On general minimax theorems. Let Abe a linear mapping between Euclidean spaces Eand F.

Minimax Methods in Critical Point Theory with Applications to Differential Equations 本文介绍了minimax theorem的含义和应用，通过数学证明和实例分析，帮助读者深入理解这一重要的理论工具。 FURTHER APPLICATIONS OF TWO MINIMAX THEOREMS. The theorem improves and generalizes the known results shown by Cheng and Lin (Acta Math Hung 73 (1–2):65–69, 1996 ), Lin and Cheng (Acta Math Hung 100 (3):177–186, 2003 ), and Frenk and Minimax Theorems - PMC. 1016/B978-0-12-165550-1. Presents basic minimax theorems starting from a quantitative deformation lemma; and demonstrates their applications to partial differential equations, particularly in problems dealing with a lack of compactness. 19,482-487. In this case, Theorem 1 reduces to the Fan's theorem on sets with convex sections involving two sets. Examples are given to Ky Fan's minimax inequality is an important tool in nonlinear functional analysis and its applications, e. In recent years the minimax theorem of John von Neumann has found numerous new extensions, due to Irle [1985], Kindler [1990], Simons [1990][1991] and others, with the aim to remove from the assumptions the last remnants of linear and convex structures, and to install assumptions of more comprehensive kinds instead. , 153 (200 6), 3308-3312. M VâN V6Ë' ìâ M There have been several generalizations of this In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. For round t= 1:::T: The algorithm plays with the distribution pt =!t In this chapter, we give an overview of various applications of a recent minimax theorem. Jurgen Kindler [1990J On a Minimax Theorem of Terkelsen's. Feb 1, 1997 · Radial and nonradial solutions for nonautonomous Kirchhoff problems. Math. game theory and economic theory. M VβN V6Λ' μβ M There have been several generalizations of this A new general min-max theorem is obtained, extending a result of Ky Fan. Paperback (Softcover reprint of the original 1st ed. Rabinowitz}, journal={Nonlinear Analysis-theory Methods \& Applications}, year={1978}, pages={161-177}, url Theorem: Von Neumann Minimax Theorem. We review and extend the main topological minimax theorems based on connectedness that have been developed over the years since the pioneering paper of Wu (1959). Using Fan's minimax inequality, Ha [6] obtained a non-compact version of Sion's minimax theorem in This chapter deals with the Minimax Theorem and its proof, which is based on elementary results from convex analysis. Many boundary value problems are equivalent to Au=O (1) where A : X --+ Y is a mapping between two Banach spaces. In general, a minimax problem can be formulated as min max f (x, y) (1) ",EX !lEY where f (x, y) is a function defined on the product of X and Y spaces. [24] have shown the minimax theorem of two vector-valued In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig ( 1931 ), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. The strong duality theorem states these are equal if they are bounded. 8 (1), 171-176, (1958) Include: Citation Only. Here is a particular case of one of the results that we obtain: Let (T,F ,μ) be a non-atomic measure space, with μ(T ) < +∞, (E,∥ ·∥) a real Banach space, I ⊆ E an unbounded set whose closure does not contain 0. The purpose of this article is to give the reader the flavor of the different kind of minimax theorems, and of the DOI: 10. Among them, there are some multiplicity theorems for nonlinear equations as well as a general well-posedness … Expand Apr 14, 2010 · Applying generalized KKM-type theorems established in our previous paper (Khanh et al. I And a close connection to the polynomial weights algorithm (and related algorithms) I Playing the polynomial weights algorithm in a zero sum game leads to equilibrium (a plausible dynamic!) I In fact, we’ll use it to prove the minimax theorem. RICCERI, Minimax theorems for limits of p arametrize d functions having at most one loc al minimum lying in a certain set , T opology Appl. 1 (weak duality). 99. The first minimax theorem was proved in a famous paper by von Neumann (cf. In 2013, Zhang et al. Feb 4, 2014 · Abstract. The theorem improves and generalizes the known results shown by Cheng and Lin (Acta Math Hung 73(1–2):65–69, 1996 ), Lin and Cheng (Acta Math Hung 100(3):177–186, 2003 ), and Frenk and Kassay (Math 在博弈论的数学领域，极大极小定理是提供条件的定理，以保证极大极小不等式也是等式。这个意义上的第一个定理是1928 年的冯诺依曼极小极大定理，它被认为是博弈论的起点。