VIVA LA CHILE

first hand qckd6htd table 2 flop 3s6sJh:

Can you explain what your rational is when you say that you would not be inclined to cbet this hand when you anticipate he will be c/r 15% on this board whereas you would be when he would c/r 10% or less. It seems perfectly reasonable that you would have a threshold for how often villain must check raise before you start checking back but I am wondering what you are basing it on. Is it just a rule of thumb based on experience? Is it what you notice other players are doing? Is it based on some sort of equity calculations? I think it's pretty cool that you have a distinct number you are looking at and I'm just looking for a general impression of how you come up with this.

Hey Alex, Thanks for your response, what you said makes sense intuitively. While I was awaiting your reply I ran some numbers using PokerJuice and what I find really interesting is that I think the results may be counterintuitive depending on the range that you assign to opponent. That is, you may do better c-betting when he c/r 15% than when he c/r 10%. Also based on my assumptions and analysis, c-betting will do better than checking back only when he has a 15% three bet and not when he has a 10% 3 bet.
I have villain flatting pre with 85%-$3b15o against your hand KdQcTd6h
Here are my assumptions for villain value c/r (hands he can get in with):
ass:6+,ass:74+,ss:457,j3+:ss,j3+:45+,66+,j6+:25+=7.35% of his pre flop range so close to what you said

adding in these hands as c/r bluff brings his total c/r to 10%:
45!3+!2!7!ss,as:75+!j+:*!ss,ahh!j+!45+:(57,74),ahh!j+:(78,qt),qtk:hh!ss!j (some dry gutters to the nuts, most dry As, some ahh without much else, some backdoor straight and flush draws)

Also adding in his super dry Jx will bring it to 15%
j!ss!25+:[T-][T-]!3!6

I have him calling this subset of these hands that are not in his c/r range:
6+:25+,99+,ss,245+,3+:45+ (6 plus with gunshot and better, 99 and better, ss, wraps, and 3 with open ender or better)

Overall this is a pretty wide flop range that has him defending close to 61% of his range at a slight overall range disadvantage (48-52% range vs range on flop) making it difficult for your weakest hands (like the hand you have) to cbet profitably.
The difference between the 10% number and the 15% number is that he moves his weakest and most dry Jx into the c/r range in the 15% instance

So my next set of calculations I make the following assumptions:
1) Your hand plays at least as well on turn vs the subset of hands that he calls the flop with against your bet as it would against the same subset of hands if you checked back the flop because you leave your range uncapped by betting
2) Given 1, you can calculate profitability of flop bet based on the equity you are able to deny him vs the equity he is able to achieve or deny you

Also it is important to note that his calling range has better equity than your hand so for the purposes of this calculation I am just having him achieve that equity, although you should be able to do better in position than allow him to achieve 100% of it

These calculations are from poker juice based on the ranges I mention above.
So: (How often he check folds)(Equity his c/f has against your hand)(pot size)-(pot size after you bet and he calls)(his equity in that pot)-your bet-his portion of the pot if you had checked-(his c/r frequency)(your bet plus initial pot)(the amount of equity your hand has vs his c/r range)

vs his 10% c/r:
(.39)(.38)(225)-559(.64)-167-144-(.10)(.28)(392)
33-24-11=-2

vs his 15% c/r
(.39)(.38)(225)-559(.57)-167-128-(.31)(.15)(392)
33-11-18=+4

Conclusion: Even if opponent is able to achieve 100% of his flop check call equity, you do better betting against his 15% c/r range than checking. Against 10% you do worse based on this calculation, but again this is assuming he achieves all of his check call equity, so this might also very well be a bet. For this to be otherwise, you would have to have different assumptions about the composition of his flop check raise range. In this instance because you achieve a lot of equity against his c/f range and because you are not really concerned about folding if he is c/r a lot of his dry Jx, a bet actually works well.
Also, I don't want to go down this rabbit hole, but I think it would be really hard for him to c/r more than this without being exploitable. And adding hands that he would otherwise be folding and do not block Jx could be really difficult to play, especially since I already gave him what I think is a reasonable amount of gutters, dry open enders, dry As, and what would amount to his stronger backdoor draws, all of which can't call vs shove on your part.

Apparently brackets function as some sore of hyperlink in red so the parts of the equation in red should look like this ((559)(.64)-167-144))(.51) and ((559)(.57)-167-128))(46)

((Pot size after you bet and he calls)(his equity)-bet-his portion of 225 had you checked))(frequency at which this occurs)

Alex thanks for your response. In the interest of not spending a ton more time on this problem, I am going to try to explain more generally what I am trying to do and then address your specific concerns.I will highlight my assumptions to make them more obvious.

I. Overall argument
1. My main point was that perhaps we should be more inclined to bet against an aggressive check raiser with our specific hand, even though we would be forced to fold it.
2. My secondary, although more implicit conclusion, was that with this specific hand, KQT6, we should probably C-bet regardless.

