## Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications

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# Bounded Rationality of Generalized Abstract Fuzzy Economies

**Academic Editor:**Nan-Jing Huang

#### Abstract

By using a nonlinear scalarization technique, the bounded rationality model for generalized abstract fuzzy economies in finite continuous spaces is established. Furthermore, by using the model , some new theorems for structural stability and robustness to ()-equilibria of generalized abstract fuzzy economies are proved.

#### 1. Introduction

In 1952, Arrow and Debreu [1] first proved the existence theorem of the Walrasian equilibrium. Borglin and Keiding [2] generalized Arrow and Debreu's result to generalized abstract economy without order preferences. Then, the result was extended to many directions, see, for example, [3–9] and the references therein.

As is known to all, Zadeh [10] established the fuzzy set theory to treat the scenarios which cannot be characterized precisely as an attempt to develop the mathematical framework. Since then, the theory of fuzzy sets has been extended to many areas. Butnariu [11] first established the characterization of the fuzzy game. Billot [4] obtained some equilibrium existence theorems of fuzzy games. Huang [12, 13] studied a new class of generalized abstract fuzzy economies with uncountable number of players and proved some new existence theorems of maximal element for the abstract fuzzy economies and qualitative fuzzy games, respectively. For more about the fuzzy games and fuzzy abstract economies, we refer to [14–18].

On the other hand, we know that both the Nash equilibrium and the Debreu equilibrium are established on the assumption of perfect rationality of the players in the game. This assumption in the economic models is too strict. Ideally, we would like to build a model of bounded rationality that could be meaningful not only in theory but also in application. For this purpose, Anderlini and Canning [19] established the abstract framework, a model , that consists of a parameterized class of general games and an associated abstract rationality function. They discussed the relationship of the robustness to bounded rationality and structural stability for the model . They also applied the result to four economic examples. Under the weak assumptions of Anderlini and Canning [19], C. Yu and J. Yu [20, 21] proved the following result: the model is structurally stable and robust to -equilibria for almost all parameter values of generalized games and multiobjective games. Recently, Wang et al. [22] improved the economic model proposed by Anderlini and Canning, and under the weaker assumptions, they also showed that the model is structurally stable if and only if it is robust to -equilibria in generalized convex spaces. Miyazaki and Azuma [23] generalize the definitions of structurally stable and robustness to -equilibria that are introduced in Anderlini and Canning [19] and C. Yu and J. Yu [20, 21]. They also studied the essentiality of equilibria in a bounded rational environment.

Motivated and inspired by the work mentioned above, in this paper, a new class of generalized abstract fuzzy economies with an uncountable number of agents with fuzzy constraint correspondences and fuzzy preference correspondence in finite continuous spaces without any convexity structure is established. Then, a new existence theorem for generalized abstract fuzzy economies with an uncountable number of agents with fuzzy constraint correspondences and fuzzy preference correspondence in locally finite continuous uniform spaces is established in Section 3. Finally, applying the result in Section 3, we obtain that the model , which consists of a parameterized class of generalized abstract fuzzy economies and the associated abstract rationality function, is structurally stable at if and only if is robust to -equilibria in Section 4. The results presented in this paper generalize some known results in [12, 13, 19–22].

#### 2. Preliminaries

Let and be the family of all subsets and the family of all nonempty finite subsets of a set , respectively. Let be the cardinality of , for arbitrary . We denote by the standard -dimensional simplex with vertices . For any nonempty set of , we denote by the convex hull of the vertices . Ding and Wang [7] first introduced the following definition of finitely continuous topological space (or FC-space) without any convexity structure.

*Definition 1. *A finitely continuous space (for short, FC-space) consists of a topological space and a mapping such that for each , where some elements in may be the same, there exists a continuous mapping . For each and for any , if , then a subset of is said to be an FC-subspace of .

It is easy to see that each FC-subspace of is also a FC-space by the definition of FC-subspaces of a FC-space, and if is a family of FC-subspaces of an FC-space and , then is also an FC-subspace of , where is any index set.

