The real answer is that you should be mostly calling and not raising much either for value or as a bluff after the chk back. You should bluff hands like this at a very small frequency probably.


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Of course, this makes sense. Thanks.

I'm pretty sure even if we were designing an AI algorithm to test out against other optimized AI algorithms that having a mixed strategy in these situations would work out to roughly zero difference in EV.

Having a mixed strategy for a specific hand is relevant basically only in spots where you're on the threshold between two different actions where you can't put more into one column without losing balance and in cases like this where ranges are so wide even these threshold hands offer trivial improvements.

It makes the analysis needlessly complex while ignoring the stuff that actually matters.


The important thing is that if you choose to check Q high that you also have a large number of ace highs (and maybe king highs, bottom pairs, etc) that also check back that you call most turns with. You'd need a solver to know exactly where the cutoff points but it's really not that important and makes a small difference to your overall strategy as long as it's done with the right directionality (ie: you aren't checking your gutshots or hands with low showdown value that have a combination of backdoor draws while betting your king high / ace highs) and proportions (given the pot size relative to the bet / the fact that his bluffs have equity you should be calling the turn bet with something between 2/3rds and 3/4 of your flop check range).


Wrt the turn raise I think you probably should almost never have a value hand so I wouldn't do it, but if you did want to balance it for theoretical purposes for the odd time you have 22 or slow played a set, I guess you can take some solace in knowing that Q4s would probably be close to the ideal hand to do it with. Why? You shouldn't have J4, T5, 95, 84 type hands that have lesser showdown value, and your K4/A4 hands have enough showdown value that they perform better as a call. Or just save your money and don't do it.

If this is true, which I think it is, it's because computers don't exploit. If we add the potential for exploitation to the problem, using correctly mixed strategies will become a necessity to maintaining a minimum ev.

I think mixing is necessary with key draws. Pure strategies in poker are particularly vulnerable to exploitation. I mix as default and if I decide that my opponent is bad enough in some way, I will use a pure strategy vs such a player.

You make it seem like I just throw away 30 years of poker theory advancement in the name of "gee um mixed strats are cool now so I play a mixed strat." I actually argued for pure strategies being superior for a long time as I also felt that mixed strategies were unnecessary. My reasoning was that because two options have the same ev(resulting in a mixed strategy), it didn't really matter which option we chose. Now, I see that mixed strategies perform well vs all player types. Pure strategies may be great for exploiting with reads, but when we don't have reads I think a mixed strategy is best.

I rambled about this topic recently here:

https://forumserver.twoplustwo.com/1...ought-1731910/

and now I'm here:

https://forumserver.twoplustwo.com/1...egies-1734702/

I agree there are fringe cases where having a mixed strategy with some number of hands adds some minute value to the EV of a decision tree - just that it's so small that it doesn't warrant as much attention as it's already gotten (at least in limit hold em). In big bet games it matters a bit more.

If having a balanced check back and optimal call down range in these spots adds 0.25bb/100 to your winrate against a typical field of opponents, adding in mixed strategies for specific hands is worth something to the tune of 0.000001bb/100. That might even be too generous.

I agree that it doesn't matter vs a non adjusting opponent; we could play a pure strategy vs a gto bot and we wouldn't lose any ev with those hands that should mix. However, I believe that the liability vs human opponents is real.
I'm not claiming that using mixed strategies will add to your winrate vs typical opponents that don't adjust, but I do think that it helps maintain a minimum ev vs players that do adjust.

Flop check is good. Turn, seems like we're gonna have way too many bluffs if we're raising with this hand. Like we'd have to be checking overpairs on the flop sometimes to balance having this in a turn bluff range, which doesn't seem appealing to me.

I'd call a hand like this on the turn and probably be calling most of my continues (raising 22).

I would bet the flop. I don't think we need a check back range on this board especially against the described opponent. Our range is still strong on this board. Villain will have a few jacks in his range but will also have a lot of suited cards that our range beats. We need to live with the fact that we may get check raised sometimes... and if our backdoor draws don't improve we will need to give up. We can't get to showdown every time and we will be calling with enough bluff catchers not to be exploited.

For those advocating checking back the flop: how do you determine which situations to include a check back range? How do you go about balancing your cont-bet range and your check back range

I call the turn as played.

Don't get me wrong, I think 100% cbet is going to perform fine here.

Hand waving but I would probably include some Ax, 55, stuff like K hi and Q hi with backdoor potential, maybe throw a stronger hand in like a top pair with a weak kicker once in a blue moon.

FWIW I've played a lot with OP and he's going to be played back at a ton on textures like this, so it's probably better for his overall strategy to have a check back range in more creative spots like this one, where most players bet 100% and don't really think about it (which as mentioned, is a fine enough strategy and one I generally employ).

Having done some solver work now I think the answer to this is more fundamental - the ranges are simply too narrow for a strategy to have pure 0%/100% options for each combo. Compare LHE to a game like omaha8 where there are tons more hand combinations - you can use a 0/100 strategy for each hand combo in o8 and be fine, but if you try that in LHE you might have some gaps in your strategy. Now against humans, its probably find to "round" your decisions to 0 or 100 and not do too poorly, but still, you will eventually end up in some narrow range spots where you can't come close to a 0/100 strategy without deviating massively from the "gto" answer.

Jeebus that took several minutes to grok. I'm sure I'm prime for a "tear it down and start over" but I was vaporlocked on but v bb is *too narrow*?

Ya DD nails it. Which is exactly why a play like in this thread can be fine, assuming it's a low frequency play.

