Sion minimax theorem. ru/7ypibkx/halo-oglasi-kuce-na-cuvanje.

Nevertheless, inspired by proof structure in Komiya (1988), we extend Sion’s theorem to nonlinear space by providing a new ap-proach based on Helly’s theorem alone. g. I would suggest to use the Frank-Wolfe algorithm described here. A SIMPLE PROOF OF THE SION MINIMAX THEOREM. Let ̃ beavalueof suchthat ̃ =argmax ∈ min ∈ ( , , ̃ )=argmin ∈ max ∈ Abstract. 5. Theorem: Sion’s Minimax Theorem Let A and Z be convex, compact spaces, and. Theorem (Von Neumann-Fan minimax theorem) Let X and Y be Banach spaces. tu-darmstadt. As LinAlg 's answer, by Sion minimax theorem, if one of x or f belongs to a bounded set, the strong duality holds. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the A minimax theorem is a theorem which states that under certain conditions on X X, Y Y and f f : infx∈Xsup y∈Y f(x, y) = sup y∈Y infx∈Xf(x, y) inf x ∈ X sup y ∈ Y f ( x, y) = sup y ∈ Y inf x ∈ X f ( x, y) All minimax theorems rely strongly on convexity: the sets X X and Y Y are usually required to be convex subsets of vector Fan-Browder fixed point theorem for multi-valued mappings. Dec 15, 2023 · Abstract: The work studies cooperative decentralized constrained POMDPs with asymmetric information. We suppose that X and Y are nonempty sets and f: X x Y →IR A minimax theorem is a theorem which asserts that, under certain conditions, $$\mathop { {\min }}\limits_ {Y} \mathop { {\max }}\limits_ {X} f = \mathop { {\max }}\limits_ {X} \mathop { {\min }}\limits_ {Y} f Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. : A Simple Proof for von Neumann’ Minimax Theorem. Let and be non-void convex and compact subsets of two linear topological spaces, and let ∶ × →ℝbe a function that is continuous and quasi-concave in the first VON NEUMANN MINIMAX THEOREM Theorem: Let A be a m×n matrix representing the payoff matrix for a two-person, zero-sum game. The minimax theorem results in numerous applications and many of them are far from being obvious. Let us recall the following definition where, for a mixed strategy pair (x,y), we define V(x,y) := Pm i=1 Pn j On general minimax theorems. March 1958. M VβN V6Λ' μβ M. 8, Iss: 1, pp 171-176. 3] and more re ned subsequent algebraic-topological treatment. Jun 1, 2010 · The proof presented by Von Neuman and Morgenstern (1944) is not Von Neumann's rather involved proof of 1928, but is based on what they called "The Theorem of the Alternative for Matrices" which is in essence a reformulation of an elegant and elementary result by Borel's student Jean Ville in 1938. ON GENERAL MINIMAX THEOREMS. Introduction. DOI: 10. 1137/22m1505475 Corpus ID: 258960221; Sion’s Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm @article{Zhang2022SionsMT, title={Sion’s Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm}, author={Peiyuan Zhang and J. Let f be a real-valued Oct 24, 2017 · About a symmetric multi-person zero-sum game we will show the following results. The purpose of this note is to present an elementary proof for Sion's minimax theorem. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy. 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. minimax theorem or linear programming duality. Oct 1, 2010 · Then, we give some generalized minimax inequalities for vector-valued functions by means of the generalized KKM theorem. Math. 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. Under Assumption 1, the modiied version of Sion’s minimax theorem with the co-incidence of the maximin strategy and the minimax strategy imply the existence of a symmetric Nash equilibrium. SION. Thus, they are equivalent. The first purpose of this paper is to tell the history of John von Neumann’s devel-opment of the minimax theorem for two-person zero-sum games from his first proof of the theorem in 1928 until 1944 when he gave a completely different proof in the first coherent book on game theory. Acta Sci. Later, John Forbes Nash Jr. However their proofs depend on topological tools such as Brouwer fixed point theorem or KKM theorem. The result is based on an intersection theorem which may be of interest 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). All have their bene ts and additional features: (1) The original proof via Brouwer's xed point theorem [4, x8. •Useful for proving Yao’s principle, which provides lower bound for randomized algorithms •Equivalent to linear programming duality John von Neumann March 1958. von Neumann (8) proved his theorem for simplexes by reducing the problem to the 1-dimensional cases. We consider the general form of (P) in geodesic metric spaces, i. Von Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero Among these result an elementary proof of the well-known Sion’s minimax theorem concerning quasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. INTRODUCTION The minimax theorem, proving that a zero-sum, two person game (a strictly competitive game) must have a solution, was the starting point of the theory of strategic games as a distinct discipline, although 1 2 VON NEUMANN, VILLE, AND THE MINIMAX THEOREM game theory soon moved on to games with n players and with nonconstant sums of Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. $$. : A × Z → R. The result is based on an intersection theorem which may be of interest on its own right. October 2023. