Topological degree and number of Nash equilibrium points of bimatrix games

In this paper, using the topological degree, we give a new proof of a well-known result: the number of Nash equilibrium points of a nondegenerate bimatrix game is odd. The calculation of the topological degree allows the localization of the whole set of non-degenerate equilibrium points.     

On differential games with a feedback nash equilibrium

This note provides a lemma on differential games which possess a feedback Nash equilibrium (FNE). In particular, it shows that (i) a class of games with a degenerate FNE can be constructucted from every game which has a nondegenerate FNE and (ii) a class of games with a nondegenerate FNE can be constructed from every game which has a degenerate FNE.The author would like to thank an anonymous referee for invaluable comments and suggestions.     

0-Epi families of mappings,topological degree,and optimization

Using the concept of 0-epi mapping defined by Furi, Martelli, and Vignoli, we define the concept of 0-epi family of quasiconvex mappings. We prove that the theory of 0-epi families is a good substitute of the degree theory in the localization of Nash equilibrium points, because it is more refined and very simple.     

Topological degree and the Sperner lemma

In this paper, we give a new proof of Sperner's lemma, in its superstrong from, using the topological degree. Thus, we point out a relation between several methods for fixed-point theorems using either the topological degree, or the KKM lemma, or the Sperner lemma.The author would like to thank Dr. G. Leitmann for his remarks and suggestions.     

抽象经济均衡问题解的存在性及其算法

本文首先研究一类新的向量均衡问题,利用截口定理与KKM定理两种不同的工具证明此类均衡问题解的存在性,接着,把这类向量均衡问题推广到更为一般的情形,随后讨论了具有上下界的均衡问题,它是由Isac,Sehgal和Singh于1999年提出的一个公开问题,本文在一定条件下获得了一个新的解的存在性定理,并构造了一个迭代算法,讨论了算法的收敛性。     

Hadamard良定性的统一研究

对一些非线性问题的Hadamard良定性给出一个统一的定理，应用这个定理，可以容易地推出KyFan点，Nash平衡点等的Hadamard良定性。此外，最优化问题和鞍点问题的通用良定性也被研究给出了两个定理。     

Equilibria of noncompact abstract economies in general almost convex strategy spaces

We first extend the concept of almost convex condition and establish a fixed point theorem for correspondences with convex values only on an almost convex subset for their ranges. This generalizes both results of Himmelberg [C.J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207] and the result of Jafari and Sehgal [F. Jafari, V.M. Sehgal, An extension to a theorem of Himmelberg, J. Math. Anal. Appl. 327 (2007) 298–301]. Furthermore, applying it, we have existence theorems for equilibria of noncompact abstract economies in general almost convex strategy spaces. Our theorems generalize the corresponding results of Zhou [Jianxin Zhou, On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 193 (1995) 839–858], Tan and Wu [K.-K. Tan, Z. Wu, A note on abstract economies with upper semicontinuous correspondence, Appl. Math. Lett. 11 (5) (1998) 21–22] in several ways. In particular, we answer the question raised by Zhou in the reference cited above in the affirmative with weaker hypotheses.     

Quasi-Equilibrium Problems and Fixed Point Theorems of Separately l.s.c and u.s.c Mappings

In this paper, we apply our new results on quasi-variational inequality problems to generalized quasi-equilibrium problems. Some su?cient conditions on the existence of solutions are shown. In particular, we establish several results on the existence of solutions to fixed points of separately lower and upper semicontinuous (lower and lower, upper and upper) mappings.These generalize some well-known fixed point theorems obtained by previous authors as Browder and Ky Fan, Wu, et al., etc.     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

Admissible points of a convex polyhedron

In this paper, we present several new properties of the admissible points of a convex polyhedron. These properties can be classified into two categories. One category concerns the characterization and generation of these points. The other category concerns the circumstances under which these points are efficient solutions for linear multiple-objective programs with nonnegative criteria matrices. To illustrate some of the potential utility of these properties, we describe their application to two practical problems. These problems are the linear multiple-objective problem with interval coefficients and the problem of optimizing over the efficient set.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.     

A minimization method for the sum of a convex function and a continuously differentiable function

This paper presents a method for finding the minimum for a class of nonconvex and nondifferentiable functions consisting of the sum of a convex function and a continuously differentiable function. The algorithm is a descent method which generates successive search directions by solving successive convex subproblems. The algorithm is shown to converge to a critical point.The authors wish to express their appreciation to the referees for their careful review and helpful comments.     

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

Monotone insertion of lattice-valued functions

Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality. This research was supported by the MEyC and FEDER under grant MTM2006-14925-C02-02/ and by UPV05/101     

Fast reconstruction of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD design environment

This paper compares the performances of two different optimisation techniques for solving inverse problems; the first one deals with the Hierarchical Asynchronous Parallel Evolutionary Algorithms software (HAPEA) and the second is implemented with a game strategy named Nash-EA. The HAPEA software is based on a hierarchical topology and asynchronous parallel computation. The Nash-EA methodology is introduced as a distributed virtual game and consists of splitting the wing design variables-aerofoil sections-supervised by players optimising their own strategy. The HAPEA and Nash-EA software methodologies are applied to a single objective aerodynamic ONERA M6 wing reconstruction. Numerical results from the two approaches are compared in terms of the quality of model and computational expense and demonstrate the superiority of the distributed Nash-EA methodology in a parallel environment for a similar design quality.     

不确定条件下n人非合作博弈均衡点集的通有稳定性

基于经典博弈模型的Nash均衡点集的通有稳定性和具有不确定参数的n人非合作博弈均衡点的概念,探讨了具有不确定参数博弈的均衡点集的通有稳定性.参照Nash均衡点集稳定性的统一模式,构造了不确定博弈的问题空间和解空间,并证明了问题空间是一个完备度量空间,解映射是上半连续的,且解集是紧集（即usco（upper semicontinuous and compact-valued）映射）,得到不确定参数博弈模型的解集通有稳定性的相关结论.     

关于一般化凸乘积空间上的极大元和平衡点

利用一般化凸乘积空间上的Fan-Browder型不动点定理给出了新的极大元存在定理,然后定义了两个概念：“类Uθ”和“类V”,并讨论了在抽象经济中平衡点的存在性问题．文中所得结论改进和推广了文献中的相应结果．     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

Remarks on quasi-variational inequalities and fixed points in locally convex topological vector spaces

In this note, we first prove existence theorems for noncompact generalized quasivariational inequalities. As applications, two fixed point theorems for upper or lower semicontinuous multivalued mappings without compact domains are given in locally Hausdorff topological vector spaces. These results generalize or improve corresponding results in the literature.     

拓扑度计算与应用

本文运用锥理论计算一类全连续场的拓扑度,极大地减弱了非线性算子下方 有界的条件, 因而本质上改进和推广了现有结果. 最后,把抽象结果应用于研究超线 性Hammerstein积分方程非平凡解的存在性.