On essential components of the set of Nash equilibrium points

Yu, Jian; Xiang, Shuwen

doi:10.1016/s0362-546x(98)00193-x

Cited by 48 publications

(18 citation statements)

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confidence: 60%

“…In this paper, by Fort theorem [3] we establish the generic stability result for SGVEP. Further, we also establish the existence result of essential components of the solution set for SGVEP, which generalizes those results of essential components in [4][5][6][7].Let I be an index set. For each i ∈ I, let X i and Y i be two Hausdorff topological vector spaces, K i a nonempty, convex and compact subset of X i , and C i a closed, convex and pointed cone of Y i with intC i = ∅, where intC i denotes the interior of C i .…”

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confidence: 52%

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Generic stability and existence of essential components of the solution set for the system of generalized vector equilibrium problems

Lin

2007

Appl. Math.- J. Chin. Univ.

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By using Fort theorem the generic stability result for the system of generalized vector equilibrium problems is established. Further, by proving the existence and connectivity of minimal essential set the existence result of essential components in the solution set is derived. §1 IntroductionThe system of generalized vector equilibrium problems (in short,SGVEP) includes as a special case the multiobjective Nash equilibrium problems [1,2] . In this paper, by Fort theorem [3] we establish the generic stability result for SGVEP. Further, we also establish the existence result of essential components of the solution set for SGVEP, which generalizes those results of essential components in [4][5][6][7].Let I be an index set. For each i ∈ I, let X i and Y i be two Hausdorff topological vector spaces, K i a nonempty, convex and compact subset of X i , and C i a closed, convex and pointed cone of Y i with intC i = ∅, where intC i denotes the interior of C i . Denote K = i∈I K i . For each i ∈ I, let F i : K × K i → 2 Yi be a set-valued map, where 2 Yi denotes the family of all nonempty subsets of Y i . The system of generalized vector equilibrium problems is: findingx is said to be a solution of SGVEP. A SGVEP is usually denoted by {K i , F i } i∈I (in short,{F } ).Throughout this paper, unless otherwise specified, for each i ∈ I let X i and Y i be two Banach spaces, K i a nonempty, convex and compact subset of X i and P i a nonempty, convex and compact subset of Y i , and C i a closed, convex and pointed cone of Y i with intC i = ∅. Moreover, we assume that index set I = {1, 2, · · · , n} is a finite set.

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confidence: 60%

supporting

confidence: 52%

Generic stability and existence of essential components of the solution set for the system of generalized vector equilibrium problems

Lin

2007

Appl. Math.- J. Chin. Univ.

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“…The second one aims to translate Nash equilibrium points into solutions of Ky Fan minimax inequality. Yu and Xiang [26] proposed an essential set for Nash equilibria and proved the existence of essential component in the second direction by considering perturbations of payoff functions of players. There is a lot of researches about using the essential component to discuss the stability of Nash equilibria (see, [1,2,8,17,18,24,25,27]).…”

Section: Introductionmentioning

confidence: 99%

Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria

Xiang¹,

Xia²,

Yang³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, we mainly focus on the stability of Nash equilibria to any perturbation of strategy sets. A larger perturbation, strong δ-perturbation, will be proposed for set-valued mapping. The class of perturbed games considered in the definition of strong δ-perturbation is richer than those considered in many other definitions of stability of Nash equilibria. The strong δ-perturbation of the best reply correspondence will be used to define an appropriate stable set for Nash equilibria, called SBR-stable set. As an SBR-stable set is stable to any strong δ-perturbation and, various perturbations of strategy sets are not beyond the range of strong δ-perturbation, it has the stability that various stable sets possess, such as fully stable set, stable set, quasistable set, and essential set. An SBR-stable set is stable to any perturbation of strategy sets, so it will provide convenience for study in strategic stability, which is even used to study any noncooperative game.

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“…Obviously, our results are established for vector-valued payoff function with weaker continuity and stronger convexity, so the results cannot include many present results in the literatures. What is more, Theorem 5.5 and Theorem 5.7 on the essential components of the equilibrium sets are derived in the uniform topology of best-reply correspondences, which are different from the existence results in [12,13,26,29,30] in the uniform topology of payoff functions and feasible strategy correspondences. The following two examples will show that the two kinds of topology are not equivalent.…”

Section: Discussionmentioning

confidence: 73%

“…(see [4,21,22,25,27]). Motivated by [11][12][13][14], we shall give some existence of essential components of the set of weakly Pareto-Nash equilibrium for such games, which have been studied in [26,29,30]. Some examples are given to investigate our results.…”

Section: Introductionmentioning

confidence: 99%

Berge's maximum theorem to vector-valued functions with some applications

Qiu¹,

Dingtao²,

Yu³

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce pseudocontinuity for Berge's maximum theorem for vector-valued functions which is weaker than semicontinuity. We prove the Berge's maximum theorem for vector-valued functions with pseudocontinuity and obtain the set-valued mapping of the solutions is upper semicontinuous with nonempty and compact values. As applications, we derive some existence results for weakly Pareto-Nash equilibrium for multiobjective games and generalized multiobjective games both with pseudocontinuous vector-valued payoffs. Moreover, we obtain the existence of essential components of the set of weakly Pareto-Nash equilibrium for these discontinuous games in the uniform topological space of best-reply correspondences. Some examples are given to investigate our results.

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On essential components of the set of Nash equilibrium points

Cited by 48 publications

References 0 publications

Generic stability and existence of essential components of the solution set for the system of generalized vector equilibrium problems

Generic stability and existence of essential components of the solution set for the system of generalized vector equilibrium problems

Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria

Berge's maximum theorem to vector-valued functions with some applications

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