5. Theorem (Von Neumann-Fan minimax theorem) Let X and Y be Banach spaces. Contents 1. Thepurpose f this short paper isto present a generalization of a minimax theorem (Ref. The proof uses a particularization of Farkas' theorem involving the expression of one vector as a convex combination of a set of vectors. De ne the unitary matrix X = x 1 x 2 x n. Let CˆEand DˆF be nonempty compact and convex sets. tu-darmstadt. But I'm unable to find a way to proceed with the direction required in the problem. Let N n be an arbitrary n-dimensional subspace of Hand consider a basis fe 1;:::;e ng of this space. Published 1 March 1958. org, revised Feb 2023. 47. Minimax is a strategy of always minimizing the maximum possible loss which can result from a choice that a player makes. . Published 1995. Press(1950), 19–25. The key ingredient is an alternative for quasiconvex/concave functions based on the separation of closed convex sets in finite dimension, a result discussed in a first course in Learn how game theory and von Neumann's minimax theorem can be applied to various fields and scenarios, from economics to warfare, in this honors thesis. Dec 24, 2016 · On a minimax theorem: an improvement, a new proof and an overview of its applications. The min-max values of qsatisfy n ( A). 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 $\begingroup$ If it is any consolation, I have used this minimax relationship for over 3 decades and still need a moment's pause to remember which direction is 'for free'. Google Scholar Wu Wen-Tsün, A remark on the fundamental theorem in the theory of games, Sci. Looking at the above plots of Chebyshevpolynomials, withtheir equi-oscillation properties, maybe you have already guessed it yourself. 3] and more re ned subsequent algebraic-topological treatment. Handle: RePEc:arx:papers:2209. 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. Let f be a real-valued function defined on K C such that. Ken Binmore 1 165 Accesses. Under the same assumptions of Sion's theorem, for any y λ and y 2 ZΞY and any real number a with α<min max(/(x, y λ), f(x, y 2)), there is with α<min Mar 26, 2021 · The first purely elementary proof of the minimax theorem came a year after von Neumann’s 1937 paper, when Borel’s student Jean Ville (1910–89) utilized convexity arguments and the concept of a supporting hyperplane to show the same result, included in one of Borel’s books: Ville, J. Let A be the payoff matrix. , bn} of n pure strategies (or actions). , [ 2,8 ]) but cannot be considered Dec 13, 2017 · Abstract. Then. [36]) in 1928 for A and B unit Feb 5, 2022 · Theorem 1 (Yao’s Minimax Lemma). 1037–1040. proof is an application of the strong duality theorem. MINIMAX THEOREM I Assume that: (1) X and Z are convex. 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. Mathematics, Computer Science. 7, D-64289Darmstadt, Germanykindler@mathematik. Optim. There are two players, P1 and P2. Zero-Sum Games 2 3. Suppose A2M n is Hermitian, and for each 1 k n, let fS k g 2I k denote the set of all k dimensional linear Jan 13, 2021 · In this episode we talk about Jon von Neuman's 1928 minimax theorem for two-player zero-sum games and partially prove it. 1. Sehie P ark. The name &quot;minimax&quot; comes from minimizing the loss involved when the opponent selects the strategy Oct 1, 2016 · A very complicated proof of the minimax theorem. Abstract Mar 1, 1994 · Abstract. such as the KKM principle [4, x8. The minimax theorem results in numerous applications and many of them are far from being obvious. Since x∗ 1 and x∗ 2 are “best responses” to each other, Jun 25, 2023 · After this, Tao presents a proof of this theorem which I believe I understand almost totally, there is just one small detail that I am not getting. A constructive proof of the minimax theorem Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan second CORE meeting, LMU Munich, 27 January, 2017 ve reproduced a variety of proofs of Theorem 2. 