从那时起，文献中出现了冯诺依曼原始定理的几个概括和替代版本。[1] [2] Bibliographic information. 摘要： The main result in this paper is the following generalization of von Neumann's minimax theorem:let M, N be compact, convex spaces and f a function on MXN that is upper semi-continuous and quasi-concave in M and lower semi-continuous and quasi-convex in N. Minimax Theorems. : A nontopological two-function minimax theorem with monotone transformations of the functional values. max min f(μ, v) = min max f(μ, v) . Introduction to Games The notion of a game in this context is similar to certain familiar games like chess or bridge. Using the level set method of Joó (Acta Math Hung 54(1–2):163–172, 1989), a general two-function topological minimax theorem are proved. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs . The minimax is called the value of the game. , 29, 229–247 (1985) Article MATH MathSciNet Google Scholar Forgó, F. ON GENERAL MINIMAX THEOREMS MAURICE SION 1. Format: Sep 11, 2006 · Wu, W. Math. g. Borwein. Fan, K. 8, Iss: 1, pp 171-176. Journal List. Ansari H. Let f f be a real-valued function on X × Y X × Y such that 1. Includes some previously unpublished Using the level set method of Joó (Acta Math Hung 54(1–2):163–172, 1989 ), a general two-function topological minimax theorem are proved. e. , Minimax Theorems, Proceedings of the National Academy of Sciences of the USA, Vol. Theorem 16. Note that this is also called the Extreme Value Theorem or EVT for short, though to stay consistent with the Lebl’s book I will be calling it the Min-Max theorem. The theorem states that for every matrix A, the average security levels of both players coincide. Learn more. If f is continuous, then f is bounded by the previous theorem. Fan-Browder fixed point theorem for multi-valued mappings. D. Lecture 7: von Neumann minimax theorem, Yao’s minimax Principle, Ellipsoid Algorithm Notes taken by Xuming He March 4, 2005 Summary: In this lecture, we prove the von Neumann’s minimax theo-rem. If A E IR and W C X we define LE(W, A):= n Valerii Krygin. LEMMA 1. 4. max min p Mq ⊤ ⊤ = min max p Mq. e. Oct 19, 2018 · A Minimax Theorem with Applications to Machine Learning, Signal Processing, and Finance S. [36]) in 1928 for A and B unit This paper is concerned with minimax theorems for two-person zero-sum games ( X, Y, f) with payoff f and as main result the minimax equality inf sup f ( x, y )=sup inf f ( x, y) is obtained under a new condition on f. The answer will be that the dichotomy cannot be Theorem (Von Neumann-Fan minimax theorem) Let X and Y be Banach spaces. Then. I. When the problem is variational, there exists a differentiable functional rand inf. It is worth mentioning that Schervish et al. The name &quot;minimax&quot; comes from minimizing the loss involved when the opponent selects the strategy In this paper, we deal with new applications of two minimax theorems of B. The main point of the Minimax Theorem is that inequality (1) is ac-tually an equality — which we now show by establishing the reverse inequality. Even though we are often consciously aware of our current emotional state, such as anger or happiness, the mechanisms giving Emotions are often felt in the body, and somatosensory feedback has been Apr 15, 2008 · Minimax theorems and cone saddle points of uniformly same-order vector-valued functions. Verdú and H. TL;DR: In this paper, the authors unify the two streams of thought by proving a minimax theorem for a function that is quasi-concave-convex and appropriately semi-continuous in each variable. 3, is very different from any argument used previously in obtaining minimax theorems. In a mixed policy, the min and max always commute. Ricceri ( [5], [9]). One direction is given historically in game theory. Expand 72 Bodily maps of emotions. Introduction to Games 1 2. , A Minimax Inequality and Semantic Scholar extracted view of "On the general minimax theorem" by H. 54,39-49. In this paper, we deal with new applications of two minimax theorems of B. Then max_(X)min_(Y)X^(T)AY=min_(Y)max_(X)X^(T)AY=v, where v is called the value of the game and X and Y are called the solutions. This paper is concerned with minimax theorems in vectorvalued optimization. Let g : X Y ! R be convex with respect to x 2 C and concave and upper-semicontinuous with respect to y 2 D, and weakly continuous in y when restricted to D. Giandinoto. Publisher. Yao [11] investigated some Ky Fan minimax