[AaronBrown]

I think A has a very small advantage, about 0.5%. It’s true he has more information, knowing B’s card, but he also has to act first.

I think the optimal strategy is if B < 0.433, A should stay if A < 1 – SQRT(1-2B). If B > 0.433, A should stay if A < 0.5.

There are two tricks to this solution. First is to notice this is a different game for every value of B, we do not need to worry about interactions between A’s strategy for one value versus another. Second is to solve for an equilibrium solution, the one where A doesn’t care if B stays or draws, and then prove it’s optimal. This often works for simple game theory problems.