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as "forecasting algorithms" evaluated by a proper scoring rule. This work answers a question of Professor Granas regarding the logical relationship between the Elementary KKM theorem and the Sion–von Neumann minimax theorem. 1037–1040. Dec 3, 2013 · Abstract. MAURICE. characterization of normality by selection theorem. 2010. In doing that, a key tool was a partial Sep 30, 2010 · Content may be subject to copyright. M VβN V6Λ' μβ M There have been several generalizations of this Jul 14, 2021 · I want to know whether Sion's Minimax Theorem is applicable to the following instance: \\begin{align*} \\max_{x\\in \\mathbb{R}^n} \\min_{w\\in S} v(x,w) &amp Minimax theorem Sources: Kneser? Sion? See Millar (1983, page 92). Sion's minimax theorem plus the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. The paper presents a self Sion's minimax theorem can be proven [34] by Helly's theorem, which is a statement in combinatorial geometry on the intersections of convex sets, and the KKM theorem of Knaster, Kuratowski, and Jan 22, 2023 · In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion. Main theorem: Minimax in Nonlinear Geometry In Euclidean space, Sion’s minimax theorem guarantees strong duality for convex-concave minimax problems. 5. Jun 4, 2022 · 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. We also prove an improved version of Impagliazzo's hardcore lemma. 1 The minimax theorem was proved by John von Neumann in 1928 [6], generalized by Maurice Sion in 1958 [5], and several times newly proved in 1988 [4], 2005 [3], and 2011 [2]. A, is equivalent to Sion™s minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A. if x is a feasible solution of P= minfhc;xijAx bgand y is a feasible Oct 1, 2003 · A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. ) Quasiconvex and. Szeged, 42, 91–94 (1980) MathSciNet Google Scholar Feb 25, 2020 · First, we use Sion's minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. First, we use Sion's minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Let K be a compact convex subset of a Hausdorff topological vector extend to space X, and C be a convex subset of a vector space Y. It states: Let be a compact convex subset of a linear topological space and a convex subset of a linear topological space. 0. 15]. We describe in detail Kakutani's proof of the minimax theorem Apr 24, 2021 · Sion's minimax theorem. Berge [C. We present a topological minimax theorem (Theorem 2. : An Extension of Sion’s Minimax Theorem with an Application to a Method for Constrained Games. 4. Convex optimization and strong duality. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. (Note - this wasn’t given explicitly in lecture, but we do use it later. The proof is based on a result of Victor Klee [ 9 ] Jan 1, 2007 · We include what we believe is the most elementary proof of Maurice Sion’s version of the minimax theorem based on a theorem of C. As applications, we obtain an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities. There have been several Jan 1, 2004 · Sion’s minimax theorem is extended for noncompact sets, and for certain two-person zero-sum games on constrained sets a sequential unconstrained solution method is given. A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. Then the game has a value and there exists a pair of mixed strategies which are optimal for the two players. Mathematics. Subscribe to Project Euclid. d by John von Neumann in the paper Zur Theorie Der Gesellschaftsspiele. This paper proves the following theorem. Minimax Theorems and Their Proofs. SIAM Journal on Optimization 33 (4):2885-2908. Proof for the theorem. such as the KKM principle [4, x8. The expected score of the forecasting version of he minimax theorem is one of the most important results in game theory. Oct 3, 2016 · $\min \max = \max \min$ by Sion's minimax theorem, since $\mathcal P$ is compact. provided an alte. It was rst introduc. There have been several Sep 30, 2010 · The von Neumann-Sion minimax theorem is fundamental in convex analysis and in game theory. Published 1995. Hartung, J. e. 1137/22M1505475. 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. heorem2. Abstract About a symmetric three-players zero-sum game we will show the following results. Let f f be a real-valued function on X × Y X × Y such that. Pacific J. Fenchel-Rockafellar duality problem: Show that weak duality holds, i. 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。 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. e. Lemma 1. , M, Nare geodesic metric spaces, f| Sion's minimax theorem is stated as: Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. Feb 10, 2024 · Sion's minimax theorem is usually stated with a condition that one of the sets is compact, e. Authors: Minimax Theorems. Let $f$ be a real-valued function on $X \times Y$ such that 1. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as “forecasting algorithms” evaluated by a certain proper scoring rule. Nikaido-Sion version of the minimax theorem which is accessible to students in anˆ undergraduate course in game theory. Prokhorov theorem on non Polish spaces. The method of our proof is inspired by the proof of [4, Theorem 2]. In this paper, we establish a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. de Pages 356-358 | Published online: 01 Feb 2018 Jan 1, 2003 · A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Possibly also some handwritten notes from other lectures by Millar. , p≥−d . 