Von Neumann Minimax Theorem Research Articles

Curvature on graphs via equilibrium measures

AbstractWe introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by have diameter bounded by (a Bonnet–Myers theorem), that implies that has constant curvature (a Cheng theorem) and that there is a spectral gap (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.

Maradona plays Minimax

This paper tests the theory of mixed strategy equilibrium using Maradona's penalty kicks during his lifetime professional career. The results are remarkably consistent with equilibrium play in every respect: (i) Maradona's scoring probabilities are statistically identical across strategies; (ii) His choices are serially independent. These results show that Maradona's behavior is consistent with Nash's predictions, specifically with both implications of von Neumann's Minimax Theorem.

On Farkas' lemma and related propositions in BISH

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.

Maradona Plays Minimax

This paper tests the theory of mixed strategy equilibrium using Maradona's penalty kicks during his lifetime professional career. The results are remarkably consistent with equilibrium play in every respect: (i) Maradona's scoring probabilities are statistically identical across strategies; (ii) His choices are serially independent. These results show that Maradona's behavior is consistent with Nash's predictions, specifically with both implications of von Neumann's Minimax Theorem.

Consistent Probabilistic Social Choice

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (1984). Fishburn's function returns so-called maximal lotteries, that is, lotteries that correspond to optimal mixed strategies in the symmetric zero-sum game induced by the pairwise majority margins. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming. [web URL: http://onlinelibrary.wiley.com/doi/10.3982/ECTA13337/abstract]

Some of Sion’s heirs and relatives

If one adds one extra assumption to the classical Knaster– Kuratowski–Mazurkiewicz (KKM) theorem, namely that the sets F i are convex, one gets the “Elementary” KKM theorem; the name is due to A. Granas and M. Lassonde (1995) who gave a simple proof of the Elementary KKM theorem and showed that despite being “elementary,” it is powerful and versatile. It is shown here that this Elementary KKM theorem is equivalent to Klee’s theorem, the Elementary Alexandroff– Pasynkov theorem, the Elementary Ky Fan theorem and the Sion–von Neumann minimax theorem, as well as a few other classical results with an extra convexity assumption; hence the adjective “elementary.” 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.

Games and Logic

The idea behind these games is to obtain an alternative characterization of logical notions cherished by logicians such as truth in a model, or provability (in a formal system). We offer a quick survey of Hintikka's evaluation games, which offer an alternative notion of truth in a model for first-order langauges. These are win-lose, extensive games of perfect information. We then consider a variation of these games, IF games, which are win-lose extensive games of imperfect information. Both games presuppose that the meaning of the basic vocabulary of the language is given. To give an account of the linguistic conventions which settle the meaning of the basic vocabulary, we consider signaling games, inspired by Lewis' work. We close with IF probabilistic games, a strategic variant of IF games which combines semantical games with von Neumann's minimax theorem.

A Simpler Proof of the Von Neumann Minimax Theorem

This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. 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 optimization or game theory.

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.

The Monty Hall problem is not a probability puzzle* (It's a challenge in mathematical modelling)

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice?The answer is ‘yes’ but the literature offers many reasons why this is the correct answer. This article argues that the most common reasoning found in introductory statistics texts, depending on making a number of ‘obvious’ or ‘natural’ assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann's minimax theorem from game theory, rather than in Bayes’ theorem.

On the equivalence of weak learnability and linear separability: new relaxations and efficient boosting algorithms

Boosting algorithms build highly accurate prediction mechanisms from a collection of low-accuracy predictors. To do so, they employ the notion of weak-learnability. The starting point of this paper is a proof which shows that weak learnability is equivalent to linear separability with l 1 margin. The equivalence is a direct consequence of von Neumann's minimax theorem. Nonetheless, we derive the equivalence directly using Fenchel duality. We then use our derivation to describe a family of relaxations to the weak-learnability assumption that readily translates to a family of relaxations of linear separability with margin. This alternative perspective sheds new light on known soft-margin boosting algorithms and also enables us to derive several new relaxations of the notion of linear separability. Last, we describe and analyze an efficient boosting framework that can be used for minimizing the loss functions derived from our family of relaxations. In particular, we obtain efficient boosting algorithms for maximizing hard and soft versions of the l 1 margin.

