[danq]

I'm pretty sure the equilibrium strategies work out as follows. (Let a and b be player A and B's first cards.)

If b < 1/2, there is no bluffing: player A draws if he's below 1/2, and player B always draws.

If b is between 1/2 and 5/8, then player A still draws if he's below 1/2 and stands when he's above 1/2; but now player B draws if A did not, and does not draw if A did.

Finally, if b is above 5/8, player A draws if a is below sqrt(2b-1), a cutoff which is bigger than 1/2 but smaller than b (so player A sometimes keeps a losing hand); player B does not draw if A did, but if A stood, then B mixes his strategies, drawing with probability (1-b)/sqrt(2b-1).

Calculting each player's EV is some messy double integrals, but a quick simulation suggests player A wins 50.5% of the time, as AaronBrown said.

Dan

Edit: Looks like AB solved for low-card-wins and I solved for high-card-wins, otherwise our strategies are basically identical. I think there's something off about my middle case, but I don't have time to fix it right now.