I'd rather check back 66 than AK since it doesn't mind facing a x/r too much.

Here is the way I think about a check back range: I want to check back hands in the middle of my range because I want to get value with my strong hands and want to bluff with my weak hands. However, I don't want a range that is going to be medium strength regardless of the turn card that hits. So I need to pick hands that have a lot of cards that change their value on the turn.

I don't think the fact that you don't like being raised is enough reason to check back, especially if you feel the board is bad enough that you can immediately fold. I bet hands that hate getting raised all the time.

I don't think that's the right way to look at it. I think you should think more about how to realize your equity with various hands. Bet folding 66 here does a poor job of realizing it's equity. The reason you bet flop with a horrible hand as a bluff isn't cuz it's at the bottom of your range it's cuz it has very low equity and it's unlikely the turn will change that. It would be preferable to bluff a hand with no backdoor draw to one with a backdoor for this reason. Unless the hand is strong enough to bet/call, then that's a whole other can of worms.
OTR: I think AJ and A9 make sense to check down.

My turn betting range here is polarized to 8x and strong draws. I don't think bluffing with low equity draws is profitable in this situation so I would continue to check with those hands as well as underpairs and my weaker middle pairs. Then on the river my range looks like total bs to most players so I can value bet those middle pairs and underpairs. Throw in some bluffs with the very bottom of my range on the river and my expectation will be near (pot) because really, who's gonna check a strong hand 3 times from the blinds there?

well yeah, if you think you have to call a c/r or to see the river.

Well... The point I'd like to make, in a sort of round about way, is that I believe that AJ, A9 type hands may play better as b/c on turn and 66 as a chk back despite the fact that AJ, A9 are weaker hands in terms of absolute strength. AJ and A9 just fare way better vs villains' continuing ranges (whether they xr or xc the turn)

AJ is an awful checkback. Way too strong.

On which street?

Flop

Never chking AJ otf is a mistake imo.

I think part of the reason the turn is so tricky in this spot is that people's flop check ranges are too weak. I think if you're gonna have a checking range, then you need at least some 99/JJ/KK/AA/J9s/KQ/QQ/88/TT type stuff. Especially if our opponents play the turn in the way described in this thread. Nobody will expect you to have those hands so you'll still get a lot of bets in when they c/r their AT or bet/call 77.

Against weakfish, you should absolutely follow PaulV's tagbook strategy. Bet everything for value, and have a weak decapitated checkback range. You will rarely get punished and get maximum value.

Against better players, this strategy still isn't that bad. Your checkback range has some weak showdown hands so they can't bluff their entire range. The problem was trying to turn a weak showdown hand into a thin value bet -- that part be disastrous vs better players.

If we have some better hands in our check back range, then we don't necessarily have to give two free cards the times we check the flop with 66 or AK, for example. We can bet at least some of those on the turn. If our range is capped at, say, 77, then I'm not sure we can. Also, our opponents can't necessarily bet for thin value if we occasionally check hands that can raise some turns.

I agree with that

Checking back such monsters gives up too much value for the sake of deception. The goal is to find hands that won't lose too much by checking, but still have a range that is capable of becoming strong on any turn card.

I'd bet AJo 100% on the flop. I think that mixed strategies are unnecessary and less profitable than pure strategies in multiway pots, but that's just a loose theory.

As villains become more terrible and plentiful your predictable ranges are less likely to get exploited. In this spot, I'd want my range to be not completely predictable.

as paul mentioned, you want hands in your xback range that can raise for every run out. So, which Jx hands are you going to xback? AJ has too much value; but you can also reason so does KJ QJ JJ TJ 9J. I'd also argue you want to xback some Qx+ hands, Tx hands, etc.

At some point, you need to forgo some flop 'value' if you want to have strong hands for any turn card. besides, you can make up lost value on the turn against agg players.

Regarding mixed ranges: I find it easier to choose which Qx / Jx to check back via suit instead of rank. If I want to check back ~5% of my Qx, pick an ordered suit combination. Check back all those combos. That's easier than delineating at a given hand, and gives us better balance.

I used to think like this but not anymore. Even if both opponents are playing their parts of the equilibrium, it's not obvious nor necessarily true that there will even be a single hand in any of the players' ranges that have more than one option of equal expected value.

You're probably right that, on average, we are losing value with the very top of our range by checking. The question is how much are we losing, and how much do the other hands in our flop checkback range gain from the addition of the strong hands. I think it's helpful to take a look at the GTO HU solution as a starting point.

On Q8Tr, Cepheus checks some very strong hands including AA (about a third of the time) and QQ about 10%. By your logic, shouldn't he be betting these so he doesn't give up too much value? Especially in a HU situation where his opponents will often be c/r with random 9x/Jx/Qx etc?

Now obviously OP is in a 3 handed situation with much stronger ranges than the HU situation (possibly he shouldn't even have a flop check back range, I don't know). But in figuring out how to construct a GTO-ish checking range on this flop, it seems smart to start with an actual GTO solution, and then try to extrapolate. The alternative is just pure speculation.

The EV of an option with a hand depends on what you do with every other hand (and how opponent views your range)

eg, on the river, if you literally only bet with the nuts and your opponent knows this, your bet isn't worth as much as it would if you had bluffs.

There are better ways to construct your range than others.

Being predictably unbalanced will influence the EV of your hands.

At the highest level, you should strive to be balanced such that even if your opponent knows your exact range, they can't do anything about it.

I don't think that correlates with finding the highest EV play of any single hand.

Checking back may induce a bluff against a steamer, which is a good reason to try this line, and call down.

Good post

We can agree to disagree here. I think that balance is a byproduct of good poker and that good poker isn't a byproduct of balance. If I wanted to reformat the above statement:

You should strive to maximize expected value such that even if your opponent knows your exact range, they can't do anything about it.

You think this statement is possible to achieve without being balance ?