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.

The Uncertainty Principle of Game Theory

In 1928, von Neumann's minimax theorem [33] on two-player, zero-sum games showed that, in general, optimal strategies of players are random, even if the rules of the game do not involve chance. In this paper we investigate the degree of uncertainty or randomness possible in optimal strategies. We show that Heisenberg's uncertainty principle of physics corresponds to a similar principle for zero-sum games and derive an explicit lower bound for the uncertainty of a game. We also demonstrate that this lower bound is a constraint on the set of possible optimal strategies and apply it to develop a simple but effective method for finding an approximately optimal strategy. A two-player, finite, zero-sum game is traditionally described by a payoff matrix A = iatj), where the rows correspond to the strategies of the first player (P) and the columns correspond to the strategies of the second player (Q). (The reason for these particular player names will become apparent later.) The individual strategies and j are called pure strategies. The real number ai} is simply the amount of money or payoff that the first player receives from the second player if the first player chooses strategy / and the second player chooses strategy /'; the chosen strategy of the opponent is unknown to each player. If

Good Randomized Sequential Probability Forecasting is Always Possible

SummaryBuilding on the game theoretic framework for probability, we show that it is possible, using randomization, to make sequential probability forecasts that will pass any given battery of statistical tests. This result, an easy consequence of von Neumann's minimax theorem, simplifies and generalizes work by earlier researchers.

Remarks on acyclic versions of generalized von Neumann and Nash equilibrium theorems

A fixed-point theorem on compact compositions of acyclic maps on admissible (in the sense of Klee) convex subset of a t.v.s. is applied to obtain a cyclic coincidence theorem for acyclic maps, generalized von Neumann type intersection theorems, the Nash type equilibrium theorems, and the von Neumann minimax theorem. Our new results generalize earlier works of Lassonde [1], Simons [2], and Park [3,4].

Information Capacity of Channels with Partially Unknown Noise. II. Infinite-Dimensional Channels

Information capacity is considered for a communication channel in which the noise is the sum of a known Gaussian component and an independent component with unknown statistical distributions. A lower bound on capacity is sought; the unknown noise component is thus assumed to be under the control of an adversary---a jammer. The problem is modeled as a zero-sum two-person game with mutual information as the payoff function. Appropriate constraints are determined on the transmitted signal and the unknown noise component. Although the usual conditions sufficient for application of the general form of the von Neumann minimax theorem are shown not to hold, a solution is obtained for the game: a saddle value, saddle point, and minimax strategy for the jammer are obtained. The essential effect of jamming is to convert the infinite-dimensional channel into a finite-dimensional channel having the same constraints, with the dimensionality depending upon the problem parameters: the covariance of the known Gaussian noise component and the constraints on the transmitted signal and the unknown noise component.

On a form of KKM principle and SupInfSup inequalities of von Neumann and of Ky Fan type

A version of KKM principle is considered together with its application to SupInfSup inequalities of von Neumann and of Ky Fan type. These inequalities which are respectively generalizations of von Neumann minimax theorem and minimax inequality of Ky Fan provide new light in obtaining either new results or new proofs for known results.

Non-compact sets with convex sections, II

This paper is a continuation of previous work ( Pacific J. Math. 119 (1985)) and contains two results on sets with convex sections involving four sets. As direct applications, we formulate generalizations of the von Neumann minimax theorem.

Experiments with a disc machine to determine possible influence of surface finish on gear-tooth performance : B. A. Shotter (British Thomson-Houston Co. Ltd.) — Proc. Intern. Conf. on Gearing, London, 1958, Inst. Mech. Engrs., London, (in the press)

This paper tests the theory of mixed strategy equilibrium using Maradona's penalty kicks during his lifetime professional career. The results are remarkably consistent with equilibrium play in every respect: (i) Maradona's scoring probabilities are statistically identical across strategies; (ii) His choices are serially independent. These results show that Maradona's behavior is consistent with Nash's predictions, specifically with both implications of von Neumann's Minimax Theorem.

An elementary proof of von Neumann's minimax theorem

An elementary proof of von Neumann's minimax theorem