*Definition 2 (see [24]). *If is a uniform space and is an FC-space such that has a basis consisting of entourages satisfying that for each , the set is an FC-subspace of whenever is an FC-subspace of , and then is said to be a locally FC-uniform space.

We denote by and both metric spaces as follows.(i)If is contained in some compact subset of , then a set-valued mapping is said to be compact.(ii)If for each and for each open set in , the set is open in , then is said to be upper semicontinuous on .(iii)If for each and for each open set in , the set is open in , then is said to be lower semicontinuous on .(iv)If is both upper semicontinuous and lower semicontinuous on , then is said to be continuous on .Furthermore, if is a compact metric space, then (i) is upper semicontinuous at if and only if for any and ; then, ;(ii) is lower semicontinuous at if and only if for any and , there exists ;(iii) is continuous at if and only if for any , where is the Hausdorff distance defined on .

#### 3. Equilibria of Generalized Abstract Fuzzy Economies

In this section, a new existence theorem for generalized abstract fuzzy economies with an uncountable number of agents with fuzzy constraint correspondences and fuzzy preference correspondence in locally FC-uniform spaces is established.

Throughout this paper, let be the collection of all fuzzy sets on , where and are two Hausdorff topological vector spaces and are two nonempty convex subsets. We call a mapping from to a fuzzy mapping, and then for each (denote by ) is a fuzzy set in and is the degree of membership of point in .

If for each , a fuzzy set on is a fuzzy convex set, then a fuzzy mapping is called convex; that is, for any .

For any , we denote by the -cut set of .

A generalized abstract fuzzy economy consists of a nonempty topological space (a choice set) , fuzzy constraint mappings (fuzzy constraint correspondences) , a fuzzy preference mapping (fuzzy preference correspondence) , and a finite or an infinite set of agents . A point is said to be an equilibrium for if each and , where .

Theorem 3. *Let be an abstract fuzzy economy, , and such that for each , the following conditions are satisfied:*(1)* is a compact locally -uniform space;*(2)*for each and are both upper semicontinuous compact mappings with nonempty closed values;*(3)*for each and are nonempty -subspaces of ;*(4)*for each is a closed mapping such that for each is an -subspace of ;*(5)*the set is open in ;*(6)*for each .** Then, there exists a point such that for each , and
*

*Proof. *It is easy to see that becomes a locally FC-uniform space by Lemma 2.1 in [24]. For each , define the following correspondence by
Next, we will show that the family is an upper semicontinuous compact mapping with nonempty compact values. First, for each is a nonempty FC-subspace of ; is an upper semicontinuous compact mapping on with nonempty closed values because of the conditions (3) and (4). On the other hand, by the conditions (2)–(5), Lemma 3 of Fan [25], and Lemma 3.1 of [26], is an upper semicontinuous compact mapping with nonempty closed values. Now, we define a set-valued mapping by
For each is an upper semicontinuous compact mapping with nonempty compact values such that for each is an FC-subspace of . By Lemma 3 of Fan [25] and Lemma 2.1 of [24], is also upper semicontinuous compact mapping with nonempty closed values such that for each is an FC-subspace of . From Lemma 2.2 of [24], we know that there exists a point such that ; that is, for each . If for some , then we have which contradicts the condition (6). Therefore, we must have that for all . It follows from the definition of that for each and . This completes the proof of Theorem 3.

*Remark 4. *Theorem 3 extends and improves Theorem 3.1 of Huang [8], Theorem 3.1 of Huang [12], and Lemma 3.1 of Wang et al. [22] in the following aspects: (1) the generalized abstract fuzzy economy model we considered in Theorem 3 is more general; (2) we generalize the related results from the convex subset of a topological vector space and the generalized convex space to FC-space without any convexity structure. Theorem 3 is also an improved variant of Theorem 3.1 of Ding [24] in locally FC-uniform spaces. Finally, we emphasize that, FC-spaces include many topological spaces as special cases, for example, the convex subset of a topological vector space and the generalized convex space that were just mentioned and the -space.

#### 4. Structural Stability and Robustness to -Equilibria

In this section, firstly, applying the result in Section 3, we study the model , which consists of a parameterized class of generalized abstract fuzzy economies and the associated abstract rationality function. Then, we obtain that the model is structurally stable at if and only if is robust to -equilibria.

