria set at is given byWe use to denote all equilibria set at as
Definition 3. The extended model is structurally stable at if is continuous at .
Definition 4. The extended model is robust to -equilibria if for all , we can find an satisfying, for all with and for all with and where is the Hausdorff distance defined on .
Theorem 1. Given a model and is its corresponding extended model. If the model satisfies the following assumptions:(1) is a complete metric space and is a compact metric space,(2) is nonempty compact valued and upper semicontinuous,(3) is lower semicontinuous,(4)for all , ,then being -stable guarantees that is robust to -equilibria.
Proof. Obviously, if is a complete metric space, then it follows from Theorem 3.3 in Henrikson  that is a complete metric space.
Suppose model is not robust to -equilibria. Consequently, there exist , and , where , , and is a sequence of compact subsets . Take the subsequence of such that , or , .
First, consider the subsequence satisfying and . Then, we have . Thus, we can choose such thatSince , then , and it follows from Definition 1 that . Moreover, since is nonempty compact valued and upper semicontinuous, it follows from Lemma 3.7 in  that the correspondence is nonempty compact valued and upper semicontinuous. Since , without loss of generality, by Lemma 2.1 in , we can assume .
Because of the lower semicontinuity of , we obtain that which implies that . Therefore, . Since the extended model is -stable, we have that . It follows from Lemma 2.5 in  and (6) thatHowever, this is in contradiction with .
Next, consider the subsequence satisfying and . We then have . Therefore, we can choose such thatSince is nonempty compact valued and upper semicontinuous and , without loss of generality, it follows from Lemma 2.1 in  that we can assume .
Because of the lower semicontinuity of , we have that . That is to say, . Therefore, . Since the extended model is -stable, we have that . It follows from Lemma 2.5 in  and (8) thatHowever, this is in contradiction with , which completes the proof.
Remark 1. Theorem 1 improves Theorem 4.6 in  and Theorem 3.4 in , where the extended model was extended. We also generalize Theorem 3.10 in  where the structurally stability at and robustness to -equilibria are extended to the structurally stability at and the robustness to -equilibria, respectively.
Here, we propose a category of new generalized abstract fuzzy economies in an -space.
We then study the equilibrium existence theorem for the models. Furthermore, we establish an extended model with an abstract rationality function and a category of generalized abstract fuzzy economies. We demonstrate that is structurally stable at . In other words, is robust to the -equilibria.
3.1. The Equilibria of Generalized Abstract Fuzzy Economic Model in an -Space
Definition 5. (see ). An abstract convex space consists of a topological space , a nonempty set , and a map with nonempty values.
For any , we denote . Let be the -convex hull of , where . If for any , we have , i.e., , then is said to be a -convex subset of related to . If , the space is defined as . Here, if , then is called -convex. That is to say, is -convex related to . When , we let .
Definition 6. (see ). If an abstract convex space has a basis of a uniformity of , then it is said to be an abstract convex uniform space . If is dense in and for any and for any -convex subset , the set is -convex, then an abstract convex uniform space is said to be an -space.
Based on Proposition 5.1 and Theorem 8.5 in Park , we can obtain Lemma 1.
Lemma 1. Let be a -space. Suppose that is nonempty closed -convex valued and compact upper semicontinuous. Then, we can find an satisfying .
Lemma 2. Let be a family of -spaces. For each , suppose that is nonempty closed -convex valued and compact upper semicontinuous. Thus, for each , we can find an satisfying .
Proof. Let and . For each , let be the projection of onto . Then, for any , is defined by . We denote as . It follows from Lemma 2 in  that is a -space. Since each is nonempty closed -convex valued and compact upper semicontinuous, it follows from Lemma 3 of Fan  that is also nonempty closed -convex valued and compact upper semicontinuous. Next, we prove that, for any , is -convex. For every and , we derive that for every . Since every is -convex, we have that for each . Therefore, . Thus, we obtain that is -convex and is nonempty closed -convex valued and compact upper semicontinuous. It follows from Lemma 1, for each , we can find an satisfying , i.e., .
Let be two nonempty convex subsets of a Hausdorff topological vector space. In this article, we use to denote all the fuzzy sets on . is said to be a fuzzy mapping. Thus, we have that for each (denote by in the sequel) is a fuzzy set in . is defined as the degree of membership of point in .
For every , we use to denote the -cut set of . If is a set of agents (finite or infinite), is a nonempty topological space (a choice set), are fuzzy constraint mappings (fuzzy constraint correspondences), and is a fuzzy preference mapping (fuzzy preference correspondence), then is said to be a generalized abstract fuzzy economy. If, for every and , where , then the point is said to be an equilibrium of .
