Sion minimax theorem. minimax theorem or linear programming duality.

Two players are in one group 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. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them 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. 15]. We will show the following results. (Note - this wasn’t given explicitly in lecture, but we do use it later. → f (x , y is concave for each ) x. 5. DOI: 10. On the compactness in Sion's minimax theorem. Fan-Browder fixed point theorem for multi-valued mappings. 7, D-64289Darmstadt, Germanykindler@mathematik. Hence the use of such applications has to be based not only on belief in the minimax theorem, but on a 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 Oct 1, 2010 · The von Neumann–Sion minimax theorem is fundamental in convex analysis and in game theory. Sci. Acad. It can be viewed as the starting point of many results of similar nature. 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. 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 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 Oct 3, 2016 · $\min \max = \max \min$ by Sion's minimax theorem, since $\mathcal P$ is compact. Theorem (Von Neumann-Fan minimax theorem) Let X and Y be Banach spaces. convexlike minimax theorem of Ky Fan [1953], and a topological theorem in the spirit of the quasiconcave-convex minimax theorem of Sion [1958]. Typically, Nash’s theorem (for the special case of 2p-zs games) is proved using the minimax theorem. Then. extend to valued f ? > <1 Theorem. 1 (weak duality). he minimax theorem is one of the most important results in game theory. characterization of normality by selection theorem. 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. 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. Jan 1, 2003 · A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. The expected score of the forecasting version of About a symmetric three-players zero-sum game we will show the following results. 4134/BKMS. 1. Apr 24, 2021 · Sion's minimax theorem. provided an alte. Citation & Abstract. Sion’s generalization [2] was proved by the aid of Helly’s theorem and the KKM theorem due to Knaster et al. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the Theorem: Sion’s Minimax Theorem Let A and Z be convex, compact spaces, and. 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. Download to read the full chapter text. , Paris 248, 2698–2699 (1959; Zbl 0092. SIAM Journal on Optimization 33 (4):2885-2908. Acta Sci. 47 (2010), No. [3]. This paper proves the following theorem. ," Pacific Journal of Mathematics, Pacific J. Minimax Theorem CSC304 - Nisarg Shah 26 •We proved it using Nash’s theorem heating. 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 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. Let C X be nonempty and convex, and let D Y be nonempty, weakly compact and con-vex. Let X be a compact Hausdorfl space and let $(Y,A)$ be a measurable space. Lemma 1. von Neumann (8) proved his theorem for simplexes by reducing the problem to the 1-dimensional cases. First, we use Sion's minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Published 1995. 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 proof is based on a result of Victor Klee [ 9 ] 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. A general minimax theorem. 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 Fan-Browder fixed point theorem for multi-valued mappings. 8 (1), 171-176, (1958) Include: Citation Only. 47. 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. Receive erratum alerts for this article. We consider the general form of (P) in geodesic metric spaces, i. d by John von Neumann in the paper Zur Theorie Der Gesellschaftsspiele. I would suggest to use the Frank-Wolfe algorithm described here. S. A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. A SIMPLE PROOF OF THE SION MINIMAX THEOREM. 5. Oct 21, 2023 · Sion’s Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm. Compactness and dimensionality. 1137/22M1505475. The result is based on an intersection theorem which may be of interest Apr 5, 2015 · The Sion–von Neumann minimax theorem itself can be proved by simple topological arguments using connectedness instead of convexity. This work answers a question of Professor Granas regarding the logical relationship between the Elementary KKM theorem and the Sion–von Neumann minimax theorem. We suppose that X and Y are nonempty sets and f: X × Y → R. 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. Bull. ON GENERAL MINIMAX THEOREMS. 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. <1> Theorem. 7. However their proofs depend on topological tools such as Brouwer fixed point theorem or KKM theorem. Proof. : An Extension of Sion’s Minimax Theorem with an Application to a Method for Constrained Games. Convex optimization and strong duality. SION. 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. ) Quasiconvex and. As applications, we obtain an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities. 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. Sion's generalization (7) was proved by the aid of Helly's theorem and the KKM theorem due to…. On general minimax theorems. Oct 5, 2016 · 1. 