An Introduction to Conway's Games and Numbers

Abstract

Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic structure: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition generalizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is understood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Winning Ways by Berlekamp, Conway and Guy [WW]; they are now appearing in their second editions. [WW] is a most beautiful book bursting with examples and results but with less stress on mathematical rigor and exactness of some statements. [ONAG] is still the definitive source of the theory, but rather difficult to read for novices; even the second edition shows that it was originally written in one week, and we feel that the order of presentation (first numbers, then games) makes it harder to read and adds unnecessary complexity to the exposition. [SN] is an entertaining story about discovering surreal numbers on an island. This note attempts to furnish an introduction to Combinatorial Game Theory that is easily accessible and yet mathematically precise and self-contained, and which provides complete statements and proofs for some of the folklore in the subject. We have written this note with readers in mind who have enjoyed looking at books like [WW] and are now eager to come to terms with the underlying mathematics, before embarking on a deeper study in [ONAG, GONC] or elsewhere. While this note should be complete enough for readers without previous experience with combinatorial game theory, we recommend looking at [WW], [GONC] or [AGBB] to pick up