I'd like to examine a few board-types here but will start with A-rag-rag boards (the other ones I think are very favorable for steals are Q-rag-rag and paired boards with the pair 4 through T, but I'll try to get back to those later).

Here's what I was thinking:

If we start off with a tight raiser (AK, JJ+), then on an A-rag-rag board, the raiser has hit his hand 12 times on AK and 3 times on AA for a total of 15. But he's missed 6 times each on JJ-KK for a total of 18. Hence, more than half the time, he does not like this flop.

So, let's say you called the raise with 99 and missed your set (or 87s and missed your draw, whatever). If we have a pot of 20 and he CBs 15, well, you're still far from ahead most of the time, but he's also weak a slight majority of the time.

What are your options?

1) Float. The real problem with this is that the play has become so common that I think KK may be tempted to check-call or else to fire the turn again. It might not be a bad idea, however, on drawy boards (straight and flush draw) where you can bluff the turn if any of those hit. If the board isn't drawy and you have a decent image, it does admittedly look a lot like some kind of AJ-ish hand. So, this might be the best option.

2) Mini-raise. Also viable.

3) Real raise, looking like AK. I think this is a good option only if you really do have some A, hence significantly reducing the number of times the board looks good to him. The problem otherwise is that you really need to raise to about 45 at minimum for an "information raise" and the tight raiser has hit sufficiently often that the bluff becomes too costly when he does like the flop. That's still 15 times vs. only 18 folds, and you're risking 45 to win 35. So, the math just doesn't work out.

Vs. a loose raiser, who is going to have a lot of AQ hands for sure, maybe some SCs and some small PPs in the range, I do think the real raise is the best option, since AQ objectively should lay down--IF it's plausible for you to have AK and your opponent is otherwise solid.

I'd be interested in hearing thoughts on floating and mini-raising, though. I think I'd probably end up laying down KK in both instances actually. But the question is what other players would be likely to do.

OOP the check-mini-raise might be the best way to go. With position, often the float, although there, too, the mini-raise can look settish...

Thoughts?