Thoughts on #4
I'm not convinced that any of the solutions to #4 yet are right (unless I've missed an important one among all the posts), for the following reasons:
The value-raise and bluff-raise criteria for B must be the same as in #3 because A has no additional options when B raises. So, I don't see how there can be any difference there.
The ONLY differences between the two scenarios arise when B limps. In that case, A has some weapons at his disposal in #4.
But in case #4, when B limps, A is actually faced with a decision very similar to the raise decision of B in the easy case, but there are some significant differences:
When B raised in #3 he was winning $1 when A folded to the bluff-raise but risking $2 whenever A won.
On #4, A is making $2 whenever B limp-folds but is risking only $1 if B wins.
So, basically, it seems to me that A is just getting 4 times the odds for his raise in #4 than B got on his raise in #3.
That should at least mean that A wants to make 4 times as many bluff-raises as B did. So, rather than bluffing 1/18 of his hands, he should want to bluff 2/9 of them, I would think.
Also, on the value-raise, where B was value-raising the top 1/6 of his hands, A should presumably value-raise 4 times that many, or the top 2/3 of his hands in the range [y,5/6], where y is the threshold where B limps.
A will obviously also want to value-raise on the hands [5/6,1] because he knows he will win, and B will have to call some of the time.
Anyhow, that's about as far as I've gotten with logic. Not being able to figure out how to get a value for y that way, I tried using the brute force method and oddly came up with 1/4 (!!!), but I can't believe that's correct.
Ankenman's argument that A's raising option should actually mean that B would want to fold his thinnest value-calls seemed pretty logical to me, and if for some odd reason the call-threshold should go down, surely it wouldn't go down THAT much.
Anyhow, I guess my main point is that I don't see how B's raising criteria or A's criteria for calling the raise can change in #4 because A has no additional options in #4 if B raises (at least if I'm understanding the problem correctly).
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