Questions tagged [infinite-games]
Infinite games. Combinatorial game theory for infinite two-player games of perfect information. Open games, clopen games. Determinacy. Transfinite game values. Topological games.
76 questions
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An infinite game played on $\{0,1\}^\mathbb{N}$ in which the players must avoid creating an algebraic dependency
This mathematical game occurred to me. It may be quite basic for experts, but it seems to lead to some interesting questions.
Turns are indexed by elements of $\mathbb{N} = \{1,2,\ldots, \}$. In turn $...
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13
votes
1
answer
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When determinacy pulls back
Let $f:\mathbb{R}\rightarrow\omega_1$ be the "usual" surjection (send $r$ to the ordinal coded by $r$ according to some fixed reasonable coding system if such an ordinal exists, and to $0$ ...
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11
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For which sets do Arthur and Nimue have a winning strategy in this game (“communicate a bit”)?
TL;DR: I define a three-player game (Arthur, Nimue, Merlin) where Nimue is shown a hidden bit $b$ chosen by Merlin and tries to communicate it to her ally Arthur, but Arthur must act computably while ...
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3
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1
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In a nonempty Baire space, the first player does not have a winning strategy in the Choquet Game
Suppose $X$ is a nonempty topological space. The Choquet Game on $X$ is a two-player infinite game defined as follows. The first player choose a nonempty open subset $U_0\subseteq X$. The second ...
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9
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0
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Does cocountable topology on $\mathbb{R}$ have the $2$-Markov Menger property?
References:
Applications of limited information strategies in Menger’s game by Clontz
Almost compatible functions and infinite length games by Clontz and Dow
Def. 3.7 of [1] $\mathcal{A}(\kappa)$ ...
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4
votes
1
answer
170
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Are W-spaces with countable pseudocharacter first countable?
Cross-post of a question originally asked by Almanzoris on Mathematics Stack Exchange.
A topological space $X$ is called W-space if P1 has a winning strategy at each point $x \in X$ for the following ...
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9
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1
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Who wins the Scrambler-Solver game for infinitary Rubik's cubes?
Given a suitable infinitary analogue $\mathcal Q$ of the Rubik's cube (as developed below), consider the two player game played between the Scrambler and the Solver wherein the Scrambler scrambles the ...
- 1,860
6
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0
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199
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A pre-Ramsey set that is not Ramsey
$\newcommand{\N}{\mathbb{N}}$
Recall that given a $A \in [\N]^\omega$ and $a \in [A]^{<\omega}$, the Ellentuck neighbourhood is defined to be:
$$
[a,A] := \{B \in [\N]^\omega : a \sqsubseteq B \...
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8
votes
0
answers
191
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Which sentences are "strategically preserved"?
Below, everything is first-order.
Say that a sentence $\varphi$ is strategically preserved iff player 2 has a winning strategy in the following game:
Players 1 and 2 alternately build a sequence of ...
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12
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0
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646
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Connection properties of a single stone on an infinite Hex board
This includes a series of questions.
One of the most typical examples is shown as the picture below.
An half-infinite Hex board with an one row of black stones. Black stones are separated by one ...
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4
votes
1
answer
580
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"Infinity": A card game based on prime factorization and a question
I have been developing a card game called "Infinity", which involves a unique play mechanic based on card interactions. In this game, each card displays a set of symbols, and players match ...
15
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3
answers
1k
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Examples of concrete games to apply Borel determinacy to
I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves ...
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6
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2
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Embeds in a topological W-group, or a W-space that embeds in a topological group?
In Theorem 3.11 of Tkachuk - A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace it's shown that if a compact Hausdorff space embeds in a topological W-...
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0
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Are gaps and loopy games interchangeable in the Surreal Numbers?
The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
6
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0
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245
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Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
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18
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1
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Let $R, S$ be infinite sets. Alice and Bob will play a game over $R\times S$
Let $R, S$ be infinite sets. Alice and Bob will play a game over $R\times S$.
