  We consider the behaviour of minimax recursions defined on random trees. Such
recursions give the value of a general class of two-player combinatorial games.
We examine in particular the case where the tree is given by a Galton-Watson
branching process, truncated at some depth $2n$, and the terminal values of the
level-$2n$ nodes are drawn independently from some common distribution. The
case of a regular tree was previously considered by Pearl, who showed that as
$n\to\infty$ the value of the game converges to a constant, and by Ali Khan,
Devroye and Neininger, who obtained a distributional limit under a suitable
rescaling.
  For a general offspring distribution, there is a surprisingly rich variety of
behaviour: the (unrescaled) value of the game may converge to a constant, or to
a discrete limit with several atoms, or to a continuous distribution. We also
give distributional limits under suitable rescalings in various cases.
  We also address questions of endogeny. Suppose the game is played on a tree
with many levels, so that the terminal values are far from the root. To be
confident of playing a good first move, do we need to see the whole tree and
its terminal values, or can we play close to optimally by inspecting just the
first few levels of the tree? The answers again depend in an interesting way on
the offspring distribution.
  We also mention several open questions.
