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checking and raising is a more expensive bluff, and betting can be cheaper.

betting also give you better implied odds on average, and 3 bets go in less often.

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Betting out can also be expensive since this line increases the probability that this will still be a 3 player confrontation on the turn, which means the hero will be significantly less likely to take this pot down unimproved.

About the implied odds argument. With a strong draw such as a flush draw in a 3 handed raised pot, I think its better if the hero takes the line that is most likely to win this pot as soon as possible, rather than try to draw cheaply. Just telling you how I approach this situation, not saying your way is wrong and my way is right.

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Are you expecting the 9 or the 7 to be an "out" that you are cleaning up? I daresay that if you fold them with a c/r, you would have folded them with a donk. At least when you would have been called anyway(whether you bet or c/r, like when an opponent has an A or decent T) you are more likely to get the 3rd player in to pad your equity, since realistically the flush is how you are going to expect to win this pot.

I think your general approach is interesting, but analyzing this particular hand IMO favors a lead, and can be swayed either way depending on how you have played recent hands, and how your opponents would expect you to play an ace.

Surf

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I see where youre coming from given this board texture. It may seem that a checkraise is unlikley to win the pot since this board is very likely to have hit the raiser. I would still go for the checkraise here though, and heres my logic, flawed or not.

Looking at the preflop action, The button raised after one player limped in....what does this tell me? To me this means that the button can have many hands that missed this board since he could easily have been attempting to isolate the lone limper. Some of the hands the button may have are KQ,QJ,KJ, other weird non-pair iso-type hands, and 99,88,77 and maybe any lower pocket pair that didnt set up. All of these hands will have a difficult time calling me unimproved on the turn.

The action preflop tells me that the button's hand is not that strong which is why I will usually checkraise with this strong of a hand(a flush draw) no matter what the board texture. I also think that if the middle guy doesnt have top pair or a strong draw he will fold which is usually typical. Given that I think there is a higher than normal chance that the button does not have a strong holding even on this board, I do not want to bet out and give the middle guy the chance to call me for one small bet on some weak hand like Tx, or weak draw like any gutshot.

About my 9 or 7 being an out or not. In these shorthanded situations many times the hero will have more outs than he thinks no matter what the texture of the board is since the likelihood of everyone missing the flop is higher than normal. When this is the case, checkraising to get the middle guy out is paramount. And when you add in the fact that a small percentage of the time(maybe 5%)the hero will induce the button to fold a strong hand like Tx or KK unimproved on the turn, this play has even more value. These are the reasons I like checkraising in this situation better than betting out.