65
votes
Accepted
A game on integers
I claim that Player A has a winning strategy in your game, and furthermore, it is a winning strategy for her simply to play the smallest available number.
Let me consider the game along with several ...
- 205k
58
votes
Accepted
Does knight behave like a king in his infinite odyssey?
Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which ...
- 205k
37
votes
Accepted
Is there a position in infinite Go for which the life of a particular stone has transfinite game value?
This is a really great question!
Previous attempts to make sense of infinite Go have sometimes had problems because it wasn't clear how to define the winner of a game of Go after transfinite play. ...
- 205k
32
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
There's a few issues that need to be distinguished here. First, one can distinguish the question of how you find the winning strategy from the question of how you define what the winning strategy even ...
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25
votes
Does knight behave like a king in his infinite odyssey?
(Moved from a comment.)
Questions 2, 3, and 4 are answered negatively by [W], which decomposes the plane into concentric annuli of width two.
OP observes that this construction is still spiral (...
- 576
22
votes
Accepted
An infinite game possibly due to Ernst Specker
I don't know about the game attributed to Specker, but here is a
simple game with your desired features.
Let us call it the Chocolatier's game. There are two players,
the Chocolatier and the Glutton. ...
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19
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
Let me address the criticism that the Nash equilibrium is of questionable real-life significance.
I'll begin by openly admitting something that theorists often are reluctant to admit: One big reason ...
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15
votes
The Chocolatier's game: can the Glutton win with a restricted form of strategy?
Yes. See Theorem 1.2 in K. Ciesielski and R. Laver, A game of D. Gale in which one of the players has limited memory, Period. Math. Hungar. 21 (1990), no. 2, 153–158
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13
votes
Accepted
The Axiom of Determinacy and the Banach-Mazur game
The claim is false. The Banach-Mazur game, also known as the $**$-game shows (and is equivalent to) that every set of reals has the Baire property. What is true, as you've noted, is that if one has a ...
- 3,628
13
votes
Accepted
Choosing subsets of $\mathbb R$ of cardinality $\frak c$, who wins?
In ZFC, the player aiming for the empty set has a winning strategy in the game played on any infinite set, including the reals. Using the axiom of choice, we can well-order the set and thereby pretend ...
- 205k
13
votes
The Chocolatier's game: can the Glutton win with a restricted form of strategy?
(Not an answer; promoted from a comment on another answer)
If we modify the game so that the glutton can remember (only) the last chocolate they ate, they have a winning strategy as follows:
Well-...
- 555
12
votes
A game on integers
Joel's answer uses heavy machinery to give a much stronger result. Below is a strong result for $k=1$ using heavy computations (not mine.)
My suspicion is that for the question as asked there might ...
- 29.7k
11
votes
The Chocolatier's game: can the Glutton win with a restricted form of strategy?
Instead of looking at the unordered set of previously eaten chocolates, let's consider a slight modification, where the Glutton knows the finite sequence of previously eaten chocolates. That is, I ...
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10
votes
Accepted
Undetermined Banach-Mazur games in ZF?
This is only a partial answer. ZF + DC + 'every Banach-Mazur game is determined' is inconsistent.
Let $X$ be the set of all functions of the form $f: \alpha \rightarrow \{0,1\}$, with $\alpha$ an ...
- 7,779
10
votes
The Chocolatier's game: can the Glutton win with a restricted form of strategy?
If the chocolate space is $[0,1]$, then Glutton can't win with a positional measurable strategy, where positional means knowing what was eaten when and what is currently on offer; and moreover we ...
- 3,864
10
votes
Strategic vs. tactical closure
The answer to the topological version of Banach-Mazur games is negative, proved by Gabriel Debs in 1985:
Debs, Gabriel, Stratégies gagnantes dans certains jeux topologiques (Winning strategies in ...
- 205k
9
votes
Accepted
Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game
Player I has a winning strategy: First play a singleton $A_0=A_1=\ldots=\{z_0\}$, for some real $z_0$, and the $x_n$'s consistent with $z_0$, until player II plays their first 1, if they ever do. ...
- 6,154
8
votes
The Sudoku game: Solver-Spoiler variation
For all $n \geq 2$, here is a simple winning strategy for Spoiler on the $n^2 \times n^2$ board, that requires at most $n^2-1$ moves to win. I assume that Solver plays first, but the strategy can ...
- 29.3k
8
votes
Choosing subsets of $\mathbb R$ of cardinality $\frak c$, who wins?
This problem (with inconsequential differences in the rules of play) was posed by S. Banach as Problem 67 in The Scottish Book, which is available here. The following text is copied from R. Daniel ...
- 9,146
7
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
The problem of finding a winning strategy (if one exists) is purely computational (with rare exceptions), and its solution very much depends on the game. So, this problem cannot be part of a general ...
- 90.3k
7
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
I recommend you read "Harrington on Cash Games, Volume I" by Dan Harrington and Bill Robertie. It is not a formal math book, but rather a book on winning strategies for poker.
What I find ...
- 7,001
7
votes
Strategic vs. tactical closure
For a partial answer, let me prove that every strategically closed partial order admits a nearly tactical winning strategy, one that depends only on the previous two moves, that is, on the previous ...
- 205k
6
votes
Determinacy of (infinite, possibly loopy) combinatorial games
This amounts to the Gale-Stewart theorem showing that open games are determined. The issue of draws can be easily finessed, as I explain below.
Specifically, a game of perfect information is open for ...
- 205k
6
votes
Solution to simple mathematical game
Not an answer.
For those who want to see the nimbers of these games, I have written the following code, which calculates the nimbers for the games with target $N$ from $2$ to $999$.
How to use it:
...
- 3,140
6
votes
Accepted
Reference for graduate-level text or monograph with focus on "the continuum"
I like your idea of such a course a lot! If it is appropriate to recommend a book in German language, I think this one could be the perfect match:
Oliver Deiser (2007): Reelle Zahlen: Das klassische ...
- 316
6
votes
Reference for graduate-level text or monograph with focus on "the continuum"
A great textbook for your course would be "The Structure of the Real Line" by Lev Bukovský. It covers all of the topics you mentioned, except for the Banach-Tarski Paradox, and provides all ...
- 3,991
6
votes
Strategic vs. tactical closure
Here is another partial answer.
Theorem. Suppose Player II has a winning strategy $\sigma$ in the Banach–Mazur game on a poset $P$. If there is a subset $D\subseteq P$ such that
(1) $\forall p\in P\ \...
- 9,146
5
votes
Solution to simple mathematical game
I think that every even target greater than $6$ is indeed a first player win. I have no proof but present evidence and speculation on what might lead to a proof.
For every even target at least up to ...
- 29.7k
5
votes
Accepted
Can I win this variant of the Banach-Mazur Game?
Sure you can win. Let's enumerate rationals as $q_n, n = 1, 2,\ldots$. Also we can WLOG assume that $\varepsilon_{i+1} \le \frac{\varepsilon_i}{20000}$. We will make it so that $d(q_n, p_{n+1}) \ge 10\...
- 2,626
5
votes
Strategic vs. tactical closure
This post is an addition to the argument posted by bof last night. In bof's post, it is proved that posets having a special kind of dense subset cannot provide a counterexample for Monroe's question. ...
- 15.8k
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