13047 [math. Min-max theorem. Expand. It is well known that John von Neumann [15] provided the first proof of the theorem, settling a problem raised by Emile B. When this work has been completed, you may remove this instance of {{ ProofWanted }} from the code. 2. (3) Foreachz ∈ Z,thefunctionφ(·,z)isconvex. Then, the minimax equality holds if and only if the function p is lower semicontinuous at u =0. It is demonstrated that the minimax theorem holds as a consequence of this opment of the minimax theorem for two-person zero-sum games from his first proof of. Published 1 October 2016. Sergiu Hart, 2022. Proof of the Minimax Theorem CSC304 - Nisarg Shah 20 •When (𝑥෤1,𝑥෤2)is a NE, 𝑥෤1 and 𝑥෤2 must be maximin and minimax strategies for P1 and P2, respectively. But von Neumann's original minimax theorem seems to be easier. Cite as: arXiv:2405. , am} of m pure strategies (or actions). M. The theorem states that if you have a closed interval I on a continuous function, then f will achieve it’s maximum value and minimum value on I. To save this book to your Kindle, first ensure coreplatform@cambridge. Secondary 31C20. When equality holds one says that f, W, and Z satisfies a strong max–min property (or a saddle-point property). A SIMPLE PROOF OF THE SION MINIMAX THEOREM. (2) Tucker's proof of T. I get the idea how to find the dual LP from primal LP, but my basic knowledge is not enough for finding dual LP for the LP in chapter "Duality and the Minimax Theorem" from the same scribe. Feb 9, 2021 · My attempt: I have been able to prove the other direction (that too with help from online lecture notes) - i. d by John von Neumann in the paper Zur Theorie Der Gesellschaftsspiele. While his second article on the minimax theorem, stating the proof, has long been translated from German, his first announcement of his result (communicated in French to the Academy of Sciences in Paris by Borel, who had Jun 1, 2000 · On a minimax theorem: An improvement, a new proof and an overview of its applications. The minimax theorem, proving that a zero-sum two-person game must have a solution, was the starting point of the. Ville [9], A. A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications". SIAM J. 1037. 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. Proof for the theorem. It was proved by John von Neumann in 1928. , Paris 248, 2698–2699 (1959; Zbl 0092. Let C X be nonempty and convex, and let D Y be nonempty, weakly compact and con-vex. , New. ): Let I = [a, b] ⊂ ℝ be a compact interval and f: I → ℝ a continuous function. Proof: We will prove this for the absolute maximum. native proof of the minimax theorem using Brouwer's xed point theo-rem. This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. For the applications I'm looking for, a minimax theorem with a context in m-dimentional Euclidean space. Subjects: Combinatorics (math. 2 Citations Von Neumann’s …rst announcement of his proof, communicated in French by Borel to the Academy of Sciences fVON NEUMANN, VILLE, AND THE MINIMAX THEOREM 15 in Paris, is translated in an appendix to this paper. Michel Willem; "The material is presented in a unified way, and the proofs are concise and elegant Sep 13, 2022 · Download a PDF of the paper titled Calibrated Forecasts: The Minimax Proof, by Sergiu Hart (1995) of the existence of calibrated forecasts by the minimax theorem, In an appendix to the Traité a note appears entitled Sur la théorie générale des jeux où intervient l’habileté des joueurs where Ville provides a remarkably simple proof of von Neumann’s minimax theorem as well as a generalization to continuous strategy sets, and for the first time, outlines the importance of convexity in the The proof of this theorem ( since its context is of linear topological spaces and your stament uses semi continuity, quasi convexity and quasi concavity) is very intricate. Wald [11], and others [1] variously extended von Neumann's result to cases where M and N were allowed to be subsets of certain infinite dimensional linear spaces. 2024. In formal terms, the theorem is as follows (Bényi, n. While his second article on the minimax theorem, stating the proof, has long been Theorem 0. Soc. After a brief discussion of partisan combinatorial games, we will discuss the zero-sum games and von Neuman's minimax theorem. In doing that, a key tool was a partial 1. The article presents a new proof of the minimax theorem. 3). If you would welcome a second opinion as to whether your work is Proofs of the minimax theorem based on the Brouwer fixed point theorem or the Knaster-Kuratowski-Mazurkiewicz (KKM) principle are elegant and short (see, e. This paper defines a class of strong local saddle points based on the lower bound properties for stability of variable selection and gives a framework to construct continuous relaxations of the discontinuous min-max problems based on the convolution. g. This provides a fine didactic example for many courses in convex analysis or functional analysis. 1, Exer. LEMMA 1. In fact, because we have nfree coe cients we may choose them in such a way that xis Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have The purpose of this note is to present an elementary proof for Sion's minimax theorem. 