Ky Fan Minimax Inequality Research Articles

Nash’s Existence Theorem for Non-Compact Strategy Sets

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.

Necessary and sufficient conditions for the existence of fuzzy doubly strong equilibria in generalized fuzzy games

The concept of doubly strong equilibrium is a refinement of the concept of strong Nash equilibrium which represents the most desirable situations for generalized games, where all players get the best among the available outcomes. In this paper, we use necessary and sufficient conditions for the existence of solutions of the Ky Fan minimax inequality to establish necessary and sufficient conditions for the existence of fuzzy doubly strong equilibria in generalized fuzzy games. As consequences, we obtain sufficient conditions for the existence of (fuzzy) strong Nash equilibria in generalized (fuzzy) games, including generalized games with countable infinitely many players, which generalize existing existence theorems for strong Nash equilibria in games.

Variational inequalities, Ky Fan minimax inequality, and strong Nash equilibria in generalized games

Variational inequalities, Ky Fan minimax inequality, and strong Nash equilibria in generalized games

Necessary and sufficient conditions for solutions of the Ky Fan minimax inequality and the non-emptiness of fuzzy cores in economies

In this paper, we provide a necessary and sufficient condition for the existence of solutions of the Ky Fan minimax inequality. We then apply this characterization to establish necessary and sufficient conditions for the non-emptiness of fuzzy cores and fuzzy α-cores in finite coalition production economies with infinite-dimensional commodity spaces. As consequences, we obtain existence theorems for cores and α-cores which generalize corresponding results in the literature, in particular, we derive the non-emptiness of fuzzy cores in finite coalition production economies with upper semicontinuous (possibly discontinuous) utility functions and compact feasible commodity allocation sets.

Fuzzy Strong Nash Equilibria in Generalized Fuzzy Games with Application in Urban Public-Sports Services

In this paper, we apply the existence of solutions of the Ky Fan minimax inequality to establish the existence of fuzzy strong Nash equilibria in generalized fuzzy games and strong Nash equilibria in fuzzy coalition generalized games. We then show some applications of our existence results about the strong Nash equilibrium in urban public-sports services.

Some Vector Ky Fan Minimax Inequalities with Nonconvex Domain

In this paper, by virtue of separation theorems of convex sets and scalarization functions, some minimax inequalities are first considered. As applications, some existence theorems of vector equilibrium problems with different order structures were also obtained.

A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique allows to avoid any monotonicity assumption. We adapt a weak coercivity condition introduced in Castellani and Giuli (J Glob Optim 75:163–176, 2019) for a generalized game which extends an older one proposed by Konnov and Dyabilkin (J Glob Optim 49:575–577, 2011) for equilibrium problems. Our main result encompasses and generalizes several existence results for equilibrium, quasiequilibrium and fixed-point problems.

Non-emptiness of the alpha-core: sufficient and necessary conditions

We give sufficient and necessary conditions for the non-emptiness of the alpha-core in the setting of strategic games with non-ordered and discontinuous preferences. In order to prove our results, we can avoid the use of Scarf’s Theorem for NTU-games, by suitably appealing to the Ky Fan minimax inequality. Examples clarify our conditions and allow the comparison of our results with the previous ones.

Existence of strong equilibrium in discontinuous games

In this paper, we give necessary and sufficient conditions for the existence of strong equilibria in games with non-ordered preferences. Our result does not require continuous preferences and, in the case of normal form games, (quasi-)concavity of payoff functions is not needed. The result has been obtained by using the Ky Fan minimax inequality. Examples illustrate our conditions in some economic situations and compare the result with the previous literature.

Ky Fan minimax inequalities for set-valued mappings on dense sets

We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.

Equilibrium existence in games: Slight single deviation property and Ky Fan minimax inequality

In this paper, we exhibit a relation between the Nash equilibrium existence problem in ordinal games and the Ky Fan minimax inequality. In this way, we obtain new sufficient conditions for the existence of equilibria. The main tools used in the paper are a necessary condition introduced in Nassah and Tian (2016) for normal form games and a generalization of the single deviation property. Examples compare our result with the previous ones.

Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity

In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities. A closely related notion is that of Ky–Fan hemicontinuity, a continuity property which arises if the famous Ky–Fan minimax inequality is applied to the variational inequality framework. It is clear from the corresponding definitions that Ky–Fan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J Optim Theory Appl 165(2):344–358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not Ky–Fan hemicontinuous.

A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite-dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite-dimensional spaces. Furthermore, this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well-known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.

Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities

The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example.

Essential equilibria of discontinuous games

We introduce spaces of discontinuous games in which games having essential Nash equilibria are the generic case. In order to prove the existence of essential Nash equilibria in such spaces, we provide new results on the Ky Fan minimax inequality. In the setting of potential games, we show that games with essential Nash equilibria are the generic case when their potentials satisfy a condition called weak upper pseudocontinuity that is weaker than upper semicontinuity.

Ky Fan minimax inequalities for set-valued mappings

In this article, by virtue of the Kakutani-Fan-Glicksberg fixed point theorem, two types of Ky Fan minimax inequalities for set-valued mappings are obtained. Some examples are given to illustrate our results.Mathematics Subject Classification (2010): 49J35; 49K35; 90C47.

Variational relation problem without the KKM property with applications

In this paper, we will study variational relation problems without the KKM property, whose proof techniques differ between Luc (2008) [1] and Pu and Yang (2012) [9]. As applications, some existence theorems of solution for the generalized KKM theorem, variational inclusion problem, generalized (vector) Ky Fan Minimax Inequality, generalized Ky Fan section theorem, n-person noncooperative generalized game and n-person noncooperative multi-objective generalized game are obtained.

Existence theorems of generalized quasi-variational-like inequalities for η-h-pseudo-monotone type I operators on non-compact sets

In this article, we prove the existence results of solutions for a new class of generalized quasi-variational-like inequalities (GQVLI) for η-h-pseudo-monotone type I operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for η-h-pseudo-monotone type I operators, we use Chowdhury and Tan's generalized version of Ky Fan's minimax inequality as the main tool.

The existence of solutions for general variational inequality and applications in FC spaces

In this article, we prove the existence of solutions for the general variational inequality ϕ(x, y) ≥ f(x) - f(y) and Minty type theorem by using the generalized KKM theorem in topological spaces without linear structure. Some properties of solutions set for the general variational inequality are studied by Minty type theorem. As applications, equivalence between Browder fixed point theorem and Ky Fan's minimax inequality are studied in topological spaces.