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. minimax theorem or linear programming duality. M VβN V6Λ' μβ M There have been several generalizations of this heorem2. Fenchel-Rockafellar duality problem: Show that weak duality holds, i. Introduction. Our proofs rely on two innovations over the classical approach of using Von Neumann's minimax theorem or linear programming duality. Possibly also some handwritten notes from other lectures by Millar. 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 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. Main theorem: Minimax in Nonlinear Geometry In Euclidean space, Sion’s minimax theorem guarantees strong duality for convex-concave minimax problems. such as the KKM principle [4, x8. 2). The paper presents a self 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). Dec 24, 2016 · On a minimax theorem: an improvement, a new proof and an overview of its applications. 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 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 We also prove an improved version of Impagliazzo's hardcore lemma. 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 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. Next, Simons [ 4] showed different kinds of minimax theorems, and Li Minimax Theorems. 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 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. 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 Sep 30, 2010 · Content may be subject to copyright. 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 and Their Proofs. That is, you can interchange min and max. Korean Math. It states: Let be a compact convex subset of a linear topological space and a convex subset of a linear topological space. 8, Iss: 1, pp 171-176. Nikaido-Sion version of the minimax theorem which is accessible to students in anˆ undergraduate course in game theory. f(a, z). 28 Feb 1958 - Pacific Journal of Mathematics (Mathematical Sciences Publishers) - Vol. Let f be a real-valued function defined on K C such that. Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. 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. VON NEUMANN MINIMAX THEOREM Theorem: Let A be a m×n matrix representing the payoff matrix for a two-person, zero-sum game. Zhang and Suvrit Sra}, journal={SIAM Journal on Optimization}, year={2022}, url={https://api Mar 31, 2019 · On compactness in Sion's minimax theorem. All have their bene ts and additional features: (1) The original proof via Brouwer's xed point theorem [4, x8. Let f f be a real-valued function on X × Y X × Y such that. Apr 30, 2015 · Sion's minimax theorem. R. e. f(x, ⋅) f ( x, ⋅) is upper semicontinuous and quasi-concave on Y Y for each x ∈ X x ∈ X. 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 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. g. 4. Subscribe to Project Euclid. Maurice Sion "On general minimax theorems. It was rst introduc. Sehie P ark. Abstract 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. 3] and more re ned subsequent algebraic-topological treatment. 5, pp. Determining whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. Expand. Theorem 16. Von Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero 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]. The purpose of this note is to present an elementary proof for Sion's minimax theorem. 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 On general minimax theorems. if x is a feasible solution of P= minfhc;xijAx bgand y is a feasible Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuous function is stated as follows. 2010. 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 Oct 1, 2010 · Then, we give some generalized minimax inequalities for vector-valued functions by means of the generalized KKM theorem. View. Thus, they are equivalent. 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. Prokhorov theorem on non Polish spaces. In this section, we establish an analog of Sion’s theorem in geodesic metric spaces. There have been several Hartung, J. Share ON GENERAL MINIMAX THEOREMS MAURICE SION 1. Minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables. An example of such a game is a We present a topological minimax theorem (Theorem 2. Berge [C. MAURICE. Feb 1, 2018 · A Simple Proof of Sion's Minimax Theorem Jürgen Kindler Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. October 2023. Dec 3, 2013 · Abstract. $f(x, \cdot)$ is upper semicontinuous and quasi-concave on $Y$ for each $x \in X$. Abstract. quasiconcave are weaker conditions than convex and concave respectively. 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. : A × Z → R. Szeged, 42, 91–94 (1980) MathSciNet Google Scholar The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. 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] Jul 20, 2018 · Minimax theorems have important applications in optimization, convex analysis, game theory and many other fields. As LinAlg 's answer, by Sion minimax theorem, if one of x or f belongs to a bounded set, the strong duality holds. We describe in detail Kakutani's proof of the minimax theorem 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. von Neumann [1] proved his theorem for simplexes by reducing the problem to the one-dimensional cases. The result is based on an intersection theorem which may be of interest 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): Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. If f(a, ·) is upper semicontinuous and quasiconcave on Z ∀a ∈ A and. 