5, pp. 8 (1), 171-176, (1958) Include: Citation Only. It can be viewed as the starting point of many results of similar nature. Abstract The article presents a new proof of the minimax theorem. 7. Next, Simons [ 4] showed different kinds of minimax theorems, and Li Dec 24, 2016 · On a minimax theorem: an improvement, a new proof and an overview of its applications. Typically, Nash’s theorem (for the special case of 2p-zs games) is proved using the minimax theorem. 2. Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. native proof of the minimax theorem using Brouwer's xed point theo-rem. In this section, we establish an analog of Sion’s theorem in geodesic metric spaces. Sep 4, 2018 · We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. , Paris 248, 2698–2699 (1959; Zbl 0092. 1, Exer. In 1953, Fan [ 2] published a minimax theorem for concave–convex functionals, while in 1957, Sion [ 3] proved the theorem for quasi-concave–convex functionals. Let X and Y be non-void convex and compact subsets of two linear topological spaces, and let f : X ×Y → Rbe a function, that is continuous and quasi-concave in the first Pacific Journal of Mathematics, A Non-profit Corporation. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the Passy–Prisman theorem and consequently of the Sion theorem, contrary to most Sions minimax theorem ( wiki, paper) can be stated 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. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them In particular, he proved in the following two-function minimax inequality (since the compactness of X is not needed, this result can in fact be strengthened to include Sion's theorem, Theorem 3, by taking g = f): A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy implies the existence of Nash equilibrium which is symmetric in each group, and they are equivalent. The first main result of the paper is a geodesic metric space version of Sion’s minimax theorem; we believe our proof is novel and broadly accessible as it relies on the finite The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. Share Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuous function is stated as follows. f(a, z). 1958 On general minimax theorems. 28 Feb 1958 - Pacific Journal of Mathematics (Mathematical Sciences Publishers) - Vol. Theorem 1 of [14], a minimax result for functions f: X × Y → R, where Y is a real interval, was partially extended to the case where Y is a convex set in a Hausdorff topological vector space ( [15], Theorem 3. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. von Neumann [1] proved his theorem for simplexes by reducing the problem to the one-dimensional cases. Let X be a compact Hausdorfl space and let $(Y,A)$ be a measurable space. 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. A further main contribution is to decompose the minimax relation into independent halfs, such that the minimax theorems quoted above and hence the bulk of the minimax Apr 1, 2005 · 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. Min-max theorem. Citation & Abstract. [3]. (2) Tucker's proof of T. In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. Let f be a real-valued function defined on K C such that. Download to read the full chapter text. Zhang and Suvrit Sra}, journal={SIAM Journal on Optimization}, year={2022}, url={https://api Feb 13, 2022 · Specifically, we study minimax problems cast in geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. LEMMA 1. Sci. The first main result of the paper is a geodesic metric space version of Sion's minimax theorem; we believe our proof is novel and transparent, as it relies on Helly's theorem only. Two players are in one group Feb 1, 2018 · A Simple Proof of Sion's Minimax Theorem Jürgen Kindler Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. ve reproduced a variety of proofs of Theorem 2. R. 8 (1): 171-176 (1958). However, not only from purely mathe-matical or theoretical motivation, but also from more practical motivation we deal with the theorem. Oct 5, 2016 · 1. Nov 30, 2023 · First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. max min f(μ, v) = min max f(μ, v) . 1 Introduction We consider the relation between Sion™s minimax theorem for a continuous function and existence of a Nash equilibrium in an asymmetric three-players zero-sum game with two groups1. Sehie P ark. 1 (weak duality). Hence the use of such applications has to be based not only on belief in the minimax theorem, but on a DOI: 10. 1037. We take a step towards understanding certain nonconvex-nonconcave minimax problems that do remain tractable 2. On general minimax theorems. Soc. The existence of a symmetric Nash equilibrium is proved by Sion's minimax theorem plus the coincidence of the maximin strategy and the minimax strategy Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuous function is stated as follows. → f (x , y is concave for each ) x. An example of such a game is a Apr 30, 2015 · Sion's minimax theorem. First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Then infx ∈K supy ∈C f(x y , ) = supy ∈C infx ∈K f(x , y ). Bull. The proof of Krein-Milman Theorem and the reason behind Jul 26, 2023 · The paper proves the minimax theorem for a specific class of functions that are defined on branching polylines in a linear space, not on convex subsets of a linear space. Sion's generalization (7) was proved by the aid of Helly's theorem and the KKM theorem due to…. We will show the following results. ," Pacific Journal of Mathematics, Pacific J. If f(a, ·) is upper semicontinuous and quasiconcave on Z ∀a ∈ A and. 