inequalities and existence Minimax theorem in a general topological space is obtained. These applications deal with: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. TLDR. 1. , 33, 145–158 (1975) MATH MathSciNet Google Scholar König, H. Trost [1989J Minimax Theorems for Interval Spaces. extend to valued f ? > <1 Theorem. The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. Michel Willem. A class of vector-valued functions which includes separated functionsf (x, y)=u (x)+v (y) as its proper subset is introduced. A minimax theorem for payoffsf is proved and the Fan-König result for concave-convex-like payoffs in a general version is obtained under a new condition onf. Proc Natl Acad Sci U S A. Sep 1, 1998 · Three recent minimax theorems of Lin and Quan, Cheng and Lin, and Chang, Cao, Wu, and Wang are generalized, where two functions are involved and where the classical convexity assumptions are replaced by connectedness properties of certain level sets. Let CˆEand DˆF be nonempty compact and convex sets. Using the level set method of Joó (Acta Math Hung 54 (1–2):163–172, 1989 ), a general two-function topological minimax theorem are proved. if x is a feasible solution of P= minfhc;xijAx bgand y is a feasible Dec 24, 2016 · In [3], we proved two general minimax theorems which, grouped together, can be stated as follows: THEOREM 1. Arch. 2. Volume 24 of Progress in nonlinear differential equations and their applications, ISSN 1421-1750. In wikipedia and a lot of research papers, Sion's minimax theorem is quoted as follows: Let X X be a compact convex subset of a linear topological space and Y Y a convex subset of a linear topological space. 2 Definitions and Proof Outline All of our games will be finite two-player zero-sum games, allowing a streamlining of notation. Acta Math. SIAM J. Proof: We will prove this for the absolute maximum. In case min and/or max are not attained the min and/or max in the above expressions are replaced by inf and/or sup. In this paper, we study the following nonautonomous Kirchhoff problem: −1+b∫ℝN|∇u|2dxΔu+V (x)u=a (x)|u|p−2u+λ|u|q−2u,x∈ℝN,u∈H1ℝN,$$…. : On the general minimax theorems. Feb 1, 1997 · Minimax Theorems. The minimax theorem by Sion (Sion (1958)) implies the existence of Nash equilibrium in the n players non zero-sum game, and the maximin strategy of each player in {1, 2, , n} with the minimax strategy of the n+1-th player is equivalent to the Nash equilibrium strategy ofthe n playersNon zero- sum game. Instant Purchase. Title. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Emotions coordinate our behavior and physiological states during survival-salient events and pleasurable interactions. Semantic Scholar extracted view of "A general minimax theorem based on connectedness" by H. 2023. 2, and 3. Jan 1, 2003 · The above proof, given originally in [19], [20], was the first one using only elementary set-theoretical arguments for establishing general topological minimax theorems. : A general minimax theorem based on connectedness. Zero-Sum Games 2 3. that is to say, $$ \inf_ {y \in Y}\sup_ {x \in X}f (x, y) = \sup_ {x \in X}\inf_ {y \in Y}f (x, y). Let K be a compact convex subset of a Hausdorff topological vector space X, and C be a convex subset of a vector space Y. MATH MathSciNet Google Scholar The book is intended to be an introduction to critical point theory and its applications to differential equations. Google Scholar I. ," Pacific Journal of Mathematics, Pacific J. 4. Citation & Abstract. Starting from a beginning point, each player performs a sequence Jun 1, 2014 · In 2014, Zhang [23] proved a general two-function topological minimax theorem by using the level set method. Jing Zhang Jianming Liu Dongdong Qin Qingfang Wu. Such a game will be a triple G= (A 1,A 2,u) where A 1,A 2 are finite sets anduis a function from A 1 ×A Minimax Theorems 2012-12-06 Michel Willem Many boundary value problems are equivalent to Au=O (1) where A : X --+ Y is a mapping between two Banach spaces. Published 1 October 2016. Moreover, a saddle point theorem on a topological space without any linear structure is established. The method of our proof is inspired by the proof of [4, Theorem 2]. 3. T. Then infx ∈K supy ∈C f(x y , ) = supy ∈C infx ∈K f(x , y ). max min f(ì, v) = min max f(ì, v) . I They have a very special property: the minimax theorem. Abstract This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. 