Use game theory when the problem involves two players making perfect decisions against each other. Keywords: "both players play perfectly", "determine the winner". If the problem has one player or asks about reachability, use standard graph search instead. If the problem has randomness, consider expectimax or stochastic game techniques.
Signals to look for: the problem involves alternating turns, best play from both sides, or finding if a winning strategy exists. If you see phrases like 'both players play optimally', game theory is the right tool.