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Questions tagged [infinite-games]

Infinite games. Combinatorial game theory for infinite two-player games of perfect information. Infinitary aspects of common recreational games. Open games, clopen games. Determinacy. Transfinite game values. Topological games.

Infinite games, including

  • Combinatorial game theory for infinite two-player games of perfect information, such as infinite chess and infinite variants of other popular games.
  • Infinitary aspects of common recreational games, particularly when the associated game tree is infinite.
  • Open games, clopen games.
  • Determinacy.
  • Transfinite game values.
  • Topological games, such as Banach–Mazur game or Choquet game.
118 questions
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In the game of Repeat-a-Number, who wins?

I devised a game recently. There is a string of numbers, and each player extends the string by appending a number to the end based on the current last number of the string. The string starts as the single number $1.$ If the last number of the string…
mathlander
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25
votes
0 answers

"Infinito", a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to solve this game? How can I find the best move each…
Matteo Perlini
  • 351
24
votes
4 answers

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff the program halts? The rules are the same as…
coffeemachine
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21
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Good text to start studying topological games?

Topological games and some similar infinite games seem to be often used used as a tool in some areas of general topology, but also some other areas, such as Ramsey theory, filters, etc. Probably the best known are Banach-Mazur and Choquet game, but…
Martin Sleziak
  • 56,060
17
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1 answer

A simple game on infinite chessboard

Player $A$ chooses two queens and an arbitrary finite number of bishops on $\infty \times \infty$ chessboard and places them wherever he/she wants. Then player $B$ chooses one knight and places him wherever he/she wants (but of course, knight cannot…
Grešnik
  • 1,812
15
votes
1 answer

Is there an undetermined Banach-Mazur game in ZF?

Given a topological space $\mathcal{X}=(X,\tau)$, the Banach-Mazur game on $\mathcal{X}$ is the (two-player, perfect information, length-$\omega$) game played as follows: Players $1$ and $2$ alternately play decreasing nonempty open sets…
Noah Schweber
  • 260,658
13
votes
2 answers

Infinite wacky race

Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the finish line is infinitely far away. To be more…
Alma Arjuna
  • 6,521
11
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0 answers

Is mate-in-$n$ problem for Trappist-1 undecidable?

Trappist-1 is a variant of infinite chess that has a piece called huygens which leaps any prime number of squares orthogonally. To actually implement this game, it should have decidable mate-in-$0$ (checkmate detection) and stalemate-in-$0$…
Ris
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11
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1 answer

Infinite combinatorial games

Hercules vs. Hydra: Recall the story where every time Hercules cuts of a head, two more heads grow instead. Now suppose the following: The hydra starts off with one head, but every time Hercules cuts off a head, $\aleph_0$ grow anew. Show that…
ctlaltdefeat
  • 750
8
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1 answer

Speedrun Probability Math in Boss Fight Video Games

Context: Speedrunning is a way of playing a video game with the aim of completing it as fast as possible. This probability question was inspired by a very simple video game's boss battle, and it is a math question at its core. I tried calculating…
user45266
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7
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The principle $S_1(\mathcal O,\mathcal O)$ versus the game $G_1(\mathcal O,\mathcal O)$

Given a topological space $X$, Let $\mathcal O$, denote the set of all open covers of $X$. We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ \mathcal U_n : n \in \mathbb N \}$, there exists a…
topsi
  • 4,292
7
votes
3 answers

Game: $\frac{1}{3}$ winning probability vs. $\frac{2}{3}$ winning probability

$A$ and $B$ play a round-based game. Each round $A$ wins with probability $\frac{1}{3}$ and $B$ with probability $\frac{2}{3}$. The loser of a round pays $1$ USD to the winner. The winner of the whole game is the one who wins all the USD from the…
Philipp
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7
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Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)

(Now asked at MO.) Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\omega$ at least one of the following holds: There is a set…
Noah Schweber
  • 260,658
7
votes
1 answer

$6$ bishops and a knight on an infinite chessboard

Player $A$ places $6$ bishops wherever he/she wants on the chessboard with infinite number of rows and columns. Player $B$ places one knight wherever he/she wants. Then $A$ makes a move, then $B$, and so on... The goal of $A$ is to checkmate $B$,…
Grešnik
  • 1,812
7
votes
2 answers

A two player game on compact topological spaces

I've though of an infinite game that two players may play on a given topological space $(X,\tau)$. It goes like this. On turn $n$ Player I selects a point $x_n\in X$ and Player II selects a neighborhood $U_n$ of $x$. They play $\omega$ times, and…
Anguepa
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