Clearly, A's strategy depends on B's holding.

If A has redrawn, he'll want to raise with any holding stronger than B's and with 1/2 that many holdings which are weaker, and will lose the rest.

That means that the value of of a redrawn hand against B holding b and a pot of p will be:
(1-b)(3/2)(p)
Which is obviously better than zero.
Ergo, for b larger than 1/3, A will want to raise with any holding larger than b's and with any holding less than (1-b)/2.

If B's holding is equal to, or worse than 1/3, B will fold to any raise, so it doesn't matter whether A redraws or not.

So, for B<1/3, the value of the game to A is the pot.
For B>1/3, the value of the game to A is (2-((1-B)(3/2)))(1-B)(3/2)(pot)