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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Mixed strategies as best responses in continuous games

I understand that in any finite two-person game, if I's mixed strategy $\sigma$ is a best response to one of II's (pure or mixed) strategies, then any pure strategy that is in the support of $\sigma$ ...
37 views

Average coinflips to reach the 6 (Custom game)

a friend of mine asked me about a certain game mechanic in a videogame and how many tries it would take for him (on average) to reach his goal. In abstract form, it can be explained as follows: You ...
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Where do winning strategies occur for Player II in the Determinacy of Computable Open Games relative to a parameter?

Moschovakis goes over various theorems proving the Determinacy of closed/open games, and I am reading into various papers regarding the characterization of ordinals where winning strategies for the ...
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Overall confusion in Moschovakis' Proof that $\Sigma_2 ^0$ games are determined (Page 221)

I'm reading through Moschovaki's proof that all $\Sigma_2^0$ games are determined, and the second part of the proof is confusing me. I follow up to the point where they prove $u\in W^{\xi}\implies$ I ...
44 views

In terms of binary games, what is meant by a "$\Sigma_2 ^{0}$ game"?

I am reading some articles on Determinacy and winning strategies in binary games, and there seems to be several notions of a "$\Sigma_{m}^{n}$ game." The ones I'm looking at in particular ...
36 views

Formulating stochastic sequential or discrete time games

I am trying to formulate a stochastic sequential game in discrete time, and was trying to do so as is common in the literature but couldn't find the appropriate setting anywhere. I'm sure that it ...
38 views

Argument for cardinality of all possible strategies for a two-player game

If I understand correctly, a strategy for a two-player game (for either player) is a function from $\omega^{<\omega}$ (i.e. the set of all finite sequences of natural numbers) to $\omega$. Jech ...
57 views

Are Banach-Mazur Games related to filter Convergence?

Is there a way to connect filter convergence to the condition for player 1/2 to win a Banach-Mazur game in an if (and only if) fashion? Thanks! Details below... A Banach-Mazur game is defined as ...
62 views

Why are Bernstein sets not determined?

I've been reading Oxtoby's Measure and Category and in chapter 6 he discusses the game of Banach-Mazur for two players $A$ and $B$ on the unit interval $I_0 \subset \mathbb{R}$. Towards the end of the ...
32 views

Infinite game (Ehrenfeucht-Fraïssé?) for Linear Temporal Logics

Imagine we have two LTL formulae: A and B. I would like to prove whether they are equivalent or not (the formulae can have the "Globally" operator, so the game is infinite). To do so I have ...
64 views

Presentation on game theory and determinacy

I have to write a short paper(about 20 pages) and prepare a presentation (about 1 hour) for an exam on Game Theory (it is a general, introductory course). I've looked up some things on the internet ...
32 views

Is there a paper or book about countable-open game?

I am studying about topological game, and I found the progression: Point-Open(X)>> Finite-Open(X)>>Compact-Open(X). I understand that a natural way to extend 'be finite' is 'be compact', ...
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"Playable" games on an uncountable set

Almost all games in real life are based on a finite set of integers (for example, we can index each possible movement in chess by an integer). But it is still pretty interesting to think about games ...
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What is the most accurate scoring system between 2 players?

Considering each player has 3 attempts to score, which of the following methodologies is the most accurate to determine a players score: Highest Score of 3: 123, 456, 789 Total Score of 3: = 123, 456, ...
61 views

Game on $[0;1]$ about repeating decimal

Two players play the following game. Before the beginning $"0."$ is written on the board. The first player writes any (finite) sequence of digits. The second one then writes only one digit. Then the ...
102 views

history-based strategy versus position-based strategy

SUMMARY: suppose a game is played that is a walk on an infinite directed graph without blind points, where the two players each make one step alternatingly. Suppose that the pay-off of a play is ...
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Can you help me find an optimal mixed strategy for this simple 2-person allocation game?

Consider the following simple 2-person game. Players 1 and 2 each have 100 dollar coins, with a barrier between them, hiding each other's moves. Each player must allocates his 100 coins into 3 piles:...
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How to win when placing colored dots on a plane against one opponent?

A‘s color is purple, B‘s color is green. They alternately set a dot in their color on a set 2D-plane. Whoever manages to construct a triangle in their own color without a dot in the other color ...
53 views

Do Hybrid games exist?

I'm new to game theory. So far, I know that we have games with finite strategy sets and games with continuous strategy sets. I was wondering if there are any games in which some players have finite ...
51 views

Coin Flipping Game -- St. Petersburg Variant?

Suppose Alice and Bob are playing a game with a fair coin. Every time the coin comes up heads, the score (initially 0) increases by 1. If the coin comes up tails, the score decreases by 1. If the ...
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Rothberger game and Meager set.

Someone know examples of topological spaces of first category and in which Player II has a winning strategy in the Rothberger game? Remember that: The Rothberger game on a topological space $X$ is ...
254 views

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 he ...