# 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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### 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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### "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 ...
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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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### 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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### 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 ...
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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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### 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 vote
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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 ...
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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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### 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 ...
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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 $\mathsf{CH}$, ...
260 views

### 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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### 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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### 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. ...
1 vote
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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 ...
1 vote
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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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1 vote
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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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### 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 ...
$A$ 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 $5$ ...