  We study the bond percolation game and the site percolation game on the
rooted Galton-Watson tree $T_{\chi}$ with offspring distribution $\chi$. We
obtain the probabilities of win, loss and draw for each player in terms of the
fixed points of functions that involve the probability generating function $G$
of $\chi$, and the parameters $p$ and $q$. Here, $p$ is the probability with
which each edge (respectively vertex) of $T_{\chi}$ is labeled a trap in the
bond (respectively site) percolation game, and $q$ is the probability with
which each edge (respectively vertex) of $T_{\chi}$ is labeled a target in the
bond (respectively site) percolation game. We obtain a necessary and sufficient
condition for the probability of draw to be $0$ in each game, and we examine
how this condition simplifies to yield very precise phase transition results
when $\chi$ is Binomial$(d,\pi)$, Poisson$(\lambda)$, or Negative
Binomial$(r,\pi)$, or when $\chi$ is supported on $\{0,d\}$ for some $d \in
\mathbb{N}$, $d \geqslant 2$. It is fascinating to note that, while all other
specific classes of offspring distributions we consider in this paper exhibit
phase transition phenomena as the parameter-pair $(p,q)$ varies, the
probability that the bond percolation game results in a draw remains $0$ for
all values of $(p,q)$ when $\chi$ is Geometric$(\pi)$, for all $0 < \pi
\leqslant 1$. By establishing a connection between these games and certain
finite state probabilistic tree automata on rooted $d$-regular trees, we obtain
a precise description of the regime (in terms of $p$, $q$ and $d$) in which
these automata exhibit ergodicity or weak spatial mixing.