II. Your concern number 1
I agree with you that opponent is not likely calling 99+ and folding some of his weakest Jx and probably some of his 6+gs. Believe it or not I did not include those hands in my initial calculations. The reason I chose to include them here is that I was trying to make it as potentially unprofitable as possible for you to bet your specific hand KQT6. The fact that he is leading 5% (which will certainly include some of his stronger hands that might detract from his value c/r range) and only continuing with 50% speaks even more in favor of my argument that you should cbet.

III. Your concern 2
I do not understand why you think this. All my calculations reference the bet size you actually made: 167. Nowhere do I include you making a pot sized cbet as far as I know.

III. Your concern 3
Referenced above

IV. Your concern 4
I do not try to do this. Here is the one fundamental assumption I make explicitly:
"Your hand plays at least as well on turn vs the subset of hands that he calls the flop with against your bet as it would against the same subset of hands if you checked back the flop because you leave your range uncapped by betting"

In other words, all I am assuming explicitly is that you do at least as well against the range he will call your cbet with as you would do against the exact same range if you checked back the flop.

Here is my implicit assumption (and this is taking into account the new set of ranges I am giving him that are more consistent with the numbers of 5%,50%,45% -lead, defend, and c/f you give me in your post):
When you check back you are not able to make more money with your specific hand against the part of his range that would c/r you off in on the flop as opposed to the amount of money he is able to achieve when you check back with the part of his range that you did not cbet him off on the flop. Because the part of his range that you did not cbet him off on the flop constitutes 46% of his range and has 35% equity against your hand and he is check raising you off of 41% equity 14% of the time, I think you can say clearly that regardless of your turn strategy you will never do better with these portions of your range by not Cbetting the flop.

V.
What I will now try to do is rewrite my formula using the actual as opposed to theoretical range you give villain, plus a high c/r frequency, and try to see if that confirms my argument that a Cbet will gain EV against his range.

Here is his new overall continuing range including all of his c/r, c/c and hands that don’t lead.

His new c/r range:
j!ss!25+!3!6,ass:6+,ass:74+,j6+:ss,ss:457,j3+:45+,66+,j6+:25+,45!3+!2!7!ss,as:75+!j+,ahh!j+!45+:(57,74),ahh!j+:(78,qt),qtk:hh!ss!j+,6:(74)!ss!3!j!qq-aa!5,6:(75)!ss!3!j!qq-aa!4 (includes the 6x and gutshot combos you mentioned)

minus

j!ss!25+:[T-][T-][T-]!3!6 which he is now c/f and j3:ss which is is now leading
=15.47% c/r of his preflop range (59% against your range)

His new calling range
j+,ss,245+,3+:45+ again minus his c/r range, leading range, and j!ss!25+:[T-][T-][T-]!3!6

which constitutes another 37.35% and has 58% equity vs your hand

Now the portion of his range that is c/f which is about 46% of his total pre flop range (which corresponds closely to what you said based on the fact that he is leading 5% and defending 50%):

*!((j3:ss,j!ss!25+!3!6,ass:6+,ass:74+,j6+:ss,ss:457,j3+:45+,66+,j6+:25+,45!3+!2!7!ss,as:75+!j+,ahh!j+!45+:(57,74),ahh!j+:(78,qt),qtk:hh!ss!j+,6:(74)!ss!3!j!qq-aa!5,6:(75)!ss!3!j!qq-aa!4,j+,ss,245+,3+:45)!(j!ss!25+:[T-][T-][T-]!3!6))

and this 46.37% of hands have 35% equity against your actual hand
so when you check back these hands will remain in his range whereas when you bet he will fold flop

So now I believe that the difference between betting and checking can be distilled to two decision trees- one involves you checking back against the 95% of his preflop calling range that does not lead and the other involves you c-betting, getting him to fold his folding range, getting him to call his calling range, and him folding out your hand with a c/r
So the equation for how much equity you get him to fold when you bet vs how much equity he gains vs you when you bet is (keep in mind this equation takes into account that you are betting 167 into 225 (for my equations I did not include rake but it is negligible)

(How often he check folds)(Equity his c/f has against your hand)(pot size)- ((Pot size after you bet and he calls)(his equity)-bet-his portion of 225 had you checked))(frequency at which this occurs)- (his c/r frequency)(your bet plus initial pot)(the amount of equity your hand has vs his c/r range)

(.46)(.35)(225)-((167+167+225)(.58)-167-(.58)(225))(.37)-(.14)(.41)(167+225)
36-10-22=4

So using numbers that correspond much more closely to the hands and numbers you have me, this bet will be more profitable than checking back as long as you meet the conditions I reference in IV which I think you undoubtedly will. And in this case, this will also extend over to the cases in which opponent is using a lower check raise frequency because he is folding an even larger percentage of preflop range.

