Strategic thinking in infinite games, CosmoCaixa Science Museum, Barcelona, March 2023

I am deeply honored to be invited by la Caixa Foundation to give a talk in “The Greats of Science” talk series, to be held 16 March 2023 at the CosmoCaixa Science Museum in Barcelona. This talk series aspires to host “prestigious figures who have contributed towards admirable milestones, studies or discoveries,” who will bring the science to a general audience, aiming to “give viewers the chance to explore the most relevant parts of contemporary sicence through the top scientists of the moment.” Previous speakers include Jane Goodall and nearly a dozen Nobel Prize winners since 2018.

I hope to rise to those high expectations!

My topic will be: Strategic thinking in infinite games.

Have you time for an infinite game? Many familiar finite games admit natural infinitary analogues, infinite games that may captivate and challenge us with intriguing patterns and 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.

The theory builds upon the classical finitary result of Zermelo (1913), the fundamental theorem of finite games, which shows that in every finite two-player game of perfect information, one of the players must have a winning strategy or both players have draw-or-better strategies. This result extends to certain infinitary games by means of the ordinal game-value analysis, which assigns transfinite ordinal values $\alpha$ to positions in a game, generalizing the familiar mate-in-$n$ idea of chess to the infinite. Current work realizes high transfinite game values in infinite chess, infinite draughts (checkers), infinite Go, and many other infinite games. The highest-known game value arising in infinite chess is the infinite ordinal $\omega^4$, and every countable ordinal arises in infinite draughts, the optimal result. Games exhibiting high transfinite ordinal game values have a surreal absurd character of play. The winning player will definitely win in finitely many moves, but the doomed losing player controls the process with absurdly long deeply nested patterns of forcing moves that must be answered, as though counting down from the infinite game value—when 0 is reached, the game is over.

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 draughts and the logic of infinitary games, Oslo, November 2021

This will be a talk 11 November 2021 for the Oslo Seminar in Mathematical Logic, meeting online via Zoom at 10:15am CET (9:15am GMT) at Zoom: 671 7500 0197

Abstract. I shall give an introduction to the logic of infinite games, including the theory of transfinite game values, using the case of infinite draughts as a principal illustrative instance. Infinite draughts, also known as infinite checkers, is played like the finite game, but on an infinite checkerboard stretching without end in all four directions. In recent joint work with Davide Leonessi, we proved that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal. This result is optimal for games having countably many options at each move—in short, the omega one of infinite draughts is true omega one.

Transfinite game values in infinite draughts

A joint paper with Davide Leonessi, in which we prove that every countable ordinal arises as the game value of a position in infinite draughts, and this result is optimal for games having countably many options at each move. In short, the omega one of infinite draughts is true omega one.

• J. D. Hamkins and D. Leonessi, “Transfinite game values in infinite draughts,” Mathematics arXiv, 2021.
[Bibtex]
@ARTICLE{HamkinsLeonessi:Transfinite-game-values-in-infinite-draughts,
author = {Joel David Hamkins and Davide Leonessi},
title = {Transfinite game values in infinite draughts},
journal = {Mathematics arXiv},
year = {2021},
volume = {},
number = {},
pages = {},
month = {},
note = {Under review},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
eprint = {2111.02053},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/transfinite-game-values-in-infinite-draughts},
}

Abstract. Infinite draughts, or checkers, is played just like the finite game, but on an infinite checkerboard extending without bound in all four directions. We prove that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal.

Davide Leonessi, MSc MFoCS, Oxford, September 2021

Mr. Davide Leonessi successfully defended his dissertation for the Masters of Science degree in Mathematics and Foundations of Computer Science, entitled “Transfinite game values in infinite games,” on 15 September 2021. Davide earned a distinction for his thesis, an outstanding result.

Abstract. The object of this study are countably infinite games with perfect information that allow players to choose among arbitrarily many moves in a turn; in particular, we focus on the generalisations of the finite board games of Hex and Draughts.

In Chapter 1 we develop the theory of transfinite ordinal game values for open infinite games following [Evans-Hamkins 2014], and we focus on the properties of the omega one, that is the supremum of the possible game values, of classes of open games; we moreover design the class of climbing-through-$T$ games as a tool to study the omega one of given game classes.

The original contributions of this research are presented in the following two chapters.

In Chapter 2 we prove classical results about finite Hex and present Infinite Hex, a well-defined infinite generalisation of Hex.

We then introduce the class of stone-placing games, which captures the key features of Infinite Hex and further generalises the class of positional games already studied in the literature within the finite setting of Combinatorial Game Theory.

The main result of this research is the characterization of open stone-placing games in terms of the property of essential locality, which leads to the conclusion that the omega one of any class of open stone-placing games is at most $\omega$. In particular, we obtain that the class of open games of Infinite Hex has the smallest infinite omega one, that is $\omega_1^{\rm Hex}=\omega$.

In Chapter 3 we show a dual result; we define the class of games of Infinite Draughts and explicitly construct open games of arbitrarily high game value with the tools of Chapter 1, concluding that the omega one of the class of open games of Infinite Draughts is as high as possible, that is $\omega_1^{\rm Draughts}=\omega_1$.

The full dissertation is available: