  We study the $k$-jump normal and $k$-jump mis\`{e}re games on rooted
Galton-Watson trees, expressing the probabilities of various outcomes of these
games as specific fixed points of certain functions that depend on $k$ and the
offspring distribution. We discuss results on phase transitions pertaining to
draw probabilities when the offspring distribution is Poisson$(\lambda)$ (i.e.
for which values of $\lambda$, the draw probability is strictly positive). We
compare the probabilities of the various outcomes of the $2$-jump normal game
with those of the $2$-jump mis\`{e}re game, and a similar comparison is drawn
between the $2$-jump normal game and the $1$-jump normal game, under the
Poisson regime. We describe the rate of decay of the probability that the first
player loses the $2$-jump normal game as $\lambda \rightarrow \infty$. Finally,
we discuss a sufficient condition for the average duration of the $k$-jump
normal game to be finite.
