Generic Stability of the Weakly Pareto-Nash Equilibrium with Strategy Transformational Barriers

Liu, Luping; Jia, Wensheng; Zhou, Li

doi:https://doi.org/10.1155/2022/1689732

Journal of Function Spaces

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Abstract Introduction Preliminaries and Model Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Unique and Non-Unique Fixed Points and their Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 1689732 | https://doi.org/10.1155/2022/1689732

Generic Stability of the Weakly Pareto-Nash Equilibrium with Strategy Transformational Barriers

Luping Liu,¹Wensheng Jia,¹and Li Zhou¹

Academic Editor: Santosh Kumar

Received17 Jun 2022

Accepted23 Sept 2022

Published06 Oct 2022

Abstract

The object of this paper is to establish a new model with strategy transformational barriers for a class of generalized multileader multifollower multiple objective games (GMLMFMOG) and further deduce some new results of the weakly Pareto-Nash equilibrium (WPNE) with strategy transformational barriers for the GMLMFMOG. First, we investigate the existence of the WPNE with strategy transformational barriers for the GMLMFMOG by using the Kakutani-Fan-Glicksberg fixed point theory. Next, we study the generic stability of the GMLMFMOG with strategy transformational barriers in Hausdorff space. Finally, we obtain that the majority of the WPNE with strategy transformational barriers for the GMLMFMOG are essential on the meaning of Baire’s category. In addition, we demonstrate that there is at least an essential component for the GMLMFMOG with strategy transformational barriers.

1. Introduction

Barriers, such as market competition [1], the Lévy risk process [2, 3], the optimal dividend problem [4], and the marketing ethics of medical schemes [5], are common in the field of economics. Transformational barriers, an important aspect of barriers, represent many factors that make the behaviour of shift strategy more difficult or costly for consumers. Furthermore, the payoff function with the strategy transformational barriers may be an abstract partial order rather than a numerical order. Game theory is an important tool for studying the interactions among the decision-making behaviours of players in many fields, such as economics, political science, psychology, and biology. Glicksberg [6] and Mas-Colell [7] provided a maximum element method to analyze the decision-making behaviours of players with the strategy transformational barriers. Therefore, the payoff function with strategy transformational barriers was introduced into game model to further study the decision-making behaviour of players based on the fact that there is a cost for players to change their strategies in practical life.

Fort [8] first presented the essential fixed point in 1950. Wu and Jiang [9] first provided the concept of essential equilibrium for a finite game through using fixed point theory for continuous mapping. Afterwards, Yu and Luo and Yu [10, 11] extended previous work to the general -person noncooperative game, generalized game, or other games by using entirely different approaches. Recently, Scalzo [12, 13] and Carbonell-Nicolau and Carbonell-Nicolau and Wohl [14, 15] provided some extensions about discontinuous payoffs and further studied the essential stability of discontinuous games. Yang and Zhang [16] proved some existence and essential stability results of cooperative equilibrium for population games. We can also refer to [17–20] for more details on the essential stability. Hence, the essential stability has become one of the important topics in nonlinear analysis and game theory.

The weakly Pareto-Nash equilibrium (WPNE) of the multiple objective game was proposed by Shapley and Rigby [21]. Pang and Fukushima [22] studied the existence of a type of multileader multifollower multiobjective game by using quasivariational inequalities. Sherali [23] obtained the existence and uniqueness results of a WPNE regarding the multileader multifollower game. Kulkarni and Shanbhag [24] considered multileader multifollower game with shared-constraint approach to obtain local Nash equilibrium (NE), Nash B-stationary point, and Nash strong-stationary point. Yu and Wang [25] verified some existence theorems for 2-leader multifollower game in locally convex topological space. Yang and Ju [26] obtained some consequences on existence and stability of solution for multileader multifollower game. Jia et al. [27] provided the existence and stability of a WPNE for the generalized multileader multifollower multiple objective game (GMLMFMOG). Inspired by the above research work, this paper establishes a new generalized multiobjective multileader multifollower model with strategy transformational barriers by considering the influence of strategy transformational barriers and analyzes the strategy selection of the players. The leaders consider multiple objectives when selecting their strategies. The followers also consider multiple objectives when selecting their strategies with complete knowledge and make optimal responses to the leaders’ strategies. The goals of all players are to maximize their own incomes. Furthermore, the existence of the WPNE with strategy transformational barriers of a GMLMFMOG is proved, and the generic stability of the GMLMFMOG with strategy transformational barriers is obtained. We prove that the solution set of the GMLMFMOG with the strategy transformational barriers is essential and that there is at least one essential component of the WPNE with the strategy transformational barriers under the meaning of the Baire’s category.