1 and 1. 42-47, 1953. [Sch+20] establish a very general complete class theorem as well as a minimax theorem using finitely additive pri-ors and finitely additive randomization. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general The present note wants to answer the main question which had been left open in the paper of the title [9] and, as far as we know, in the subsequent literature. Gong [5] obtained a strong minimax theorem and established an equivalent relationship between the strong minimax notion of equilibrium, as well as an elementary proof of the theorem. The talk wants to present an extension and unification, due to the author in Aug 24, 2020 · In this paper, we give an overview of some recent applications of a minimax theorem. v. This provides a fine didactic example for many courses in convex analysis or functional analysis. Here is a particular case of one of the results that we obtain: Let (T,F ,μ) be a non-atomic measure…. Jul 1, 2000 · Moreover, we obtain a minimax theorem and establish an equivalent relationship between the minimax theorem and a saddle point theorem for the scalar set-valued mappings, in which the minimization Minimax (sometimes Minmax, MM [1] or saddle point [2]) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for minimizing the possible loss for a worst case ( max imum loss) scenario. However their proofs depend on topological tools such as Brouwer fixed point theorem or KKM theorem. → f (x , y is concave for each ) x. 1, 3. Qualifies for Free Shipping. The minimax theorem results in numerous applications and many of them are far from being obvious. 1 Review: On-line Learning with Experts (Actions) Setting Given nexperts (actions), the general on-line setting involves Trounds. The article presents a new proof of the minimax theorem. As applications, we obtain minimax theorems in various settings and saddle-point theorems in particular. Let A be the payoff matrix. It was proved by John von Neumann in 1928. König A general minimax theorem. Kindler and R. The condition (c) in Theorem 1 is obviously fulfilled if either X or Y is compact. Jan 1, 2004 · Request PDF | MINIMAX THEOREMS REVISITED | Very general conditions are established that ensure the existence of a saddle-value for a function F (x, y) : C × D → R, where C, D are sub-sets Oct 12, 2016 · 1. 39 (1); 1953 Jan. In the second part of lecture, By relaxing the compact assumption to the Lindelöf one, the noncompact minimax theorem is established, which complements existing studies of the minimAX theorem. p∈∆n q∈∆m q∈∆m p∈∆n. LetL Pacific Journal of Mathematics, A Non-profit Corporation. Such a reduction is often possible with some form of compactness of the space X or Y and a suitable continuity and convexity or quasi‐convexity of the kernel K . Maurice Sion "On general minimax theorems. So a natural follow-up question is what's the example that prevent minimax theorem to drop compactness totally in this setting? optimization; convex-optimization; Feb 15, 2010 · Irle, I. Irle, Minimax theorems in convex situations, Game Theory and Mathematical Economics, O. We define several kinds of cone-convexities for set-valued mappings, give some…. Each player can prevent the other from doing any better than this. Among them, there are some multiplicity theorems for nonlinear equations as well as a general well-posedness result for functionals with locally Lipschitzian derivative. Author. Moeschlin and D. Here I reproduce the most complex one I am aware of. Theorem 1 (Concrete von Neumann minimax theorem). Thus,theset E = ff(x) jx 2[a;b]g isboundedabove. Tuy In the present paper, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. THEOREM. Minimax Theorems When X and Y Are More General Convex Sets It will be convenient at this point to give a more compact notation for the various "level sets" associated with f. Aug 1, 2011 · An elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students is provided. Progress in Nonlinear Differential Equat Series. Yen-Cherng Lin Q. The aim of this section is Theorem 3. 39, pp. Then d:= max y2D min x2C hAx;yi= min x2C max y2D hAx;yi=: p: (1) In this chapter, we give an overview of various applications of a recent minimax theorem. Minimax Theorem CSC304 - Nisarg Shah 26 •We proved it using Nash’s theorem heating. Let Abe a linear mapping between Euclidean spaces Eand F. