Playing to win strategy algorithm
Your game is equivalent to the PSPACE-complete problem TQBF. The quantified Boolean formula corresponding to your example is
$$
\forall x \forall y \exists z (x \land y) \lor (y \land z) \lor (\lnot y \land \lnot z).
$$
Since it is PSPACE-complete, in particular it is NP-hard, and you there is no efficient general way to find out who wins in this game.

There is an exponential time strategy which involves constructing the game tree. While this strategy is feasible when the number of variables is small (like in your example), it is not practical when there are many variables.