Here's my take on it, show me where I'm wrong.

Terminology:

Avalue is Player A's card. Bvalue is player B's card.

Axioms:

First off- your odds of winnign a hand is 1-value. A 1 has no chance, a 0 has 100%, a .5 is even money.

Secondly- a randomly drawn card, assuming even distribution, has an average value over the long term of .5.
This means if a player draws, his hand basicly becomes a .5 (averaged over many many hands).



Strategy no bluffing:

Starting hand 1: A>B, B<.5. A should redraw. B should stand pat, as he is expected to win over A's .5. B wins 1-Bvalue percent of the time. Bvalue can be assumed to be the average value for the bottom half- .25. So B wins 75% of the time. This combo occurs .5*.75=.375 of the hands. So B gets .375*.75=.28125= $28.125 of expectation.

Starting hand 2: A<B, B<.5. A should stand pat. B sees the stand pat and redraws. A wins 1-Avalue percent of the time, B wins Avalue percent of the times. THis occurs .5*.25=.125 of the hands. B value's average hand is .25. A is less than him, so his average hand is .125. So B here gets .125*.125=.015625, or 1.5625 in expectation

Starting hand 3: B>.5. This is the tricky one. If A stands pat, B will redraw as he is beat. If A redraws, B has to redraw as A is now a .5. So B has to redraw either way. This puts him at .5 A will decide what to do based on his card. If he is <.5, he stands pat, if >.5, he redraws. When he redraws (50% of the time, or 25% of the total of the game), he has a .5 and thus a 0 expectation. When he is under .5, B will win 1-Avalue of the time. A here over the long run is .25 (middle of the range). This occurs .25 of the time. So B's expected value here is .25*.25=.0625=$6.25 in expected value.


In short, without bluffing B will win 35.9375% of the time. Huge advantage for A.


Adding in bluffing is more complicated, I'll try and post later. It may need to wait untila fter work.