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. prove a more general version of the minimax theorem with more general strategy spaces. If A {), A minimax theorem is a theorem that asserts that, under certain conditions, $$ \inf_ {Y}\sup_ {X}f = \sup_ {X}\inf_ {Y}f, $$. M VβN V6Λ' μβ M There have been several generalizations of this ON GENERAL MINIMAX THEOREMS MAURICE SION 1. Irle, A general minimax theorem, Zeitschrift für Operations Research 29 (1985), 229–247. The method of proof, making use of 3. The theorem improves and generalizes the known results shown by Cheng and Lin (Acta Math Hung 73(1–2):65–69 Sep 26, 2011 · Minimax Theorems. -Let X be a topological space, Y a convex set in a Lecture 18: Nash’s Theorem and Von Neumann’s Minimax Theorem Lecturer: Jacob Abernethy Scribes: Lili Su, Editors: Weiqing Yu and Andrew Mel 18. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. Minimax theorems for quasi-concave-convex functions. Introduction, von Neumann's minimax theorem [10] can be stated as follows : if M and N are finite dimensional simplices and / is a bilinear function on MxN, then / has a saddle point, i. Poor, On minimax robustness: A general approach and A Topological Minimax Theorem. Let S be an n-dimensional simplex with vertices n a {),, a n. by Michel Willem. 2012. f(x, ⋅) f ( x, ⋅) is upper semicontinuous and quasi-concave on Y Y Remark5. Springer Science & Business Media, Dec 6, 2012 - Mathematics - 165 pages. Here is the coarse structure of the argument. In this paper, by relaxing the compact assumption to the Lindelöf one, we establish a noncompact minimax theorem, which complements existing studies of the minimax theorem. It is shown that our result is independent of the most known min-max theorems. Pallaschke, eds. The purpose of this note is to present an elementary proof for Sion's minimax theorem. Formally, let X and Y be mixed strategies for players A and B. It is shown in particular that the topological minimax theorems of Geraghty and Lin (1984) are essentially a rediscovery of much earlier results of Tuy (1974), while the latter can be derived from a minimax theorem recently developed Sep 30, 2010 · Park [7][8] [9] [10] obtained existence theorems of Nash equilibrium by applying minimax inequalities and general KKM theory. Let f be a real-valued function defined on K C such that. PMC1063722. In the paper, a new kind of concavity of a function defined on a set without linear structure is introduced and a generalization of Fan Ky inequality is gi The minimax theorem can then be stated as follows: Theorem 1 (Minimax Theorem) For any finite two-player zero-sum gameG, max σ 1∈Σ 1 min σ 2∈Σ 2 u(σ 1,σ 2) = min σ 2∈Σ 2 max σ 1∈Σ 1 u(σ 1,σ 2) (1) Note that when we work in an arbitrary F, there is no immediate reason that either side of (1) must be well-defined. Proof for the theorem. PICK UP IN STORE. : A General minimax theorem. A ( [3], Theorems 1. In a similar fashion, ON GENERAL MINIMAX THEOREMS MAURICE SION 1. This condition is based on the concept of averaging functions, i. The results in the Aug 16, 2004 · A minimax result is a theorem which asserts that (1) max a∈A min b∈B f ( a, b )= min b∈B max a∈A f ( a, b ). 55,573-583. Mathematical Methods in the Applied Sciences. Lecture 16: Duality and the Minimax theorem 16-3 says that the optimum of the dual is a lower bound for the optimum of the primal (if the primal is a minimization problem). A standard technique for proving general minimax theorems is to reduce the problem to the minimax theorem for matrix games. We present a topological minimax theorem (Theorem 2. Oct 21, 2018 · [8] B. J. The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. Since Fan gave his minimax inequality in [2], various extensions of this interesting result have been obtained (see [4,11] and the references therein). It more general Hausdor topological vector spaces [20]. The Minimax Theorem 3 References 5 1. The justly celebrated von Neumann minimax theorem has many proofs. Dec 6, 2012 · Minimax Theorems. Heinz Konig [1991J A General Minimax Theorem based on Connectedness. , North-Holland (1981), 321–331. Although the related material can be found in other books, the authors of this volume have had the following goals in mind: To present a survey of existing minimax theorems, To give applications to elliptic differential equations in bounded domains, To consider the dual Apr 1, 2008 · Ferro [3,4] studied minimax theorems for general vector-valued functions. The minimax theorem implies that if there is a good response pq to any individual q, then there is a silver bullet Jul 13, 2024 · The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. Lai. