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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. …

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 …

I have a question about the Chocolatier's game, which I had introduced in my recent answer to a question of Richard Stanley. To recap the game quickly, the Chocolatier offers up at each stage a finite …

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. T …

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the fi …

This is a great question! I've now managed to eliminate the use of countable choice. Theorem. Without using any choice principle, it follows that player I can have no winning strategy in the game. …

I don't know the answer to your question, but perhaps we might gain insight from the following Interview with Jason Simmons, a professional rock/paper/scissors player, which appeared a few years ago o …

It is a very nice question. I claim that it is not sufficient that $C$ is determined, and indeed, there are counterexamples where $C$ is a game with only two moves. Consider the two-dimensional gam …

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 mo …

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 cer …

Update. (Oct 28, 2015) See below, for a position with game value $\omega^4$. This is a great question, which I have been pondering for some time. I have just completed a joint article Transfinit …

Thanks for your kind words about my blog. In the general-size square Sudoku board, you have an $\kappa\times\kappa$ array of $\kappa\times\kappa$ local blocks for some (possibly infinite) cardinal $\ …

The game of infinite Hex, proceeding from an arbitrary position, is a good example with all the features you seek. The game was the subject of my Oxford student Davide Leonessi's masters MFoCS dissert …

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 …

Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt …