What's the name of the theory that the player with the most options has the advantage?
This sounds a lot like a strategy-stealing argument. It's often used to prove that one side of a game has an advantage, even if optimal strategy is not known. As a broad example, consider a game with the following properties:


It is a turn-based game
The starting positions are identical except Player A goes first
Player A has the option of skipping the first turn


Assuming optimal play, the first player cannot be at a disadvantage. The proof is a strategy-stealing argument: Any hypothetical strategy that Player B could use to get an advantage could be countered by Player A just skipping the first turn and then using that exact strategy.