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general . As a library, NLM provides access to scientific literature. C = {⃗u(σ2) : σ2 ∈ ∆(S2)} ⊆ Rn1, and observe that C is a compact and convex set. Volume 24 of Progress in Theoretical Computer Science. The minimax theorem can then be stated as follows: Theorem 1 (Minimax Theorem) For any finite two-player zero-sum gameG, max σ 1∈Σ 1 min σ 2∈Σ 2 u(σ 1,σ 2) = min σ 2∈Σ 2 max σ 1∈Σ 1 u(σ 1,σ 2) (1) Note that when we work in an arbitrary F, there is no immediate reason that either side of (1) must be well-defined. This paper studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems and produces a geodesically complete Riemannian manifolds version of Sion's minimax theorem. Mathematics. Heinz Konig [1968J Uber das von Neumannsche Minimax-Theorem. 2). Jul 1, 2000 · Minimax Theorems for Set-Valued Mappings under Cone-Convexities. We relate it to questions about the performance of randomized algo-rithms, and prove Yao’s minimax principle. Zeitschrift für Operations Research. Receive erratum alerts for this article. The key ingredient is an alternative for quasiconvex/concave functions based on the Apr 1, 2005 · TLDR. Science Record, 8, 229–233 (1959) Google Scholar Tuy, H. It is whether the typical dichotomy "finite intersections are connected in one variable, all intersections are connected in the other variable" is in the nature of the matter or not. SHIP THIS ITEM. Moreover, let p,q,r,s be four numbers such that 0 < s < q ≤ p, p ≥ 1, r > 1 In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. Hung. …. of their nonstandard source. Expand. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. Colloq. Let C X be nonempty and convex, and let D Y be nonempty, weakly compact and con-vex. 71:1227–1234, 2009), we prove the existence of solutions to a general variational inclusion problem, which contains most of the existing results of this type. $$. in Nonlinear Anal. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the This discussion is continued in the sections Quantitative minimax theorems and Minimax theorems and weak compactness. Optim. Contents 1. 1996) $159. In this manner, we establish several standard minimax theorems based on the push-down of nonstandard priors. A textbook for an advanced graduate course in partial differential equations. 28 Feb 1958 - Pacific Journal of Mathematics (Mathematical Sciences Publishers) - Vol. Maurice Sion. : A remark on the fundamental theorem in the theory of games. Among various ap-plications given in [1, 2], Fan used his theorem to derive in a direct and simple way the Sion's minimax theorem [5]. real-valued functions ϕ defined on some subset of the plane On general minimax theorems. Feb 4, 2014 · A general two-function topological minimax theorem is proved using the level set method of Joó to improve and generalize the known results. •Useful for proving Yao’s principle, which provides lower bound for randomized algorithms •Equivalent to linear programming duality John von Neumann Oct 1, 2016 · A very complicated proof of the minimax theorem. 50016-1 Corpus ID: 118873846; Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations @article{Rabinowitz1978SomeMT, title={Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations}, author={Paul H. Typically, Nash’s theorem (for the special case of 2p-zs games) is proved using the minimax theorem. ON GENERAL MINIMΛX THEOREM 173 3. In game theory, minimax is a decision rule used to minimize the worst-case potential loss; in other words, a player considers all of the best opponent responses to his strategies, and selects the strategy such that the opponent&#x27;s best strategy gives a payoff as large as possible. Ricceri ([5],[9]). Nash's result on the existence of Nash equilibria, essentially a fixed point theorem, implies the minimax theorem of von Neumann. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf (x,y) = maxminf (x,y). Google Scholar . Subscribe to Project Euclid. Let ni = |Si| and write, for each fixed σ2 ∈ ∆(S2), the function s1 7→ u(s1, σ2) as a vector ⃗u(σ2) ∈ Rn1. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. fo zk ym fd ph cr nw yl si ap