Infinite-Games Workshop

Welcome to the Infinite-Games Workshop, beginning Autumn 2023. The past ten years has seen an explosion in the study of infinite games, for researchers are now investigating diverse infinite games, including infinite chess, infinite draughts, infinite Hex, infinite Othello, infinite Go, indeed, we seem to have research projects involving infinitary analogues of all our familiar finite games. It is an emerging research area with many new exciting results.

This autumn, we shall set the workshop off with talks on several exciting new results in infinite chess, results which settle what had been some of the big open questions in the topic, including the question of the omega one of chess—the supremum of the ordinal game values that arise—as well as a finite position with game value $\omega^2$.

The workshop talks will be run at a high level of sophistication, aimed for the most part at serious researchers currently working in this emerging area. Mathematicians, computer scientists, infinitary game theorists, all serious researchers are welcome.

All talks will take place on Zoom at meeting 968 0186 3645 (password = latex code for the first uncountable ordinal). Contact for further information.

Talks will be 90 minutes, with a workshop style welcoming questions. All talks will be recorded and placed on our YouTube channel. Talks will generally be held on Thursdays at 11:00 am New York time.

Add our calendar: Infinite-Games Workshop Calendar

The workshop is being organized by myself with the assistance of Davide Leonessi.

FAll 2023 Talks

21 September 2023 11am ET

Infinite draughts: a solved open game

Davide Leonessi, The Graduate Center of the City University of New York

Davide Leonessi, CUNY GC

Abstract: In this talk I will introduce open infinite games, and then define a natural generalization of draughts (checkers) to the infinite planar board. Infinite draughts is an open game, giving rise to the game value phenomenon and expressing it fully—the omega one of draughts is at least true $\omega_1$ and every possible defensive strategy of the losing player can be implemented. 

5 October 2023 11:00 am ET

Introduction to infinite games

Joel David Hamkins, Professor of Logic, University of Notre Dame

Abstract: I shall give a general introduction to the subject and theory of infinite games, drawing upon diverse examples of infinitary games, but including also infinite chess, infinite Hex, infinite draughts, and others.

2 November 2023 11:00 am ET

Complexity of the winning condition of infinite Hex

Ilkka Törmä, University of Turku, Finland

Abstract: Hex is a two-player game where the goal is to form a contiguous path of tokens from one side of a finite rectangular board to the opposite side. It is a famous classical result that Hex admits no draws: a completely filled board is a win for exactly one player. Infinite Hex is a variant introduced recently by Hamkins and Leonessi. It is played on the infinite two-dimensional grid $\mathbb{Z}^2$, and a player wins by forming a certain kind of two-way infinite contiguous path. Hamkins and Leonessi left open the complexity of the winning condition, in particular whether it is Borel. We present a proof that it is in fact arithmetic.

16 NOvember 2023 11:00 am ET

A finite position in infinite chess with game value $\omega^2+k$

Andreas Tsevas, Physics, Ludwig Maximalians Universität München

Abstract: I present a position in infinite chess with finitely many pieces and a game value of $\omega^2+k$ for $k\in\mathbb N$, thereby improving the previously known best result in the finite case of $\omega\cdot n$ for arbitrary $n \in\mathbb N$. This is achieved by exercising control over the movement of a white queen along two rows on the chessboard via precise tempo manipulation and utilization of the uniquely crucial ability of the queen to interlace horizontal threats with diagonal moves.

7 December 2023 11:00 am ET

All Countable Ordinals Arise as Game Values in Infinite Chess

Matthew Bolan, University of Toronto

Matthew Bolan, Infinite-Games Workshop

Abstract: For every countable ordinal $\alpha$, we show that there exists a position in infinite chess with infinitely many pieces having game value $\alpha$.

An exploration of infinite games—infinite Wordle and the Mastermind numbers, Harvard, October 2023

This will be a talk 16 October 2023 (Note new date!) for the Colloquium of the Harvard Center for Mathematical Sciences and Applications (CMSA).

Abstract: Let us explore the nature of strategic reasoning in infinite games, focusing on the cases of infinite Wordle and infinite Mastermind. The familiar game of Wordle extends naturally to longer words or even infinite words in an idealized language, and Mastermind similarly has natural infinitary analogues. What is the nature of play in these infinite games? Can the codebreaker play so as to win always at a finite stage of play? The analysis emerges gradually, and in the talk I shall begin slowly with some easy elementary observations. By the end, however, we shall engage with sophisticated ideas in descriptive set theory, a kind of infinitary information theory. Some assertions about the minimal size of winning sets of guesses, for example, turn out to be independent of the Zermelo-Fraenkel ZFC axioms of set theory. Some questions remain open.

Infinite games—strategies, logic, theory, and computation, Northeastern, June 2023

This will be an online Zoom talk for the Boston Computaton Club, a graduate seminar in computer science at Northeastern University, 16 June 12pm EST (note change in date/time). Contact the organizers for the Zoom link.

Abstract: Many familiar finite games admit natural infinitary analogues, which may captivate and challenge us with sublime complexity. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Wordle, or infinite Sudoku? In the Chocolatier’s game, the Chocolatier serves up an infinite stream of delicious morsels, while the Glutton aims to eat every one. These games and others illustrate the often subtle strategic aspects of infinite games, and sometimes their downright logical peculiarity. Does every infinite game admit of a winning strategy? Must optimal play be in principle computable? Let us discover the fascinating nature of infinitary strategic thinking.

Infinite Games, Frivolities of the Gods, Logic at Large Lecture, May 2022

The Dutch Association for Logic and Philosophy of the Exact Sciences (VvL) has organized a major annual public online lecture series called LOGIC AT LARGE, where “well-known logicians give public audience talks to a wide audience,” and I am truly honored to have been invited to give this year’s lecture. This will be an online event, the second of the series, scheduled for May 31, 2022 (note change in date!), and further access details will be posted when they become available. Free registration can be made on the VvL Logic at Large web page.

Abstract. Many familiar finite games admit natural infinitary analogues, which often highlight intriguing issues in infinite game theory. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Go, infinite Wordle, or infinite Sudoku? Let me introduce these games and use them to illustrate various fascinating concepts in the theory of infinite games.

Come enjoy the lecture, and stay for the online socializing event afterwards. Hope to see you there!

Infinite Wordle and the Mastermind numbers

[bibtex key=”Hamkins:Infinite-Wordle-and-the-mastermind-numbers”]

Download article at arXiv:2203.06804

Abstract. I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number 𝕞, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.