1. DOI 10. This theorem requires a proof. 2. Introduction to Games 1 2. The paper distils the essence of Owen&#8217;s elementary proof of the minimax theorem by using transfinite induction in an abstract setting. Let Abe a linear mapping between Euclidean spaces Eand F. S. LetL I They have a very special property: the minimax theorem. Formally, let X and Y be mixed strategies for players A and B. coherent book on game 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. The Minimax Theorem 3 References 5 1. 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. INTRODUCTION. 1 Theorem (Courant-Fischer). The justly celebrated von Neumann minimax theorem has many proofs. Korean Math. If n <( A) then n is the n-th eigenvalue of A counted with multiplicity from the lowest value. Outline of proof. Starting from a beginning point, each player performs a sequence Oct 11, 2012 · vectors. We also extend Dantzig Feb 1, 2018 · A Simple Proof of Sion's Minimax Theorem Jürgen Kindler Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. Acad. hat Commented Dec 30, 2017 at 21:13 Tucker’s proof of his Lemma is indeed short and novel, but in this light we agree with Adler’sview of Tucker’s Lemmaas a“variant of Farkas’s Lemma”[1, p. The key ingredient is an alternative for quasiconvex/concave functions based on the H. extend to valued f ? > <1 Theorem. Then d:= max. You can help Pr∞fWiki P r ∞ f W i k i by crafting such a proof. 2010. Finally, we are ready to solve the key minimax problem that will reveal optimal in-terpolation points. Simons. Since the Aug 8, 2019 · Zero-Sum Games: Proof of the Minimax Theorem The course will start with the discussion of impartial combinatorial games: subtraction game, Nim, and Chomp, will discuss the Sprague-Grundy value. Proof: Let x∗ = (x∗ 1,x ∗ 2) ∈ X be a NE of the 2-player zero-sum game Γ, with matrix A. For the converse direction, the standard proof by Dantzig (1951) is known to be incomplete. P1 has a set A = {a1, a2, . LP duality and the minimax theorem are closely related to solving, respec-tively, inhomogeneous and homogeneous linear equations in nonnegative vari-ables. Jan 1, 2002 · the minimax theorem to be appeared se ven years later. 05863, arXiv. This second inequality is equivalent to the Brouwer Fixed Point Theorem, but is often more convenient. While the importance of minimax theorems in various fields is well established since long time, the I. Rec. The min max theorem is a specific case of the duality theorem for linear programs (the feasible set is a polygon or a polytope). (2) p(0) = inf x∈X sup z∈Z φ(x,z) < ∞. The proof I showed you last time comes from Andrew Colman's book Game Theory and its Applications in the Social and Biological Jun 1, 2010 · Von Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero sum games) in 1928. Abstract. Univ. Borel wrote sev eral papers on tw o-person games since 1921, b ut none of these claimed the general existence of the ”best” strate gies. Since this is May 23, 2022 · The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. CO] for this version) Theorem 1 (The min-max Theorem). e. 5 (The Minimax Theorem [Neu28]). Sci. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs . Feb 1, 2004 · 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. Let Abe the self-adjoint operator corresponding to a closed semi-bounded quadratic form. Remark5. In mathematics, the max–min inequality is as follows: For any function. d. e in optimization or game theory. de Pages 356-358 | Published online: 01 Feb 2018 Dec 24, 2016 · Theorem 1 of [14], a minimax result for functions $f:X\times Y\to {\bf R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set Proof of the Minimax Theorem The Minimax Theorem follows directly from Nash’s Theorem (but historically, it predates Nash). It Jul 23, 2020 · Min-Max Theorem for Continuous Functions. That's just a fixed point theorem. Let Sbe any subspace of dimension i)there is an x 2Ssuch that x ?x 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。 On general minimax theorems. Theorem 16. We describe in detail Kakutani's proof of the minimax theorem Part of the book series: Springer Optimization and Its Applications ( (SOIA,volume 17)) A unified framework is presented for studying existence and stability conditions for minimax of quasiconvex quasiconcave functions. Mathematics. 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. The general proof relies on separating hyperplane theorems. Below I will show an outline of the proof presented in the link above (just the topics) and be more detailed in the part that I didn't get. Let v∗:= (x∗ 1) TAx∗ 2 = U1(x∗) = −U2(x∗). Ricceri and was given in [2]; see Theorem 2 below. Valerii Krygin. 