This paper is outlined as follows. We present necessary preliminaries and the GMLMFMOG model with strategy transformational barriers in Section 2. In Section 3, we provide the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG. In Section 4, we investigate some generic stability results of the GMLMFMOG with strategy transformational barriers. In Section 5, we show that the majority of WPNE with strategy transformational barriers of the GMLMFMOG are essential, and then there is at least an essential component. Finally, some brief and concise conclusions are given.

2. Preliminaries and Model

2.1. Preliminaries

In this paragraph, we introduce some substantial definitions, lemmas, and game models.

Definition 1 (see [28]). Suppose that is not empty subset of Hausdorff topological vector space (HTVS) , is not empty convex cone, and a vector-valued correspondence is denoted by . We define is -usc (resp. -lsc) at if, for each open neighbourhood of the 0 element in , there exists an open neighbourhood of such that (resp. ), . Furthermore, we say is -usc (resp. -lsc) on , if is -usc (resp. -lsc) for all . We call is -continuous on , if is -usc and -lsc on . is closed if is closed on .

Definition 2 (see [29]). Let and be two HTVSs, be a closed convex pointed cone, , be not empty convex subset, and be a vector-valued correspondence. If, and , holds, then is - concave, and is - convex. If, , , and , , such that , then is - quasiconcave-like, and is - quasiconvex-like.

Remark 3. For , , if is - quasiconcave-like, then is obviously quasiconcave. However, , , , and , . We know that is - concave but not - quasiconcave-like, and is - quasiconcave-like but not - concave. Thus, - quasiconcave-like and - concave do not include each other.

Definition 4 (maximal element theorem, see [30]). Let be not empty compact convex subset (NECCS) of HTVS and with the following conditions, where denotes all nonempty subsets of : (1), , where denotes the convex hull of (2), is open in Then, there is such that .

2.2. Model

A model of the GMLMFMOG with strategy transformational barrier is denoted by a tuple , where (i) and indicate the index set of leaders and followers, respectively(ii), , , and denote the strategy set of the th leader and the th follower, separately. The leaders’ strategy represents , where , . Meanwhile, the strategy of the followers denotes , where , (iii), , , and . Let be the payoff function of the th leader. Let , be a payoff function of the th follower and be a constraint correspondence of the th follower(iv)Let be the strategy transformational barrier function of the leader . , there exists such thatwhere denotes the strategy transformational barriers of the leader changing from strategy to strategy . In particular, denotes that the th leader has no transformational strategy (v)The followers are a generalized constraint multiobjective with the strategy parametric game (PGCMOG) after fixing the strategy of the leaders. Let be the solution mapping of the WPNE with strategy transformational barriers for the PGCMOG. Particularly, such that there is , and we have , . Furthermore, if there is such that , satisfyingthen is called a WPNE with strategy transformational barriers of the GMLMFMOG, where , denotes the leader ’s cost changing from strategy to strategy

Let , , and be elements in , , and , respectively. By Definition 4, then we have the best response of the th leader with strategy transformation barriers to the other players, i.e., where is independent of .