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. Acad. 1. Fan-Browder fixed point theorem for multi-valued mappings. That is, you can interchange min and max. Receive erratum alerts for this article. S. Compactness and dimensionality. Sion's minimax theorem for a pair of playes in each group imply the existence of a Nash equilibrium which is symmetric in each group. Jul 20, 2018 · Minimax theorems have important applications in optimization, convex analysis, game theory and many other fields. extend to valued f ? > <1 Theorem. $f(x, \cdot)$ is upper semicontinuous and quasi-concave on $Y$ for each $x \in X$. 4134/BKMS. Oct 1, 2010 · The von Neumann–Sion minimax theorem is fundamental in convex analysis and in game theory. prove Sion’s minimax theorem in Euclidean space based on an elementary proof (without Helly’s theorem or KKM theorem), we found his proof to have a gap. 2). Expand. Simons. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. The existence of a saddle point for such functions does not follow directly from the classical minimax theorem and needs individual consideration based both on convex analysis and graph theory. Determining whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. Maurice Sion "On general minimax theorems. We consider the relation between Sion's minimax theorem for a continuous function and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in A geodesic metric space version of Sion’s mini-max theorem is presented, which is believed to be novel and transparent, as it relies on Helly’s theorem only. The strong duality theorem states these are equal if they are bounded. Generally solving saddle point problems are hard. Let C X be nonempty and convex, and let D Y be nonempty, weakly compact and con-vex. Finally, we prove a cone-saddle point theorem as an application of our results. 3. quasiconcave are weaker conditions than convex and concave respectively. Maurice Sion. In particular, you don't need $\mathcal Q$ to be compact. <1> Theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle Downloadable (with restrictions)! About a symmetric three-players zero-sum game we will show the following results. Theorem 16. Apr 5, 2015 · The Sion–von Neumann minimax theorem itself can be proved by simple topological arguments using connectedness instead of convexity. 47. A minimax theorem is a theorem that asserts that, under certain conditions, that is to say, The purpose of this article is to give the reader the flavor of the different kind of minimax theorems, and of the techniques that have been used to prove them. The existence of Nash equilibrium which is symmetric in each group implies Sion's minimax theorem with the coincidence of the maximin strategy and the minimax convexlike minimax theorem of Ky Fan [1953], and a topological theorem in the spirit of the quasiconcave-convex minimax theorem of Sion [1958]. Using an extension of Sion's Minimax theorem for functions with positive infinity and results on weak-convergence of measures, strong duality and existence of a saddle point are established for the setting of infinite-horizon expected total discounted costs when the observations lie in a 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. Sion’s generalization [2] was proved by the aid of Helly’s theorem and the KKM theorem due to Knaster et al. The first main result of the paper is a geodesic metric space version of Sion’s minimax theorem; we believe our proof is novel and broadly accessible as it relies on the finite Minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables. 7, D-64289Darmstadt, Germanykindler@mathematik. We suppose that X and Y are nonempty sets and f: X x Y →IR A minimax theorem is a theorem which asserts that, under certain conditions, $$\mathop { {\min }}\limits_ {Y} \mathop { {\max }}\limits_ {X} f = \mathop { {\max }}\limits_ {X} \mathop { {\min }}\limits_ {Y} f. ABOUT. Mar 31, 2019 · On compactness in Sion's minimax theorem. We suppose that X and Y are nonempty sets and f: X × Y → R. The result is based on an intersection theorem which may be of interest Oct 21, 2023 · Sion’s Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm. f(x, ⋅) f ( x, ⋅) is upper semicontinuous and quasi-concave on Y Y for each x ∈ X x ∈ X. In the present paper, we show quantum minimax theorem, which is also an exten-sion of a well-known result, minimax theorem in statisticaldecision theory, first shown by Wald [38] and generalized by Le Cam [26]. Our proofs rely on two innovations over the classical approach of using Von Neumann's minimax theorem or linear programming duality. Then. Proof. from Wikipedia, Let $X$ be a compact convex subset of a linear ray. , 103(2), 401–408 (1982) MathSciNet Google Scholar Joo, L. DOI 10. If the feasible sets for x and f are polytope, you can easily solve the . Korean Math. 47 (2010), No. Format: Minimax Theorem CSC304 - Nisarg Shah 26 •We proved it using Nash’s theorem heating. The first main result of the paper is a geodesic metric space version of Sion’s minimax theorem; we believe our proof is novel and broadly accessible as it relies on the finite 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] ON GENERAL MINIMAX THEOREMS MAURICE SION 1. In a recent paper, Kindler [4 A general minimax theorem. On the compactness in Sion's minimax theorem. View. fm wp vw qe az pr vr kz nt vz