Let be a generalized abstract fuzzy economy and satisfy all the conditions of Theorem 3. Then, from Theorem 3, we know that is an upper semicontinuous compact mapping on . Now we define that . By the conclusion from Theorem 3, we have that there exists a point such that , for each , and is an equilibrium point of the generalized abstract fuzzy economy . We denote by the set of all equilibrium points of generalized abstract fuzzy economy for each . By Theorem 3, we know that .

Let be a metric space. For any and , we define that where is the Hausdorff metric induced on induced by a metric for each .

Lemma 5. * is a complete metric space.*

*Proof. *We denote by any Cauchy sequence in . Then, for any , there exists a positive integer such that ; we have
It is easy to see that is complete and is compact by the compactness of . On the other hand, under the Hausdorff distance, the family of compact subsets of is a complete metric space. Since is a upper semicontinuous compact mapping on , by Proposition in [27], we have that there exists a compact set , such that , for all . It follows from (7) that ,
Because of , there exists , such that , for each . Without loss of generality, applying the conclusion mentioned above that is compact, let . It is easy to check that since is upper semicontinuous on , there exists a positive integer such that, ,
Then, we have ,
Let ; we have that . Since can be arbitrary small, we get and . Then, we have . Therefore, is a complete metric space. This completes the proof.

Now, we consider the model , where(i) is a complete metric space;(ii) is a metric space;(iii)for any and for all , define . For all , the continuous correspondence denoted by and for all , is a nonempty compact;(iv)for any , define by where is the distance on , for each .

is a rationality function, and means the full rationality.

For any and any , we define the set of -equilibria of generalized abstract fuzzy economy at by As a special case, we define the set of all equilibrium points of generalized abstract fuzzy economy at by It is easy to see that for all , if is lower semicontinuous, then is compact.

Note that and if and only if and for every .

Lemma 6. * is lower semicontinuous at .*

* Proof. *First, we need to prove that with , where and with ; the following inequality holds:
It is easy to see that is compact upper semicontinuous, and from Proposition in [27], we obtain that is lower semicontinuous at . Thus, we get , where is a positive integer, such that, ,
Since for any , is compact, we get that such that
On the other hand, because of and , we know that there exists a positive integer such that, , with for each . Letting , , we obtain that
Now, we obtain the conclusion that is lower semicontinuous at . This completes the proof.

The following concepts of robust to -equilibria and structurally stable at were introduced by Miyazaki and Azuma [23].

*Definition 7. *If , there exists such that, with and with and ; the model is said to be robust to -equilibria, where is the Hausdorff distance defined on .

*Definition 8. *If the bounded rational equilibrium correspondence is continuous at , the model is said to be structurally stable at .

Theorem 9. *Under the assumptions of model , then, is structurally stable at which implies that is robust to -equilibria.*

* Proof. *Suppose that is not robust to -equilibria. Then, there are , sequence with for all and , and such that
Because of , we can choose such that
Let , because is compact. Since is continuous at , we get that . By the Lemma 3.2 in [20], we obtain that
On the other hand, since
we obtain that is upper semicontinuous from Lemma 6 in [28]. Since , then . Since is lower semicontinuous and , then . Thus we obtain which contradicts that . Thus, we have that is robust to -equilibria. This completes the proof.

*Remark 10. *Theorem 9 improves and generalizes Theorem 3.1 of [19] in the following aspects: (1) is a complete metric space, is a metric space, and we dropped the compactness of ; (2) the continuity of is weakened to the upper semicontinuity; (3) the continuity of is weakened to the lower semicontinuity. Compared with Theorem 3.1 of C. Yu and J. Yu [20, 21] and Theorem 3.1 of Wang et al. [22], the model we considered in Theorem 9 is more general and we generalize the related conclusions from structural stability at to structural stability at and from robustness to -equilibria to robustness to -equilibria.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (11126346, 11201379) and the Fundamental Research Funds for the Central Universities of China (JBK130401).

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#### Copyright

Copyright © 2014 Lei Wang and Yu Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.