Also, as i mentioned before I think this hand can easily be bet as part of a, balanced and difficult to exploit betting strategy on your part but showing that would require many more calculations.

I realize that the way I analyzed the hand was somewhat strange, but I believe viable, although certainly difficult to articulate; I probably could have done a better job If you think there is something off about the assumptions, please let me know.

Hey Alex, thanks for responding, I understand that it must take you a lot of time to reply, particularly given the language barrier. My Spanish is pretty good but no doubt no better than your English, particularly discussing something with so many uncommon words.

Ok so in the course of thinking about what we have discussed I realize that we've changed topics and that may be driving some of the confusion. I think there are still a few things left to clarify.

The first point, with which you agreed in your last post, is that a C-bet is good on this flop with this hand. So we maintain a consensus on that point. Just to make sure, do you think that this is a decent C-betting range?

j6+,ass:3+,ass:25+,ss:45+,aa-qq:ss,j+:25+:ss,j+:457+,j3+:25+,63+:ss,qq-aa:45+,qq-aa,j:ahh!(57,54),6!45+!3,57!ss,99-TT,77!ss,55!ss,57!6+!ss!4,47!6+!ss!5,qtk:hh,jtq:hh

It's about 55% and includes the hand we have been looking at. It also includes a fair number of SS and straight draw check backs, although most j and better hands with nothing else are being bet, because all bets, based on his defending frequency, are immediately profitable, and these would be the hands that we most like to just win on the flop. It also includes enough value hands (conveniently listed first and mostly in order of how well they can be defended against a check raise) that I think he would have a hard time attacking us very light.

I realize now that the argument has shifted. The reason I chose to use these equations in the first place is because in your video you make the argument that you would be more inclined to C-bet against a lower c/r frequency and against an opponent less likely to attack you light. When I initially heard it, it made sense. But testing the assumption against an opponent most inclined to not let you C-bet profitably with any four cards (that is he has a high c/r frequency and a wide calling frequency), I established that based on the nature of your hand and the nature of his potential c/r range, that you actually did better by C-betting when he was more inclined to c/r light and defend wide than when he was less inclined to do so. My calculations in my first post are in favor of C-betting against someone who will attack your bet more aggressively. In the course of conducting this analysis I realized that the reason for this is that if he is going to widen his c/r range, the most likely candidates to do so with will be his really dry Jx hands against which your hand does poorly anyway. In which case C-betting your hand is good not only as a range play but also in a vacuum.

I've never heard anyone say, "I think we should be more likely to bet this hand against a higher c/r frequency even though we will have to fold to the check raise."
Which is why to me this conclusion should be an important building block in constructing your overall range. Obviously you want to have a more global perspective on your frequencies, but you also want to know how each individual component works. Knowing that your hand actually works better against a high c/r frequency may mean in the future that you use that stat differently than the standard way must of us are accustomed to.This will depend on your actual hand, the board, and the composition of your opponent's potential check-raising range.

I think the argument shifted in your second post because you did not accept the composition of my theoretical range in the first post. So all I was trying to establish in my second post was that even against his actual range, that the equation I was using in the first post still holds. Granted this is a micro perspective. Showing that C-betting your hand works as part of your overall range is another matter. But again, if you think that the C-betting range I have listed above works against the range he is playing, this should not be a problem.

As for my equation. I think that the equation I was using is different than establishing whether a bet is profitable or not and works better in the context of the first post.Obviously a bet remains profitable as does likely a check back, but in the context of the first post I think the equation helps establish that you actually gain more against an aggressive check raiser than you do against a less aggressive check raise. I think you are right in asserting that my equation is not as good for the second post. And I think some of the conclusions I made in the second post are overly broad and that I state them too strongly. The equation does not definitely establish that betting gains more EV than checking, but I think it shows more than you think. First of all, to reiterate, the equation is different than an equation to establish whether or not the bet is immediately profitable. As you recognize, you could do that much more simply .75/1.75= .42. This time, as will usually happen, the two equations correlate, but this will not always be the case. For instance, in the first post, in both instances he is continuing with about 62% of his range but one equation gives a negative output and the other a positive output. What the equation does show is that you gain more immediate EV against his range than he gains against yours by you betting. So you are not actually too worried about him calling so frequently with such a strong hand relative to yours that you are actually costing yourself by betting in a way that you would not cost yourself by checking. And I think my analysis points more generally to the fact that you have less to lose against his 15% c/r range that folds you off on the flop and 37% of hands against which you are an equity underdog and calls the flop then you have to gain against the 45% of his range that will just c/f the flop. My equation says he folds equity immediately. Your hand does not have good back draws and by checking back you are capping your range. When you bet will be be able able to fold him off a lot of his equity immediately and your range will remain uncapped which gives you more room to navigate against his 37% calling range.

