Infinite zero-sum two-person games

Yanovskaya, Elena

doi:10.1007/bf01085016

Cited by 12 publications

(7 citation statements)

References 44 publications

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“…Since π A 1 , π A 2 ∈ P(A) and α ∈ (0, 1) are arbitrary, then (22) implies that the worst-loss functionĉ ♯ is convex on P(A). Statement (a) is proved.…”

Section: Remark 13mentioning

confidence: 99%

“…Then the game {A, B, c} satisfies the conditions of Theorem 18 and v = 0. Example 2 admits the following interpretation in the form of a simple game of timing (see Yanovskaya [22,Section 6]) with noncompact decision sets. Two teams work on a project consisting of two independent tasks, each performed by one of the teams.…”

Section: The Existence Of a Lopsided Valuementioning

confidence: 99%

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Continuity of equilibria for two-person zero-sum games with noncompact action sets and unbounded payoffs

2017

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This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.

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“…Since π A 1 , π A 2 ∈ P(A) and α ∈ (0, 1) are arbitrary, then (22) implies that the worst-loss functionĉ ♯ is convex on P(A). Statement (a) is proved.…”

Section: Remark 13mentioning

confidence: 99%

Section: The Existence Of a Lopsided Valuementioning

confidence: 99%

Continuity of equilibria for two-person zero-sum games with noncompact action sets and unbounded payoffs

2017

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“…This theorem and Corollary 3.4 also describe the properties of the solution sets under these conditions. In general, an infinite game may not have a value; see Yanovskaya (1974, p. 527) and the references to counterexamples by Ville, by Wald, and by Sion and Wolfe cited there. Therefore, some additional conditions for the existence of a value and solutions are needed.…”

Section: Main Results and Discussionmentioning

confidence: 99%

“…Two players select nonnegative numbers a and b, and Player I pays the amount of c(a, b) = 𝜑(a − b) to Player II. For example, if the player, who selects the larger number wins, that is, 𝜑(a − b) = I(a > b), this game does not have a value; see for example, Yanovskaya (1974). We apply the results of our article to such games.…”

Section: Number Guessing Gamementioning

confidence: 99%

Solutions for zero‐sum two‐player games with noncompact decision sets and unbounded payoffs

Feinberg

Kasyanov

Zgurovsky

2023

Naval Research Logistics

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This article provides sufficient conditions for the existence of solutions for two-person zero-sum games with inf/sup-compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this article imply several classic facts. The article also provides sufficient conditions for the existence of a value and solutions for each player. The results of this article are illustrated with the number guessing game.

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“…Under these assumptions, we show that the minmax and maxmin values of the linearized game in Equation ( 4) are equal to each other. Such results are known as minimax theorems and have been studied in the past [24,25,26]. However, unlike past works that rely on fixed point theorems, we rely on a constructive learning-style proof to prove the minimax theorem, where we present an algorithm which outputs an approximate NE of the statistical game.…”

Section: Minimax Estimation and Statistical Gamesmentioning

confidence: 99%

Learning Minimax Estimators via Online Learning

Gupta,

Suggala,

Prasad

et al. 2020

Preprint

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We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.

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Infinite zero-sum two-person games

Cited by 12 publications

References 44 publications

Continuity of equilibria for two-person zero-sum games with noncompact action sets and unbounded payoffs

Continuity of equilibria for two-person zero-sum games with noncompact action sets and unbounded payoffs

Solutions for zero‐sum two‐player games with noncompact decision sets and unbounded payoffs

Learning Minimax Estimators via Online Learning

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