  For a collection of papers in memory of Elwyn Berlekamp (1940-2019), John
Conway (1937-2020), and Richard Guy (1916-2020).
  The Sprague-Grundy theory for finite games without cycles was extended to
general finite games by Cedric Smith and by Aviezri Fraenkel and coauthors. We
observe that the same framework used to classify finite games also covers the
case of locally finite games (that is, games where any position has only
finitely many options). In particular, any locally finite game is equivalent to
some finite game. We then study cases where the directed graph of a game is
chosen randomly, and is given by the tree of a Galton-Watson branching process.
Natural families of offspring distributions display a surprisingly wide range
of behaviour. The setting shows a nice interplay between ideas from
combinatorial game theory and ideas from probability.
