[Bozeman]

I get that B should bet his best 1/6 hands, bluff his worst 1/18, and A should call with his best 1/3.

I may have made an arithmetic mistake, but the value of the game for B is 5/18=.278, so this is better than ProfessorJC's sol'n.

Justin, just to show you are wrong, I solved for the best non-bluffing strategy for B.

Here, let B bet his hand if it is better than b, and A call with hands better than a. Here a>b, since A knows he is throwing away money calling with a hand worse than b.

a=3b/4+1/4

The value of the game for B is

1/4-1/8*(1-b)^2

This function has a maximum at b=1: namely b never bets, he calls (with hands above 1/2) or folds.

Craig