Alice's color is red and Bob's color is blue. In each step, for each $s\in S$, a player will choose finitely many ...
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1
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0
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95
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EF-games with scrambling
This question is motivated both by the notion of zero-knowledge proofs and by general curiosity about versions of the infinitely-long Ehrenfeucht-Fraisse game which don't trivialize (= Duplicator win ...
- 19.6k
7
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0
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263
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Strengthening Determinacy in constructive set theory?
Recall how games work. Let $X$ be a set (the "game space") and $\alpha$ an ordinal (the "game clock"). Alice and Bob take turns naming elements of $X$. We write them down in order ...
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4
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2
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223
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Does there exist a non-hemicompact regular space for which the 2nd player in the $K$-Rothberger game has a winning Markov strategy?
Assume spaces are regular.
A space is $\sigma$-compact if and only if the second player in the Menger game has a winning Markov strategy (relying on only the most recent move of the opponent and the ...
- 1,522
10
votes
0
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363
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Determinacy coincidence at $\omega_1$: is CH needed?
This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
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2
votes
0
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317
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Is determinacy of (some) very long open games consistent?
For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...
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7
votes
0
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Is this determinacy principle consistent?
Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):
If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
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5
votes
1
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237
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Are the completeness Games $G_{\lambda+1}(P)$ and $G_{\lambda^+}(P)$ equivalent for INC?
The completeness game $G_{\gamma}(P)$ for a partial order $P$ has players COM and INC play alternatingly and descendingly elements of $P$ with player INC playing first and player COM playing at limit ...
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3
votes
0
answers
385
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Poker with infinite stack size
In two-player No Limit Texas Hold 'Em (NLHE), the optimal strategy depends on the "effective stack size," that is maximum amount of money held by one of the players (in the sequel I'll just ...
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17
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6
answers
1k
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Strategic vs. tactical closure
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$. Player II wins when ...
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24
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2
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1k
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What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?
Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration.
Red wins after infinite play, ...
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2
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0
answers
176
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Weakening of open determinacy for uncountably long games
For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$."
Say that a ...
- 19.6k
3
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0
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199
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A p-point game with infinitely many ultrafilters
The following game-theoretic characterization of p-points is well known:
Theorem A. An ultrafilter $D$ on the set $\omega$ of natural numbers is a p-point if and only if player I does not have a ...
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28
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7
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7k
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Why is game theory formulated in terms of equilibrium instead of winning strategies?
Game theory, on the outset, seems to invite the questions,
"what can I do to win" or "how do I beat my opponent?"
So many people who are not familiar with game theory look to game ...
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3
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1
answer
256
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Game versions of the tower number $\mathfrak t$
Let us recall that $\mathfrak t$ is the smallest cardinal $\kappa$ for which there exists a family $(T_\alpha)_{\alpha\in\kappa}$ of infinite subsets of $\omega$ such that
$\bullet$ for any ordinals $\...
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6
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1
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337
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Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game
Consider the following infinite perfect information game with two players (the name I gave in the title of the post it totally made up): at each round $i \in \omega$, player $\mathrm{I}$ picks a ...
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25
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4
answers
3k
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The Chocolatier's game: can the Glutton win with a restricted form of strategy?
I have a question about the Chocolatier's game, which I had
introduced in my recent answer to a question of Richard
Stanley.
To recap the game quickly, the Chocolatier offers up at each stage
a finite ...
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15
votes
1
answer
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An infinite game possibly due to Ernst Specker
I have a vague memory of an infinite game due to Ernst Specker with
the following properties:
(1) It is a two-person perfect information game, where the players
move alternately.
(2) The possible ...
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6
votes
1
answer
431
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Are there "very narrow" undetermined games?
I'd like to close a gap left open in an old question of mine; I've tweaked the terminology to be a bit nicer.
For a (boldface) pointclass $\Gamma$ and a payoff set $G\subseteq\omega^\omega$, say that $...
