Assume cash game 100bb effective stacks.

Hero is UTG and raises 2.75bb.
SB posts 0.5bb
BB calls 2.75bb.

(Pot 6bb)
Flop: 962

Hero's range is: 55+,AT+,KJs+,QJs.
BB's range is 100%.

Can you construct a theoretically perfect range that cbets 2/3 pot?

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For example, Hero bets 2/3p to win p so if hero wins more than [(2/3)/(1+(2/3))]=2/5 or 40% of the time then hero can cbet 100% of his range. Thus BB must defend at least 40% of their range on this flop. What do we do next? Do we construct a value range that is ahead of a 40% continuing range and then balance with bluffs? But how many bluff combos do we choose? Say for example if we are betting 99+ which is 33 combos, how many bluff combos do we need?

Thanks for any help, I have only vague ideas about how many bluffs combos we need and I've never seen an example like this completely solved.

Hmmmm....
We bet for value Sets(6)+Overpairs(30)=36 combos. From what I can figure out we need about 60ish bluff combos on the flop.

These seem reasonable bluff candidates:
{55,AcKc,AcQc,KhQh,KcQc,AcJc,KhJh,KcJc,QhJh,QcJc,A cTc,ATo+}=64combos.

We check back:
{air,Ahi FD's,88,77}

So on this exact flop if we were to raise this simplified UTG range we are cbetting 74% with value and bluffs and checking back medium and weak hands. But now I'm wondering if the check back range is too weak... and maybe some Ahi flush draws need to be in both check back and betting ranges so as to not be exploitable... anyway can anyone critique this process? Is this way off?

I think you're on the right lines, if you've going with Janda's approximation of having 2 bluffs for each value hand on the flop. I haven't run this through a solver, but it's a pretty bad flop for your range (only 6 combos of sets really love this flop), and a decent one for villain (so you also need to think about his continuance range, including his check-raising range).
Since your range is so tight, you only actually have 7 front door flush draw combos, but you have 55 combos of BDFDs if you include all your single heart hands. e.g. AhTs has 2 overs and a BDNFD. You've also got a few backdoor straight draws like JTs/QJs/KJs, and these also block villain from having J9s or T9s (for top pair).
In short, you don't have a high concentration of "robust equity hands". Even some of your overpairs will become weak on many turn cards, so you have very few combos that will be betting all three streets.
The way I would construct my "air" range is to check back some of the ace highs that are dominating villain's Ax, since these have some SDV and equity. I'd be more inclined to bet with QJ/JT, since these have a legit shot to fold out hands that are currently winning (villain's ace highs), and can turn an 8-out draw. If you bet with AQ, you're almost never folding out any better hands, most turn cards don't help you, and you run the risk of being check-raise bluffed out of the pot by 8-high.
Without using a solver, I don't instantly know whether it's actually better to bet the hands containing a high heart, or to check them back. The solution is presumably to mix with a lot of combos. I'm fairly sure that the best ace highs belong in a check back range, however, but as I say I'm not sure whether having a flush blocker/backdoor should make you more inclined to bet them or check them in this instance. It might even be "optimal" to occasionally check back some overpairs.
On the whole, this might mean I don't "bluff" all that much in this spot, as I don't have enough combos that naturally fall into the "obvious triple barrels" bracket (like 87s would be). With so many overcards in my range, I'll often check back and make it a two-street pot instead. You really need a solver for spots like this.

Hi youareawesome,

I think your checking range is too weak and including flush draw to check back range is good but exploitable if turn is or is not a flush. Also highly exploitable if the turn is or is not an A (v. Can overfold or overbet vs your capped range).

About the initial question I'm sorry if I dont reply but I think your assumptions are wrong unless you re trying to experiment with something original. Because BB cant have a 100% range or he would never 3bet AA KK QQ AK etc... bb never 3bet. Well these assumptions are very possible at a 1$-2$ Live game.

P.S. I don't think it's a "simple" question, like the thread title suggests. The board is very dynamic (the turn card will often radically change the strength of many combos), and you've got an SPR of over 16. All kinds of stuff can happen whether you bet or check the flop, and it's basically impossible for a human to calculate the best lines. In fact, the only combos I'm absolutely certain should be c-bet are 99 and 66. A solver might say that checking back some combos of AA (or even JThh, which initially appears to be a slam-dunk c-bet) is best.
Oh wait. JTs isn't even in your range for some reason. You have so few good semi-bluffs that maybe you shouldn't be betting very much at all in this spot.

Thanks for the replies.

So our cbet ranges, on any board, are always approximations and there's no perfect solution? (Solvers included).

Also, @kingkong352 I am trying to simplify the question. That's why I've removed combos from the UTG range. Rather than the solution itself being helpful, I'm trying to learn the process of creating the solution. And it seems I'm on the right track, but you've pointed out some problems about having a weak check back range. So as @ArtyMcfly said, maybe some AA needs to be in the check back range.

A question about this "You have so few good semi-bluffs that maybe you shouldn't be betting very much at all in this spot." How do we choose cbets? I was thinking we choose value first, and then balance. But are you saying we find value, balance, and if there's not enough good semi-bluffs then remove some of the value?

Bet everything and print

Perhaps, we have about 45% equity vs a 40% continuing range and assuming the big blind plays perfectly and realises this equity then

They fold 0.6 and we win 6bb.
They call 0.4 and 55% of the time we lose 4bb.
While 45% of the time we win 10bb.

0.6*6+0.4(-4*55%+10*45%)=4.52bb per cbet.


(is this equation correct?). Pretty awesome result. But how to test if this is optimal? I might fiddle around with it tomorrow night because I think I know how but time for zzz...

There's always a GTO solution (if you know both players's ranges). It just won't always work out to have a predictable ratio of X amount of "value hands" to Y amount of "bluffs", partly because so many combos can't be described strictly as "value hands" or "bluffs". e.g. If you bet AQ high and get called by A8s but you fold out 33, you're simultaneously getting called by worse (value-betting) and folding out better (bluffing).

Working out what to do with each hand in your range is strongly affected by how villain plays vs your action. If he check-raised every pair+ or draw, you wouldn't want to bluff very often (or bet your thin value/SDV hands), unless villain has very few pairs and draws (so would be folding very often). If he folds quite often and mostly check-calls the flop, then you make more money by betting at a high frequency, since you'll seldom get raised, and will be able to deny his equity, while realizing yours fairly cheaply.