The disjunctive theory of combinatorial games can trace its roots to the work
of Sprague and Grundy in the 1930s, but its modern form was born with the
arrival of Conway’s On Numbers and Games in 1976 and the classic Winning
Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy in 1982.
In the ensuing three decades, combinatorial game theory has blossomed into
a serious and active branch of combinatorics, with connections to coding
theory, computational complexity, and commutative algebra.
This book is intended as a second course on combinatorial games, at
the ﬁrst- or second-year graduate level, and most readers will beneﬁt from
some prior exposure to the subject. Winning Ways is a ﬁne introduction;
in addition, an excellent new textbook by Albert, Nowakowski, and Wolfe,
titled Lessons in Play: An Introduction to Combinatorial Game Theory,
has recently appeared. Either (or both) of these references should serve as
adequate preparation for this volume.
Nonetheless, this book is completely self-contained and traces the devel-
opment of the theory from ﬁrst principles and examples through many of its
most recent advances. It should serve those who have read Winning Ways
and crave a more rigorous development of the theory, as well as professionals
seeking a cohesive reference for the many new ideas that have emerged in re-
cent years. Among those advances appearing for the ﬁrst time in textbook
form (as far as I know) are Berlekamp’s generalized temperature theory,
Thane Plambeck’s elegant theory of mis`ere quotients, David Moews’ results
on the group structure of G, and the construction of mis`ere canonical forms
for partizan games.