Fixing , we know that the player’s set-valued mapping provides the order relation “” as follows:

In general, the order relation is not transitive, and we give a sufficient condition for the transitivity of the order relation “” with the following propositions.

Proposition 5. Let be a GMLMFMOG with strategy transformational barriers, if, for any , and Then, the order relation “” has transitivity.

Proof. Setting which are three elements in and such that holds, we obtain by , , and the definition of best response mapping . Then, we attain Since is not dependent on , we can see that is equal to zero element of . Therefore, ; then, the order relation “” has transitivity.

Example 1. Considering the Hotelling model [31], the influence of the strategy transformational barrier function can be added. Assume that consumers are evenly distributed on a street and that businessmen () choose their shop location on the street. Suppose that the street can be abstracted to a line segment with a length of 1, namely, [0, 1]. Meanwhile, and represent the positions of the two businessmen. The strategy set of the businessmen is , and the payoff functions are expressed as

It is a well-known fact that is the unique NE point of the Hotelling game [31], which can better explain the phenomenon of shop centralization. However, it is worth noting that shops may not be concentrated in the centre because of the influence of relocation costs and other factors. In reality, the distribution of shop locations corresponds to a WPNE with strategy transformational barriers, which means a state of equilibrium under weaker conditions.

Suppose that the strategy transformational barrier functions are and , respectively. If and are

Setting businessmen 1 taking , , , and , we have

If , , then but

Furthermore,

Since , “” has no negative subadditivity. Then, “” does not hold; thus, the order relation “” is not satisfied to transitive.

Remark 6. When the strategy transformational barrier function does not have negative subadditivity, the order relationship “” does not have transitivity. Furthermore, the game with a strategy transformational barrier function may not have a numerical payoff function since the strategy transformational barrier function often possesses subadditivity rather than negative subadditivity.

3. Existence

In this paragraph, the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG is demonstrated.

Lemma 7 (Kakutani-Fan-Glicksberg, see [6]). Assume that is a NECCS of locally convex Hausdorff space , is a set-valued mapping, , is a nonempty, convex, compact set, and is usc on . Then, there exists such that .

Lemma 8 (see [17]). Assume that is a nonempty subset of Hausdorff space and is a vector value correspondence, where . In that case, is -continuous if is continuous.

Lemma 9 (see [28]). Suppose that and are two Hausdorff spaces and is compact. If a set-valued correspondence is closed, then is usc.

Lemma 10 (see [29]). Assume that and are two NECCSs of locally convex Hausdorff space and , respectively. is continuous correspondence; is a continuous set-valued correspondence on , , is not empty and compact subset of , as well as . Then, we obtain that is a compact, nonempty set as well as is usc on .

Theorem 11 (Fort theorem, see [8]). Suppose that and are Hausdorff and metric spaces, respectively. Given a set-valued mapping is usc on with nonempty compact value (briefly, usco), then there is a residual subset in such that is lsc on .

Remark 12 (see [29]). If is Baire space, then the residual set in is dense.

Theorem 13. Suppose that and are two NECCSs of locally convex Hausdorff space and , respectively. If satisfies the following conditions. (1), is -continuous(2), is -continuous, , is convex(3), , is - quasiconcave-like(4), is continuous, and , is a nonempty and compact subset of (5), the set-valued correspondence is convex (i.e., , , )Then, the GMLMFMOG with strategy transformational barriers contains at least a point such that , satisfying

