# 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.

35 questions
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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 ...
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### The Connect Infinity game

Recently Joel David Hamkins posted an entry on the Connect Infinity game. Connect-$\omega$ is Connect Four but played on an $\omega\times n$ grid ($n$ finite)! The above shows $n=6$. The difference ...
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### Explanation for Oxtoby's proof: a nonempty topological space $X$ is Baire iff player (I) has no winning strategy in the Choquet game

A nonempty topological space $X$ is a Baire space iff player I has no winning strategy in the Choquet game $G_X$. Oxtoby's proof I have several questions about this proof. $(\Leftarrow)$ How can ...
1answer
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### A question on gaming theory with variables [closed]

We have two stacks of coins, each with a and b coins respectively. On each step, the player is allowed to remove as many coins as they want(but at least one) from either of the two stacks he wants to (...
0answers
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### Value of an infinite game

I'm having some trouble showing that the following zero-sum game with 2 players has a value (and in consequence, computing it's value). The strategies of both players are in $\mathbb{N}-\{0\}$ and ...
2answers
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### Probability of winning a Craps game

A Craps game consists of throwing 2 dices. If the sum is either 7 or 11, you win. Else, if the sum is either 2,3 or 12, then you loose. If the sum is either 4, 5, 6, 8, 9 or 10, then let's call the ...
1answer
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### Who has a winning strategy in Choquet game on rational numbers?

I know that in a real-numbers-variant of Choquet game, the player aiming for non-empty intersection has a winning strategy. Is the same true for rational numbers?
2answers
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### 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 ...
1answer
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### A question on repeated game theory

I recently have come across a business problem which could be convereted into a game problem as follows: Imagine an infinitely repeated game between two players in which the firts player (leader) ...
1answer
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### Proof that “$\uparrow$ is the unique solution of $tiny(G) = G$”

Tiny & miny games can be defined as: $$tiny(G) = \{0||0|-G\}$$ $$miny(G) = -tiny(G) = \{G|0||0\}$$ From the Wikipedia page for tiny and miny: Similarly curious, mathematician John Horton ...
1answer
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### Game Theory Reccomendation, Mean Field Theory

I'm about to do a sort of reading course with a mathematics professor wherein I read and teach him about Game Theory. He claims not to know Game Theory. After that, we aim to read about Mean Field ...
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### Is it true player II must have a winning strategy, if the winning set is a closed but not open set?

Suppose, in a Gale-Stewart game, player I and player II choose from $\omega$ in a alternating fashion. If the outcome is in the winning set $W$, then player I wins. Otherwise player II wins. If $W$ is ...
1answer
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### The difference between winning tactic and winning strategy

In the topological game, what is the difference between winning tactic and winning strategy? Why the author (in this paper: LEFT SEPARATED SPACES WITH POINT-COUNTABLE BASES) said that, the first ...
1answer
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### A question on topological game

This is from the paper: LEFT SEPARATED SPACES WITH POINT-COUNTABLE BASES by WILLIAM G. FLEISSNER. It is a little difficult for me to understand that what's the meaning of $I$ wins, and $II$ wins. ...
1answer
37 views

### Game theory applied to invoicing

There is a traditional practise in precuement of almost any kind of paying an invoice at l ast thirty days after it has been received. Is there any application of game theory here that can justify ...
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412 views

### Question about probability in infinite game of chance

Imagine a game where you start with \$100 and toss a coin repeatedly. If it's heads - you lose \$1, if its tails - you double your money. Game ends when you lose all the money. Given infinite amount ...
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411 views

### 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 ...
1answer
121 views

### Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don't know why $X$, the set of all outcomes generated by a fixed strategy ...
1answer
196 views

### Question on proof that Gale-Stewart game is determined for open sets

An infinite game of perfect information consists of a finite set $A$ of moves, two players and a set $X \subseteq A^{\mathbb N}$ of winning conditions for player I. By convention player I makes the ...
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### Determinacy of complement of set

Here I described the Gale-Stewart game on Baire space $\mathbb{N}^{\mathbb {N}}$. If the set A is determined (one of the players has a winning strategy), what can we say about the determinacy of ...
1answer
156 views

### A chomp game with infinite board size.

Given a chomp game with board size $m * \infty$. So I've already understood that this game is final, and has a finite number of states. However I'm having problems with figuring out whether there ...
1answer
109 views

### Gale-Stewart game

Let's look at the Gale-Stewart game on $\mathbb N^{\mathbb N}$ space (it consists of infinite sequences of natural numbers). There are 2 players: $A$ and $B$. They write one natural number by turns (A ...
0answers
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### Strong Choquet preimage implies strong Choquet?

Recall that a strong Choquet space is one where player II has a winning strategy in the game where two players take turns: player I chooses an open set and a point inside, then player II chooses a ...
0answers
385 views

### 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 stationary st.) in $BM(X)$. Then can ...
1answer
79 views

### Set of winning strategies for union of winning sets

Suppose that $G( \omega, A, X)$ denotes a sequential game of perfect information in which player I and player II play an element in $A$ in each turn with a total number of $\omega$ moves. $X$, the ...
1answer
98 views

### Dealing with an infinitely repeated game

I have been playing around with problems related to game theory, and I ran into this issue related to an infinitely repeated game. Consider this game repeated an infinite number of times: \begin{...
0answers
676 views

### “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 ...
4answers
3k views

### 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 ...
2answers
3k views

### King and knight moving on an infinite chess board

There are actually two separate problems: King problem: How many squares can a king moving on an infinite chess board reach in N moves? Knight problem: How many squares can a knight ...
1answer
974 views

### Math and Logic of Infinite Chess

Two players (White and Black) are playing on an infinite chess board (extending infinitely in all directions). First, White places a certain number of queens (and no other pieces) on the board. ...