Three patterns you've seen:
Score difference: Track my score opponent's score. Opponent's gain is my loss (subtraction).
Extra parameters: Some games have changing rules (like in Stone Game II). Include them in the state.
Bitmask + game theory: When tracking used elements from a small set, combine bitmask DP with minimax thinking. The core stays the same: I win if I have any move that puts opponent in a losing state. I lose if all my moves put opponent in a winning state.