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.

43 questions
92 views

$6$ 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 $B$, ...
73 views

Covering game in Borel Determinacy proof

I was reading the introduction to the actual proof of Borel Determinacy in Kechris' "Classical Descriptive Set Theory". Here is an extract: What I don't get is why we define $\varphi$ in this way ...
274 views

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 ...
51 views

Showing that Gale-Stewart Theorem on Determinacy of Open and Closed Games is equivalent to AC

I'm studying from Kechris' "Classical Descriptive Set Theory" and I'm trying to solve exercise 20.3, which asks to show that the Gale-Stewart theorem is equivalent to the axiom of choice AC in ZF. I'...
59 views

Three-player duel of complete information. Optimal strategy and Nash equilibrium

I have a three-player duel in which players A,B and C pick a time t in the interval [0,1] to fire at a common target and they can only fire once. When player A fires at time t, he will hit with ...
28 views

Product of Paracompact spaces being Paracompact

I'm interested in the game-characterization proposed by Telgarsky (paper) of the class of paracompact spaces that preserve paracompactness under cartesian product with another paracompact space. He ...
26 views

Play Cards Game Tournament Algorithm

I am currently trying to find algorithm to minimize the total time of a tournament. The game requires $2$ teams of $2$ players in each team (total $4$ players). Then, the perfect number of ...
114 views

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 ...
59 views

Mixed strategy equilibrium in Cournot Duopoly

This maybe a trivial question to most. I am fairly new to game theory. The usual cournot duopoly (same constant marginal cost for both players) is solved using pure strategies. Are there mixed ...
59 views

95 views

96 views

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 ...
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 ...
88 views

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 ...
414 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 ...
685 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 ...
122 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 ...
84 views

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 ...
86 views

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. ...