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. heory of strategic games as a distinct discipline. Joó, A simple proof for von Neumann's minimax theorem,Acta Sci. 47 (2010), No. 3. If n = ( A) then Ahas at most n 1 eigenvalues counted with multiplicity below ( A). 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. 2 The Courant-Fischer Theorem 4. This paper offers an alternative proof of the so-called fundamental theorem of the theory of games or the minimax theorem. 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 . (4) For each x ∈ X, the function −φ(x,·):Z → is closed and convex. J. Theorem 1 (Concrete von Neumann minimax theorem). The minimax theorem was proven by John von Neumann in 1928. notion of equilibrium, as well as an elementary proof of the theorem. the theorem in 1928 until 1944 when he gav e a completely different proof in the first. •The reverse direction is also easy to prove. When managing picks up, it is alluded to as "maximin"— to augment the base pick up. A theorem giving conditions on f, W, and Z which guarantee the saddle Sion's minimax theorem. 1 was originally proved by John von Neumann in the 1920s, using xed-point-style arguments. Abstract This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. 4134/BKMS. Mar 26, 2021 · The first purely elementary proof of the minimax theorem came a year after von Neumann’s 1937 paper, when Borel’s student Jean Ville (1910–89) utilized convexity arguments and the concept of a supporting hyperplane to show the same result, included in one of Borel’s books: Ville, J. Before we examine minimax, though, let's look at 知乎专栏提供一个平台，让用户可以随心所欲地写作和自由表达自己的观点。 The justly celebrated von Neumann minimax theorem has many proofs. Von Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero sum games) in 1928. "Calibrated Forecasts: The Minimax Proof," Papers 2209. $\endgroup$ – copper. We will discuss various methods for solving such games. It can be viewed as the starting point of many results of similar nature. All have their bene ts and additional features: (1) The original proof via Brouwer's xed point theorem [4, x8. Consider the number ˝ n = supfminfhAx;xi: x2N ng: N n ˆH;dimN n = ng Then ˝ n = n. CO] (or arXiv:2405. (1938). For every two-person zero-sum game (X;Y;A) there is a mixed strategy x for player I and a mixed strategy y for player (II) such that, max x min y xT Ay = min y max x xT Ay = xT Ay; (16. R. . 15]. Borwein. The first minimax theorem was proved in a famous paper by von Neumann (cf. 7) y i Jul 13, 2024 · The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. If f is continuous, then f is bounded by the previous theorem. The proof uses Farkas Lemma which is a separating theorem for convex cones. The left-hand side of the in-equality is what will will try to lower-bound: It is the worst-case performance of Guillermo Owen's Proof Of The Minimax Theorem Download PDF. P2 has a set B = {b1, b2, . A general minimax theorem. These theorems include as special cases refinements of several known results from game theory, optimization, and nonlinear Strategies of Play. In fact, due to the following theorem by Courant and Fischer, we can obtain any eigenvalue of a Hermitian matrix through the "min-max" or "max-min" formula. Introduction. 5, pp. 2) and fixed-point theorems (Ref. CO) MSC classes: Primary 05C99. It's crucial to watch lecture videos Nov 24, 2018 · Proof of Courant-Fischer minimax theorem through deformation lemma. Modified 5 years, 7 months ago. 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. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. Viewed 232 times I review LP duality by reading Lecture 7: The LP Duality Theorem. Jan 1, 2016 · Second, we look at the classical minimax theorem. Jun 22, 2024 · Proof. he minimax theorem is one of the most important results in game theory. We de ned the Chebyshev polynomials so that Tn+1(x) = 2xTn(x) Tn 1(x) with T0(x) = 1 and T1(x Proof: A Hermitian A has orthonormal eigenvectors x 1, , x n. The first proof was due to Von-Neumann [vN28], and later generalisations and different proofs appeared in [Bor15, Fan53,Sio58,Kin05,BZ86] and 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. Sep 30, 2010 · Content may be subject to copyright. Minimax Theorems and Their Proofs. Berge [C. There have been several generalizations of this theorem. Proof: Theconvexity $\begingroup$ @Ovi Hm not so sure now what I meant. Proof. Ask Question Asked 5 years, 7 months ago. 4. Google Scholar Minimax Theorems Download book PDF Linking theorem. Pacific Journal of Mathematics. It was rst introduc. Then, max x2X Ec(A;x) min a2A Ec(a;X) : Before proving the theorem, let us interpret what it means. Math. