I believe there is either only one solution or multiple equivalent solutions (e.g. in board where all squares are an equal probability, it doesn't matter which

I believe there is either only one solution or multiple equivalent solutions (e.g. in board where all squares are an equal probability, it doesn't matter which corner you start with.) You should assume an optimal counter-strategy when figuring out your move. I don't have the time to rigorously prove there's always an optimal solution, but it seems to be the case.