Proof. , the set-valued correspondence is defined, , , we have where is independent of .
By Lemma 7, we only need to prove that the set-valued mapping is usc mapping with nonempty convex compact value. (1). Because is compact and is a continuous correspondence with compact value, is compact. is -continuous from Lemma 8; then, , is -continuous and is also -continuous. Thus, from Lemma 7(2) is convex. , i.e., , , , , and , we obtaini.e., we have Since is convex, , , and by Theorem 13 (5), we have .
Since , is - quasiconcave-like, and is convex, we obtain i.e., Thus, , is convex. (3) is a usc mapping. According to Lemma 9, we just need to verify that is closed. Thus, we next demonstrate that the set-valued correspondence is continuousSuppose that is any net on , and , , . Because is a usc mapping with compact value and , , from Theorem 16.17 in [32], we attain . Therefore, , is closed. Since is compact from Lemma 9, is usc on .
Meanwhile, assume that is any net on , , , then , . For any , we set , since is continuous, from Theorem 16.19 in [32] if there is some , , , and hold. Thus, is lsc on .
Hence, we have proved that is continuous with compact values. is compact and is a usc mapping from Lemma 10. On the basis of the above proof, we know that is a usco correspondence.
A set-valued correspondence is defined, and contains .
Because is a NECCS of locally convex Hausdorff space, is a usco mapping and Lemma 7, if there is , then holds. We obtain . Consequently, there is such that , , . This concludes the proof.

Remark 14. In this paper, the WPNE with strategy transformational barriers are more broadly concepts than the WPNE in literature [27] in practical life, which means that the player needs to consider the impact of other some factors, such as the cost of changing strategies. In particular, if the leaders have no transformational strategy barriers, then the WPNE can be considered as the WPNE with strategy transformational barriers.

4. Generic Stability

In this paragraph, we prove the generic stability of the WPNE with the strategy transformational barriers of the GMLMFMOG.

Let and be two NECCSs of Banach space and , respectively, and and satisfy all conditions provided in Theorem 13.

For and , the distance on is defined as follows: where is the Hausdorff distance between and on .

Theorem 15. is a complete metric space.

Proof. It is easy to see that serves as a metric space. Then, we just need to check that is complete.
Setting , , we need to prove . (1)Let be any Cauchy sequence in . , there is a positive whole number such that . On the one hand, , and , when , , thus . We know that is -continuous by means of Theorem 13 (1); then, there is , ; when , we obtain . Similarly, Thus, is -continuous on , and is also -continuous by proving the same method on . Meanwhile, and , there is a positive integer and , we obtain Then, , there is such that , and , we have Since the set-valued correspondence is continuous on , it is easy to know that is continuous on . (2)Since is - quasiconcave-like, is convex, fixing and , if , holds, then , , we haveSince , , and the strategy space is closed, we conclude that This indicates that , is - quasiconcave-likeand is convex (3)Since is convex, , , and , we have When is sufficiently large number, we have Thus, We take because is arbitrary, and we can obtain . Hence, is convex on . In conclusion, , and is a complete metric space.

, we define , where . By Theorem 13, there is such that . Then, is also called an equilibrium mapping.

Next, we denote to verify the generic stability result of the WPNE with the strategy transformational barriers of the GMLMFMOG.

Lemma 16. An equilibrium mapping is a usco correspondence.

Proof. By means of the compactness of and Lemma 9, we need to demonstrate that the is closed. In other words, if , , , , then we only need to prove (1)Since is compact, we assume that , is continuous, , . Let be the distance on ; since , , and , we have . Thus, (2)We verify that , , and we haveBy contradiction, suppose that formula (30) is not true, then there is some such that , . Therefore, there exists some open neighbourhood of the 0 element of satisfying Because , there is a positive integer such that , Furthermore, since , is lsc at with , there is a positive integer and such that , Then, , and we can obtain that It is a contradiction with . Thus, we can obtain ; i.e., is a closed correspondence and is a usco correspondence on by means of Lemma 9.
Next, we define a set-valued map , wherein , . It is obvious that is continuous on .
Finally, we define a set-valued mapping , where , represents the set of WPNE with strategy transformational barriers for the GMLMFMOG. According to Theorem 13, , then .

Lemma 17. A set-valued mapping is a usco correspondence.

Proof. According to Lemma 16, is usc on , and is compact . Since is continuous on , it is obvious to check that is also a usco correspondence on .