Theorem 2. Suppose that is an abstract fuzzy model of economics, , and . Suppose for every , satisfies the following conditions:(1) is a compact -space(2)for every and are compact and nonempty closed -convex valued(3)for every is closed -convex valued(4) is open in (5)for every Then, for each , we can find an satisfying and .
Proof. Let , where is a projection of onto . By Lemma 2 in , we have that is an -space. For each , is defined asFrom conditions (2) and (3) and Theorem 3.18 in , we have that is nonempty closed valued and compact upper semicontinuous. On the other hand, from conditions (2)–(4), Lemma 3 in , and Lemma 1 in , we have that is nonempty closed -convex valued and compact upper semicontinuous. From Lemma 2, for each , we can find an satisfying . If, for some , we obtain that . However, this is in contradiction with condition (5). Therefore, for each , . By the definition of , we have that for each , and . The proof is complete.
Remark 2. Theorem 2 improves Theorem 2 in  and Theorem 2 in . We also generalize the related results in  from generalized convex space to -space. Theorem 2 can be considered as an extended variant of Theorem 2 in  with respect to -spaces. We note that -spaces include a variety of topological spaces such as the -spaces and the -spaces (see [21, 22] and references therein).
3.2. Structural Stability and Robustness to Generalized Abstract Fuzzy Economies
Let be a generalized abstract fuzzy economic model satisfying all the conditions of Theorem 2. We introduce the definition , where
By the proof of Theorem 2, we have that is compact upper semicontinuous on and there exists a point such that .
Let be a metric space induced by a metric . For each and , we definewhere is the Hausdorff metric induced by on , for each .
Next, we consider the model , where , for all and for any . Here, we denote by and is a rationality function, which is defined asfor any . Here, for each , is the distance on .
For every , the -equilibria set of the generalized abstract fuzzy model of economics at is defined as follows:
We use to denote all equilibria set of the generalized abstract fuzzy model of economics at as
We notice that for all , is compact if is lower semicontinuous. It follows from Theorem 1 that .
Obviously, if and only if for each , , and , and .
Theorem 3. Let be the extended model of ; if is structurally stable at , then is robust to -equilibria.
Proof. First, we prove that is a complete metric space. Let be any Cauchy sequence in . Consequently, for any , we can find a positive integer such thatSince is compact, is complete. As a result, is compact. Obviously, under the Hausdorff distance, the family of compact subsets of is a complete metric space. We notice that is compact upper semicontinuous. Based on Proposition 3.1.11 in , for each , we can always find a compact set satisfying . From (16), we obtain thatSince , we can always find an , such that , for each . Because of the compactness of , we can let . Since is upper semicontinuous on , we can find a positive integer satisfyingTherefore, we getLet , we thus have that . Because of the arbitrariness of , we obtain that and as such . That is to say, . Thus, is a complete metric space.
In what follows, we aim to prove that is lower semicontinuous at .
For every with , in which , and for any with , we obtain thatSince is compact upper semicontinuous, it follows from Proposition 3.1.19 in  that is a lower semicontinuous mapping at . Hence, we can find a positive integer satisfying, for all ,For any , since is compact, then we can find a satisfyingFrom and , there is always a positive integer such that, for all , there exists with for every . Let . Then, for all , we haveNow, we prove that is lower semicontinuous at .
Consider the correspondenceIt follows from Lemma 6 in  that is an upper semicontinuous mapping. Therefore, according to Theorem 2, we know that if is structurally stable at , then is robust to -equilibria. Thus, the proof is complete.
Remark 3. Theorem 3 improves Theorem 2 of . First, the compactness of is dropped. Second, is upper semicontinuous weakened compared with the continuity. Third, is lower semicontinuity weakened than the continuity. Compared with Theorem 2 in , we introduce a more general model in Theorem 3. The structural stability at and robustness to -equilibria are extended to the structural stability at and the robustness to -equilibria, respectively.
In this paper, we discuss the relationship between -stability and -robustness of using a lower semicontinuous rationality function. As an application, we introduce the model consisting of a category of the parameterized generalized abstract fuzzy economic model with related abstract rationality function. We prove that the model is structurally stable at . Therefore, we can conclude the robustness of to -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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (11401484) and the Fundamental Research Funds for the Central Universities of China (JBK2102011). This research was also supported by Shanghai Social Sciences Planning Project (2020BJB013).
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