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Besides Game Theory for Social Sciences (von Neumann, Morgenstern, Nash) and Nim-like Games (Berlekamp, Conway, Guy) there are at least three more flavors of game theory. … Infinitely Long Discrete Games or "Polish games" introduced by Mazur, Banach, Ulam, Steinhaus, Mycielski, with applications to descriptive set theory and general topology. …

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. … The Banach-Mazur game on $\mathbb P$ is $\omega$-strategically closed when there is a strategy $\sigma$ such that II wins whenever II plays according to $\sigma$, meaning if the sequence of plays so far …

Now, if $\tau$ was a winning tactic for II in the Banach-Mazur game on $\mathbb P$, there would be a different way to define $\prec$ with properties 1.-5.: let $q\prec p$ iff $q < \tau(p)$. …

Intuitively, a sufficiently $\mathbb{P}$-generic filter yields a sufficiently good stationary Banach-Mazur strategy for player $2$ (and both stationarity and player-$2$-winning are appropriately WLOG here … There is some set $A\subseteq\omega^\omega$ whose Banach-Mazur game $G_{BM}(A)$ is determined such that any winning strategy for $G_{BM}(A)$ computes $\alpha$. …

The answer to the topological version of Banach-Mazur games is negative, proved by Gabriel Debs in 1985: Debs, Gabriel, Stratégies gagnantes dans certains jeux topologiques (Winning strategies in certain … www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/126/1/104687/strategies-gagnantes-dans-certains-jeux-topologiques The paper is in French, but here is the MathScinet review: A Banach-Mazur …

In this paper, we look at a conjecture of Telgarsky concerning the topological Banach-Mazur game. … It is interesting to me that the same principle used to resolve this conjecture for the topological Banach-Mazur game seems so relevant to Monroe's question for the poset Banach-Mazur game. …

As JDH observed, there are topological spaces $X$ for which Player II has a winning strategy (and in fact a winning 2-tactic), but no winning tactic in the topological Banach-Mazur game. … It remains to be shown that winning strategies in the poset game produce winning strategies in the topological game. …

Suppose Player II has a winning strategy $\sigma$ in the Banach–Mazur game on a poset $P$. …

player I's move in that game, compute the response $r$ of $\sigma$ to that play, and then extend $r$ to a chosen condition $r^+$ whose label codes the sequence $p^{\to},q$. … $\Box$ This answer amounts to the partial-order analogue of the Debs result I mentioned in my other answer, which he proved for the topological Banach-Mazur games. …

Given $\mathcal{A}\subseteq\omega^\omega$, the Banach-Mazur game with payoff set $\mathcal{A}$ consists of players $1$ and $2$ alternately playing nonempty finite strings of naturals with player $1$ winning … On the other hand, Banach-Mazur determinacy principles are relatively weak on a set-theoretic level: "Every Banach-Mazur game is determined" adds no consistency strength to $\mathsf{ZF+DC}$. …

Kunen), then, using the Banach-Mazur game, Player I has a winning strategy in $\textsf{BM}(X^{\omega})$. … I was trying to show that Player I has a winning strategy in the game $\textsf{MB}(X^{\kappa})$, but unfortunately I still haven't got it. …

strategy in $\textsf{BM} (\prod_{i\in I}X_{i}, \prod_{i\in I} \tau_{i} ).$ Where $\textsf{BM}(X)$ denotes the Banach-Mazur game played on a topological space $X$. … Remember that The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. …

Player $II$ wins the run $\langle U_0, V_0, U_1, V_1, \dots \rangle$ of the Banach-Mazur game on $X$ iff $\bigcap_{n\in\omega}V_n \not = \emptyset$. … Is there a description of the class of spaces in which $II$ has a winning strategy in the Banach-Mazur game, in terms of continuous maps? …

The Banach-Mazur game (or Choquet game) played on $X$ and the Banach-Mazur game played on $\mathcal{K}(X)$ are equivalents. …

So we can separately ask: Version 2: Does ZF prove "There is an un-quasidetermined Banach-Mazur game?" … Version 3: Is the statement "Every Banach-Mazur game is weakly determined" consistent with ZF? …