  When normal and mis\`{e}re games are played on bi-type binary Galton-Watson
trees (with vertices coloured blue or red and each having either no child or
precisely $2$ children), with one player allowed to move along monochromatic
edges and the other along non-monochromatic edges, the draw probabilities equal
$0$ unless every vertex gives birth to one blue and one red child. On bi-type
Poisson trees where each vertex gives birth to Poisson$(\lambda)$ offspring in
total, the draw probabilities approach $1$ as $\lambda \rightarrow \infty$. We
study such \emph{novel} versions of normal, mis\`{e}re and escape games on
rooted multi-type Galton-Watson trees, with the "permissible" edges for one
player being disjoint from those of her opponent. The probabilities of the
games' outcomes are analyzed, compared with each other, and their behaviours as
functions of the underlying law explored.