But you are right in asserting I did not prove that you definitely make more money across three streets by betting rather than checking.And I think you are also correct that most of your analysis should be directed at establishing whether or not the bet fits well into an overall value/bluff, board coverage C-betting strategy. But it is also useful to establish what your bet is accomplishing against different parts of his range so that when decisions are close or when you are deciding more generally what types of hands to include in your overall C-betting strategy against his range, you know which hands work best. For instance, in the future if you still feel that as part of your overall strategy with your range this hand would be a borderline C-bet and you want to play a mixed strategy with it, after conducting this analysis you may actually be more inclined to bet it when he has a high c/r stat and more likely to check it back when he has a lower c/r stat even though on first glance this reasoning seems somewhat counter-intuitive.

Hey Alex, looks like we now agree on most stuff.
That's actually probably an important part of the equation to explain.

Ok. So again when you bet, you can subdivide his range into three parts. One includes the hands that he will c/f. Another includes the hands that he will c/r. And the third is hands that he will check call. As stated previously, his check call range has an equity advantage vs. your hand. So when you bet the flop he gains equity with his check call range on the flop that he would otherwise not gain if you checked back. So when you are calculating the equity of a C-bet you stand to gain immediate equity against the part of his range that c/f the flop.However, he stands to gain immediate equity against you when he c/r and makes you fold your hand. He also gains equity when he calls you with a range of hands that have better equity than your hand.

When he folds:
The pot size is 225 and his c/f range has 35% equity vs your hand and he will fold around 45-46% of the time so this is equity you stand to gain: (225)(.35)(.46)

When he calls:
After he calls the pot size will now be 225+167+167 and he will have 58% equity. To get to that point, though, he needed to call your bet of 167. Also he only gains equity with this part of his range because the pot size is larger so you needed to discount the amount of equity he would already have in the pot anyway if you instead had checked back so (225)(.58). And overall, he check calls 37% of the time vs your bet. So here is the equation for the equity he gains with his check call range when you bet the flop.
((225+167+167)(.58)-167-(225)(.58))(.37)

When he c/r and you fold:
After you bet the pot size will be 225+167. You will have 41% equity vs his c/r range and he will be c/r 14% of the time.So the equation for the immediate equity you stand to lose is:
(225)(167)(.41)(.14)

So the overall equation for the equity you stand to gain immediately looks like this:

(225)(.35)(.46)-((225+167+167)(.58)-167-(225)(.58))(.37)-(225)(167)(.41)(.14)

Just reran it to check. Sometimes, when I am running many sims, Poker Juice will shoot out slightly distorted numbers. It appears that number is wrong. Your hand only have 33% against his c/r range.

However, I don't actually think it would matter even if it were 41%. Obviously, it appears worse the more equity your hand has vs his range. But as shown in my first post, even against a low c/r percentage and a wide continuing range you stand to lose immediate equity by betting (that's the -2 number from my first long post). And in that case your hand has only 28% equity vs his check raise range. So comparing it to this output demonstrates that you would still rather bet your hand in this instance, despite the fact that he check raises more frequently and you would be folding significantly more equity when he check raises. Again, it is counterintuitive but I am using the exact same equation so it is comparing apples to apples.

It would matter in a more general sense if it were 41% but significantly less so for the type of analysis I am conducting. As an extreme although plausible hypothetical,because I don't have time to do a more practical one, let's say you have the option to C-bet or check back the flop and when you do C-bet he will check raise 15% of the time and you will have to fold 50% equity. But he will also fold 50% of the time to your bet with a range that has 50% equity vs your hand.In this case, you would have to believe very strongly, and probably unrealistically, in your ability to take equity away from your opponent on subsequent streets to justify checking back the flop. Our particular hand definitely lies somewhere in the middle so is much more of a gray area because he is folding 35% equity 45% of the time. What my equation does show is that even if it is the case that he check raises you off your 41% equity 15% of the time, it is still a better scenario than the case in which he check raises you less frequently but calls you with a very wide continuing range that has good equity against your hand. And because this is a profitable bet and fits into a balanced strategy the onus would be up to the person inclined to check back to demonstrate that with his check back strategy he can achieve more equity on subsequent streets by checking back, than the equity he can achieve immediately by betting- which makes the equation I use for the immediate equity that can be gained with a bet, a valuable one.

Thanks for the video!

In the AAQ3dd hand (around minute 11) you estimate your equity against his range to be ~40%, so its -EV to shove. But what about calling? If we assume that he always shoves the turn and you call on any A,2,3,4,5 or diamond and fold on any other card (you especulated something similar), you'd need to have at least around 55% equity on the "calling turns" to make the flop call +EV. Do you expect your equity to be smaller than that against his range?