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof, more directly than Adler (2013). provided an alte. TLDR. If is a real-valued function on with. Later, John Forbes Nash Jr. Any help is appreciated. org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. The method of our proof is inspired by the proof of [4, Theorem 2]. This second proof is the reason that, 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. Bull. The utility for P1 is denoted U1(ai, bj) and the utility for P2 is denoted U2(ai, bj). how to use the separating hyperplane theorem to prove the minmax theorem. It states: Let be a compact convex subset of a linear topological space and a convex subset of a linear topological space. → f (x , y is concave for each ) x. 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. Introduction to Games The notion of a game in this context is similar to certain familiar games like chess or bridge. Probably, it was "better" only for me for notation purposes. Weyl, Elementary proof of a minimax theorem due to von Neumann, Contributions to the theory of games 1, Princeton. The example function illustrates that the equality does not hold for every function. 4 From LP Duality to Minimax Theorem 1. A new brief proof of Fan's minimax theorem for convex-concave like functions is established using separation arguments. 174]. (Szeged),42 (1980), 91–94. Here I reproduce the most complex one I am aware of. Then infx ∈K supy ∈C f(x y , ) = supy ∈C infx ∈K f(x , y ). Let S be a compact convex subset of a finite-dimensional vector space V, and 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 ). Any vector x2N n can be written as x= P n j=1 c je j. The Ky Fan inequality in game theory. 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. Thus,theset E = ff(x) jx 2[a;b]g isboundedabove. The connection between the minimax theorem and the solvability of systems of lin- ear inequalities and the crucial role played by convexity were first outlined by Jean Andr´e Ville in 1938, when he published the first elementary proof of the minimax theorem in an appendix to lecture notes of Emile Borel’s Sorbonne course on the ap- plication 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. 2023. Dec 24, 2016 · The proof relies on the Hilbert structure of the space Y, which leads to the application of a min-max theorem, which is due to B. To discuss this page in more detail, feel free to use the talk page. 1, Theorem 2)of Fan, who first proved minimax theorems valid for spaces that no have linear st ucture, andto provide an alternate and simple proof avoiding VonNeumann's minimax theorem (Ref. Ser3(1959), 229–233. It is more common to have a problem like $\min_x \max_y f(x, y)$, where you do minimisation for the primal and maximisation for the dual variables. max 𝑥1 𝑥1 𝑇𝐴𝑥෤ 2=𝑣෤=min 𝑥2 𝑥෤1𝑇𝐴𝑥2 =max 𝑥1 min 𝑥2 𝑥1 𝑇∗𝐴∗𝑥 May 19, 2024 · A one-line proof of a minimax theorem due to Steinerberger is given. Let Abe any random variable with values in Aand let Xbe any random variable with values in X. Jean Ville’s 1938 note, also translated below, greatly simpli…ed the proof by drawing on considerations of convexity, pointing the Dec 2, 2009 · Decision Making Using Game Theory - March 2003. The theorem shows up in Game Theory in a few places. 13047v1 [math. Sion. Much later, in the 1940s, von Neumann proved it again using arguments equivalent to strong LP duality (as we’ll do here). Each player has a utility for each (ai, bj) pair of actions. Maybe von Neumann was a bit jealous? I don't know a proof of Nash's theorem that doesn't use a fixed-point theorem. The Minimax algorithm is the most well-known strategy of play of two-player, zero-sum games. 2). Google Scholar Formalization of a 2 Person Zero-Sum Game 1. 05863 Minimax (now and again MinMax or MM) is a choice administer utilized as a part of choice theory, game theory, insights and reasoning for limiting the conceivable damage for a most pessimistic scenario (misere gameplay) situation. While his second article on the minimax theorem, stating the proof, has long been translated from German, his first announcement of his result (communicated in French to the Academy of Sciences in Paris by Borel, who had posed the problem settled by Von Neumann&#39;s proof) is 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 The punchline. nc ey wo nt ek no mm nt gu li