- 19.6k
2
votes
0
answers
191
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Banach–Mazur game and mappings
The Banach-Mazur game on a nonempty space $X$ is defined as follows: two players, $I$ and $II$, alternately choose nonempty open sets
\begin{matrix}
I & U_0 && U_1 && \cdots ...
- 115
7
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5
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Reference for graduate-level text or monograph with focus on "the continuum"
I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the ...
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4
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0
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229
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Infinite positions in 3D chomp
I've recently come back to investigating ordinal chomp. See A winning move for the first player in $3 \times 3 \times \omega$ Ordinal Chomp for a definition. I made a new discovery, that the position \...
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10
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0
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Undetermined Banach-Mazur games: beyond DC
This question is a follow-up to this one; see that question for the definition of Banach-Mazur games. There James Hanson showed that ZF+DC proves that there is an undetermined Banach-Mazur game; ...
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22
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1
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Undetermined Banach-Mazur games in ZF?
This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question.
Given a ...
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4
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1
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250
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Can I win this variant of the Banach-Mazur Game?
Suppose I play the following game against the Opponent. My moves are rational numbers $p_i$ and the Opponent's moves are real numbers $\epsilon_i>0$.
On turn $n+1$ the past move sequence is $...
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2
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1
answer
224
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Question about almost locally ccc and the Krom space
Definition 1. A family $\mathcal{B}$ of non-empty open sets in a topological space will be called $\pi$-base (or pseudo-base) if every non-empty open set contains at least one member of $\mathcal{B}$. ...
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6
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0
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246
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Spaces where the Banach-Mazur game is undetermined
Let $X$ be a non-empty topological space. The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I
and II play an inning per positive integer. In the $n$-th inning Player I ...
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28
votes
2
answers
1k
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Solution to simple mathematical game
Consider the following game (that I made up). Two players each attempt to name a target number. The first player begins by naming 1. On each subsequent turn, a player can name any larger number that ...
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5
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1
answer
683
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Banach-Mazur game and infinite products
Studying the article "Games that involve set theory or topology" of Marion Scheepers, I found the following result
Theorem 46 Let $\{(X_{i}, \tau_{i}) : i\in I \}$ be a family of topological spaces. ...
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0
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115
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Is there a well-posed definition of game on a graph? Or a well defined category of games on graphs?
All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions:
1) Is there a well-posed mathematical definition of game on ...
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1
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0
answers
243
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Winning strategies for a game on the natural numbers
Define $$F=\{(l_n, k_n)_{n=1}^t: t,l_n, k_n \in \mathbb{N}, l_1<\ldots <l_t, m_1<\ldots <m_t\}.$$ Suppose that I have a collection $G\subset F$, which is a set of ''good'' sequences.
...
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2
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0
answers
224
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An analytic game and Ramsey's theorem
Let $$P=\{(l_n, m_n)_{n=1}^t: l_n, m_n,t\in\mathbb{N}, l_1<\ldots <l_t, m_1<\ldots <m_t\}$$ and suppose that $T$ is some subset of $P$. Suppose that $T$ also has the following properties: ...
- 59
1
vote
2
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268
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Convergence and winning strategies
Suppose we have a set $B\subseteq 2^\omega\times\omega^\omega$ and a sequence $(x_n)$ in $2^\omega$ such that for each $n$, Player I (the one trying to get into the payoff set) has a winning strategy ...
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2
votes
1
answer
157
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Rothberger game and Meager in itself sets
On $(\mathbb{R}, \tau)$ the euclidean space of real numbers, we define a new topology by letting $\tau^{*}=\{X\subseteq \mathbb{R}: X=\emptyset \hspace{0.1cm}\mbox{or}\hspace{0.1cm}\mathbb{R}\setminus ...
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1
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Is there a three valued logic whose game semantics corresponds to potentially infinite games?
Consider game trees with the following properties:
Each node in the tree is one of the following:
Verifier Choice: Has one or more children
Falsifier Choice: Has one or more children
No Choice: Has ...
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