Definition 18. (1)An equilibrium point of the game is referred to essential if for every of , there is one of such that , and there exists at least an equilibrium point of with . If all equilibria points of the game are essential, then the game is an essential game(2)A set of the game is referred to essential set if for each open set of is associated with , and there is an satisfying , , and . Given that is one minimal element in total essential sets of which are ordered by inclusion relations, then is a minimal essential set(3), is composed of the union of the pairing of disjoint connected subsets [33], i.e.,wherein signifies one index set. Given a component of is essential, then is one essential set

Theorem 19. , there is a dense in such that is essential.

Proof. is complete by using Theorem 15, and is a usco correspondence by means of Lemma 17. By Theorem 11 and Remark 12, serves as lsc on one dense of such that is essential.

Remark 20. By Theorem 19, we proved that most of have a stable solution set in the dense of on the meaning of Baire’s category.

5. Essential Component

In this paragraph, we derive the essential component results of the WPNE with the strategy transformation barrier solution sets of the GMLMFMOG.

Theorem 21. encompasses at least one minimal essential set , where .

Proof. For , is usco mapping by Lemma 17, and then, is one essential set of itself. Suppose that is the collections of all essential sets of , which is defined by the set inclusion order relation, we obtain . Assume that any total order subset be on , where denote the index set. Let , then serves as compact. If , then . Note that is one open set as well as is compact, then there are such that by using the open covering theorem. It is obvious that from . It means that is in contradiction with . Thus, . Given any open set with , if , there exists with ; then, we can assume that . Because is compact and is totally order set, then when and . Hence, , which contradicts with and . Therefore, there exists such that . Since is an essential set of , , there is such that with , with , . Thus, is essential, and there must be a lower bound of in . According to Zorn’s lemma, there is one minimal element in such that includes at least one minimal essential set .

Theorem 22. , each minimal essential set of is connected.

Proof. Let be a minimum essential set of . By contradiction, we assume that is disconnected. There are two not empty closed sets and with , as well as two disjoint open sets and with such that and .
Since is the minimum essential set, and are not essential set for . Therefore, there exist two open sets, namely, and , with and such that , ; we obtain and , but , . Suppose that and are open sets and that and . Beacuse and are compact, there are two open sets, namely, and , such that and . Since is one essential set of and , there is such that with , and Since and , there exist and such that and with and .
We define a GMLMFMOG with strategy transformational barrier by a linear combination function between and as follows: where and represents the distance function on . Note that and are continuous and nonnegative; furthermore, .
We can check that . Noting that we obtain since . Next, we assume that ; then there exists . By , we attain , , , , and . Then, we obtain , which implies Thus, . This contradicts the fact that . Then, is connected.

Theorem 23. , if there exists (single point set), then is essential.

Theorem 24. , there is at least an essential connected component of .

Proof. According to Theorems 21 and 22, encompasses at least a minimum essential set and is connected. Aiming at a component of as well as , we obtain that is one essential connected component of by Definition 18 (3).

6. Summaries

In this paper, we have investigated a new generalized multileader multifollower multiple objective game (GMLMFMOG) model with strategy transformational barriers and obtained some new stability results of the WPNE with the strategy transformational barriers for the GMLMFMOG. Furthermore, we have proved the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG and studied its generic stability. In fact, we have obtained that most of the WPNE with the strategy transformational barriers of the GMLMFMOG serve as essential on the meaning of Baire’s category. In addition, we have demonstrated that there is at least an essential connected component of the GMLMFMOG with the strategy transformational barriers. These results extend the corresponding results obtained in reference [27] by introducing strategy transformational barrier function into the decision-making behaviour of players.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares 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 (Grant Nos. 12061020 and 71961003), the Science and Technology Foundation of Guizhou Province (Grant Nos. 20201Y284, 20205016, 2021088, and 20215640), and the Foundation of Guizhou University (Grant Nos. 201405 and 201811). The authors acknowledge these supports.

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