Some new versions of an old game

Vladimir Vladimirovich Tkachuk

Displaying similar documents to “Some new versions of an old game”

Infinite $G_{δ}$ -games with imperfect information

D. Blackwell (1969)

Applicationes Mathematicae

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Topological games and product spaces

Salvador García-Ferreira, R. A. González-Silva, Artur Hideyuki Tomita (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with the product of spaces which are either $𝒢$ -spaces or $𝒢_{p}$ -spaces, for some $p \in ω^{*}$ . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$ -spaces, and every $𝒢_{p}$ -space is a $𝒢$ -space, for every $p \in ω^{*}$ . We prove that if ${X_{μ} : μ < ω_{1}}$ is a set of spaces whose product $X = \prod_{μ < ω_{1}} X_{μ}$ is a $𝒢$ -space, then there is $A \in {[ω_{1}]}^{\leq ω}$ such that $X_{μ}$ is countably compact for every $μ \in ω_{1} ∖ A$ . As a consequence, $X^{ω_{1}}$ is a $𝒢$ -space iff $X^{ω_{1}}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$ -space,...

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle, G. Pieri (2010)

Bollettino dell'Unione Matematica Italiana

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Consider a M-player game in strategic form $G=(X_{1},\cdots,X_{M},g_{1},\cdots,g_{M})$ where the set $X_{i}$ is a closed interval of real numbers and the payoff function $g_{i}$ is concave and differentiable with respect to the variable $x_{i}\in X_{i}$ , for any $i=1,\cdots,M$ . The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence,...

More on the Ehrenfeucht-Fraisse game of length ω₁

Tapani Hyttinen, Saharon Shelah, Jouko Vaananen (2002)

Fundamenta Mathematicae

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By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $E F G_{ω ₁} (,)$ , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and $E F G_{ω ₁} (,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ ₀} < 2^{ℵ ₃}$ , T is a countable...

Some applications of the point-open subbase game

D. Guerrero Sánchez, Vladimir Vladimirovich Tkachuk (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given a subbase $𝒮$ of a space $X$ , the game $P O (𝒮, X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$ -th move, a point $x_{n} \in X$ and a set $U_{n} \in 𝒮$ such that $x_{n} \in U_{n}$ . The game stops after the moves ${x_{n}, U_{n} : n \in ø}$ have been made and the player $P$ wins if $⋃_{n \in ø} U_{n} = X$ ; otherwise $O$ is the winner. Since $P O (𝒮, X)$ is an evident modification of the well-known point-open game $P O (X)$ , the primary line of research is to describe the relationship between $P O (X)$ and $P O (𝒮, X)$ for a given subbase $𝒮$ . It turns out that, for any subbase $𝒮$ , the player $P$ has a winning...

On remote points, non-normality and $π$ -weight $ω_{1}$

Sergei Logunov (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show, in particular, that every remote point of $X$ is a nonnormality point of $β X$ if $X$ is a locally compact Lindelöf separable space without isolated points and $π w (X) \leq ω_{1}$ .

Displaying similar documents to “Some new versions of an old game”

Infinite G δ -games with imperfect information

Topological games and product spaces

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

More on the Ehrenfeucht-Fraisse game of length ω₁

Some applications of the point-open subbase game

On remote points, non-normality and π -weight ω 1

Infinite $G_{δ}$ -games with imperfect information

On remote points, non-normality and $π$ -weight $ω_{1}$