# 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.

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### Family of graphs characterized by their eigenvalues

Studying the convergence/divergence of certain processes on simple graphs (processes similar to Kostant games on graphs), I'm confronted with the task of characterizing graphs whose spectra must ...
84 views

### Topological game on $(0,1)$

I consider a « game » on a topological space with $2$ players. I will describe the game and tried to prove that one of the player has no winning strategy in the sense that the other player can always «...
• 1,391
1 vote
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### Existence of Nash Equilibrium in a Game with Mixed Strategy Spaces

I am considering formulating an applied research problem as a simultaneous zero-sum game with two players. The first player's set of actions is an infinite and compact subset of $\mathbb{R}^n$, while ...
114 views

### How is the set $C(f)\cap V$ of second category in $V$?

I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the ...
• 63
1 vote
70 views

### Mathematically prove whether it's possible to survive indefinitely in an unbounded tag game

Suppose we have a tag game, with an unbounded (2D) map, where the player possesses a character in the map represented by it's (x,y) position, that can, at any frame, move 1 unit left, right, up or ...
827 views

### 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 ...
• 3,901
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### Axiom of Choice and Borel determinacy for Polish space

Given a set $A$, Borel determinacy for $A$ is the theorem (of $\mathsf{ZFC}$) asserting that every Borel subset of $A^\omega$ is determined. That is, if I and II take turns playing members of $A$, and ...
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174 views

### Infinite game of an unfair coin toss

Following game between person A and B is proposed: Person A has an unfair coin with probability $p \in (\frac{1}{3},\frac{1}{2})$ of heads. Person B starts with a captial of 100€. For each time B ...
73 views

### Finite Games winning strategy exercises

I'm currently studying finite games from the textbook "Automata Theory and its Applications" by Bakhadyr Khoussainov and Anil Nerode and am having trouble with this exercise with the ...
1 vote
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### Finite Games winning strategy exercises [duplicate]

I'm currently studying finite games from the textbook "Automata Theory and its Applications" by Bakhadyr Khoussainov and Anil Nerode and am having trouble with this exercise with the ...
1 vote
121 views

### "Explicit" undetermined set of reals

Recall that a set $X \subseteq \omega^\omega$ is determined if the Gale-Stewart game for $X$ is determined. It's well-known that the axiom of choice implies the existence of an undetermined set. The ...
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1 vote
105 views

### (Combinatorial) Game Theory: Determinacy and Determinism

I am struggling with the concepts of Determinacy and Determinism. Are the following statements correct(for 2-player, zero-sum games)? Or am I getting something mixed up in my head? A game has the ...
• 13
108 views

### Why can't a Lusin set be sigma-compact?

In A Direct Proof of a Theorem of Telgársky, Scheepers asserts that the Menger game is indeteremined for a Lusin (aka Luzin) subset of the reals. That is, the first player lacks a winning strategy in ...
• 6,404
74 views

### Can P1 improve an open cover to an omega-cover in the finite-open game?

Definitions/terms taken from https://www.sciencedirect.com/science/article/abs/pii/S016686411830470X (https://arxiv.org/abs/1806.06001). In the finite-open game $FO(X)$, during each round $n<\omega$...
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85 views

### Do uncountable spaces admit Markov strategies in Rothberger-style games?

Consider the selection game $G_1(\mathcal C,\mathcal C)$ where $\mathcal C$ is associated with a class of open covers. Considering various possibilities for $\mathcal C$, do uncountable spaces admit ...
• 6,404
1 vote