Winning strategy using Dynamic Programing
For a vertex $v \in V$, let $W(v)$ be the following predicate: "a player starting at $v$ wins". If $N(v)$ is the set of outward neighbors of $v$, then
$$
W(v) = \bigvee_{u \in N(v)} \lnot W(u),
$$
where the empty disjunction is simply false.

This suggests a strategy to compute $W(v_0)$ using dynamic programming.