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. de Pages 356-358 | Published online: 01 Feb 2018 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. Maurice Sion. Generally solving saddle point problems are hard. 8 (1): 171-176 (1958). e. , p≥−d . •Useful for proving Yao’s principle, which provides lower bound for randomized algorithms •Equivalent to linear programming duality John von Neumann March 1958. $$. (2) Tucker's proof of T. 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. Soc. Dec 15, 2023 · Abstract: The work studies cooperative decentralized constrained POMDPs with asymmetric information. Authors: Oct 1, 2003 · A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. There have been several 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. Pacific J. 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. 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. Min-max theorem. LEMMA 1. Let $f$ be a real-valued function on $X \times Y$ such that 1. Let ̃ beavalueof suchthat ̃ =argmax ∈ min ∈ ( , , ̃ )=argmin ∈ max ∈ . Format: March 1958. 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. Math. In a recent paper, Kindler [4 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. 1037. Proof for the theorem. We take a step towards understanding certain nonconvex-nonconcave minimax problems that do remain tractable Pacific Journal of Mathematics, A Non-profit Corporation. 0. 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. 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. 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. Later, John Forbes Nash Jr. from Wikipedia, Let $X$ be a compact convex subset of a linear 1958 On general minimax theorems. native proof of the minimax theorem using Brouwer's xed point theo-rem. The proof of Krein-Milman Theorem and the reason behind 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]. 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. Simons. 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. ABOUT. Abstract The article presents a new proof of the minimax theorem. In particular, you don't need $\mathcal Q$ to be compact. Let us recall the following definition where, for a mixed strategy pair (x,y), we define V(x,y) := Pm i=1 Pn j Downloadable (with restrictions)! About a symmetric three-players zero-sum game we will show the following results. ve reproduced a variety of proofs of Theorem 2. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. 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. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. 2. In doing that, a key tool was a partial ray. 1037–1040. The minimax theorem results in numerous applications and many of them are far from being obvious. Let f be a real-valued A, is equivalent to Sion™s minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A. Then infx ∈K supy ∈C f(x y , ) = supy ∈C infx ∈K f(x , y ). : A Simple Proof for von Neumann’ Minimax Theorem. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as "forecasting algorithms" evaluated by a proper scoring rule. 3. 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. 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. Minimax theorem Sources: Kneser? Sion? See Millar (1983, page 92). Then the game has a value and there exists a pair of mixed strategies which are optimal for the two players. DOI 10. The result is based on an intersection theorem which may be of interest on its own right. Finally, we prove a cone-saddle point theorem as an application of our results. The strong duality theorem states these are equal if they are bounded. Sion’s minimax theorem (Sion (1958), Komiya (1988), Kindler (2005)) for a continuous function is stated as follows. , M, Nare geodesic metric spaces, f| 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。 Sep 30, 2010 · The von Neumann-Sion minimax theorem is fundamental in convex analysis and in game theory. , 103(2), 401–408 (1982) MathSciNet Google Scholar Joo, L. If the feasible sets for x and f are polytope, you can easily solve the Oct 24, 2017 · About a symmetric multi-person zero-sum game we will show the following results. tu-darmstadt. The method of our proof is inspired by the proof of [4, Theorem 2]. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. M VβN V6Λ' μβ M. Mathematics. 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 Feb 10, 2024 · Sion's minimax theorem is usually stated with a condition that one of the sets is compact, e. max min f(μ, v) = min max f(μ, v) . 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. However, not only from purely mathe-matical or theoretical motivation, but also from more practical motivation we deal with the theorem. 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 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. 1, Exer. 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. ul tm uq vh ib dk ok yf zt se