# 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.

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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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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 $...

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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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: ...

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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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### 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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### 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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### Is each Choquet topological group strong Choquet?

A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.
It is known that a metrizable space $X$ is
$\bullet$ Choquet if and only if it ...

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### final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...

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### A community effort: equilibrium in quitting games [closed]

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...

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### Continuous limits of Hex on larger and larger boards

Consider the game of Hex on a variable sized $n \times n$ board. Remap each board to be a discrete subset of the unit square $[0,1]^2$. Optimal games are now finite sequences on the unit square. We ...

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### How to describe the common boundaries between regions in a infinite Sudoku?

This relates to the answer to a question "Who wins two player sudoku?" and this awesome blog.
A Sudoku can be $N \times N$ where $\sqrt{N}$ is a natural number because $N \times N / \sqrt{N} \times \...

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### Is there a position in infinite Go for which the life of a particular stone has transfinite game value?

As follow up to Checkmate in $\omega$ moves?, we can ask the same question about go. Is there a position on a $\mathbb Z \times \mathbb Z$ goban such that either black can kill a white group, but ...

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### The Sudoku game: Solver-Spoiler variation

Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt ...

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### The arithmetic progression game and its variations: can you find optimal play?

Consider the arithmetic progression game, a two-player game of
perfect information, in which the players take turns playing
natural numbers, or finite sets of natural numbers, all distinct,
and the ...

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### A game on integers

$A$ and $B$ take turns to pick integers: $A$ picks one integer and then $B$ picks $k > 1$ integers ($k$ being fixed). A player cannot pick a number that his opponent has picked. If $A$ has $5$ ...

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### The Axiom of Determinacy and the Banach-Mazur game

The Wikipedia article on the Axiom of Determinacy (AD) claims:
Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is ...

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### Does knight behave like a king in his infinite odyssey?

The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ...

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### On the topological complexity of the set of winning strategies for Gale-Stewart Games

Given a set $A \subseteq \omega^\omega$, let $G_A$ denote the Gale-Stewart game with payoff set $A$ (so player $I$ wants the real built over the course of play to be in $A$ and player $II$ wants it ...

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### If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?

Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationery st.) in $BM(X)$. Then can ...

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### Products and Gale-Stewart games

For the purpose of this post, I will say that the Gale-Stewart game is the infinite two-player game of perfect information where players I and II alternate playing natural numbers, with I going first. ...

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### Undetermined copy/diagonalize games without CH

This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've ...

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### Choosing subsets of $\mathbb R$ of cardinality $\frak c$, who wins?

Consider the following infinite game: two players, I and II, are alternating and choosing a descending sequence of subsets of $\mathbb R$ of cardinality $\frak c$, so I chooses a set $A_1\subseteq\...

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### Determinacy of (infinite, possibly loopy) combinatorial games

I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...

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### On the Theory of Infinite Step Processes of Sequential Decision Making

On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future ...

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### Infinite-dimensional hex

Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...

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### A game on sets of reals

A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all ...