Classic Type Game Theory Problem

[BBB]

I thought about it more, and I think it makes sense that A's edge is very small.

If A's strategy were to never pat-bluff, and B was aware of this, then B would always draw if A stood pat. This would be equivalent to a game in which both A's and B's first card were dealt face up. In this case, neither player has any advantage (either player would automatically draw if his card is worse than 0.5, and if both cards are > 0.5, the better card would stand and the worse card would draw).

So A is able to achieve a small edge only because of his ability to pat-bluff.