Transfinite game values in infinite games

TL;DR

This paper develops the theory of transfinite ordinal game values for infinite perfect-information games, analyzing classes like Infinite Hex and Infinite Draughts, and determines their omega one values, revealing fundamental differences in their complexity.

Contribution

It introduces a new framework for analyzing transfinite game values in infinite games and characterizes the omega one of classes like Infinite Hex and Infinite Draughts.

Findings

Infinite Hex has omega one equal to ω.

Infinite Draughts has omega one equal to ω₁.

Class of stone-placing games has omega one at most ω.

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 and Hamkins (arXiv:1302.4377), 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 $\omega_1^{\rm Draughts}=\omega_1$.