I've said from my first post that I agree there are edge cases where it adds value. My contention is how small that value is, and i should add, that without using a solver we're basically just blindly speculating which hands would be mixed. A greater percentage of hands would be mixed in hold em than 08 but it would still not be the majority. Most hands would still be a pure strategy.


What do you think the value of not having a mixed strategy for any hands would be in this spot against even the worlds best human player? I would guess it's less than 1/1,000 of a big bet for the flop decision.

The difference on the turn is probably more noticeable but still very small. If you have a check back range the mixing may be seen in how you treat something like K9 high where you're supposed to call 25% and fold 75% (or something similar). And so if you choose to fold 100%, then all of sudden if your opponent has perfect knowledge of that error, his bluffs show an incremental increase in profit and then he adds an additional few combinations to his bluff range, and reduces his value range by one or two combinations.

And in that exact scenario - given that you have the exact edge case where a mixed strategy is valid, AND your opponent is an expert player AND somehow has perfect knowledge of your error, THEN maybe it's worth a tenth of big bet.

Jesus there are still some limit wizards playing limit

I agree that 0ev call/fold thresholds(such as the K9 example) don't offer a lot of gain nor loss for neither the deviating player nor the exploiting player. However, I would think that the potential gain or loss with the stronger draws (which realize a substantial fraction of the pot as a call or bet/raise) is substantial if we're assumed to have value hands. The exclusion of draws from the betting range may cause a loss in value for both the betting range(which is now value heavy) and the drawing/bluffcatching range(which is now draw heavy).

For the posted hand, I think this means very little in terms of the overall profitability of a raising range; we raise the turn so infrequently that it's not going to have a big impact on our winrate if we don't have draws in our raising range.

However, on a more coordinated board? I'm checking the flop much more often than the flop Jlot posted, thus I will be raising much more frequently on the turn and having draws in my range will become a priority. If I play a pure strategy on such a board(starting with the flop decision and including the turn raise) there will just be too many draws to choose from to include them at 100% frequency(else I would be extremely draw heavy). I will have to exclude draws from the turn raising range in some order of preference. This will limit my draw coverage on some river cards; If I prioritize my turn raising selection process with draws to either (the least showdownable) or (the highest profitability) then there are exploits for both approaches:

(the least showdownable draws) are vulnerable to more frequent turn 3 bets, as well as expanded turn and river bluffcatching(note that you will have a ton of unshowdownable hands in your range if you choose this method).

(the highest profitability draws) are not vulnerable in themselves, but if included in the betting range at 100% frequency these hands leave a void in the checking range(which has much fewer strong draws now), and the resulting river range after raising the turn will not have as many unshowdownable hands to bluff with.

So let's say there's a 789 flop with 2 diamonds, you check, the turn is a J that completes a flush, and you're saying that you need to have some Ts or flush draws that check back because otherwise you'd only have draws if you raised the turn? I don't think this is a problem - if you don't have any strong hands in that particular run off it's just, like the hand in the op, you wouldn't have a value range and so you don't bluff often if ever. Some run outs just favor one players range more than they do the other. You can have no raises in certain spots.

As for the river, it's also not really that much of a problem if you have hands that are 'unshowdownable' - it just redefines what you bluff with. If the board is 789J and your turn value range is Tx, your bluffs are Adx and the river brings a 4 flush, your straights become your bluffs and you'd only do them with limited frequency... to push him to being indifferent to calling with lower flushes (where he calls with a fraction of his range to make you indifferent to bluffing). I don't see the problem in him having some hands in his range that are worse than your bluffs though maybe I'm wrong.

I have T8, T7s, KT, AT all at frequency in my flop check range for 987r btn vs bb.

This would influence the value of checking those busted high equity draws on the river. If taken to an extreme, there may be zero hands in your range that earn ev bluff > ev check.

I would think that a straight in the supposed scenario with the four flush board is a very profitable checkback on the river vs all but the tightest strategies.

I think it's a problem on boards that I do have a raise range on.

I agree, but 987 J(with 3 flush) btn vs bb isn't one of those spots. Due to the necessity of having a flop check range(which keeps us from being draw heavy on the flop), which leads to having straights and draws in our available turn range after checking the flop, our available draw raising range on the turn is huge relative to our value hands. Including the flushing Aces at anywhere near 100% will lead to two immediate problems:

turn betting range draw supersaturation(bluffing too much)

checking range weakness(too many bluffcatchers and weak draws* not enough strong draws)

*too many weak draws means too many available river bluffs after turn checks through and river checks to me.

This would only happen in scenarios where you have no value range either.

If you check back straights in that scenario the optimal response is to just fold 100% for him. By betting a single combination of a straight in that scenario you improve by getting him to still fold all of his flushes (and straights that would chop). At some point if you bet enough combinations of straights his optimal response shifts to calling with worse flushes and your optimal bet frequency with straights lies right on the cusp of that occurring.

If you insist on having a raise range in a spot where you can't have a hand worth raising for value it's a problem. But whether it's a problem to not have a raise range is kind of a separate issue.

You say that you check back some number of hands that include a T so it's not really an issue for you - you can raise all those for value. But even if you bet all Tx hands on the flop, the fact that you can't have a raise range on the turn doesn't imply there's a flaw in your overall strategy.

The hand in the original post is a more clear illustration of that principle but it should come up in a lot of spots. Like when you check back the turn on a 3 flush board, the 4 flush comes in - you probably don't ever have the nut. You might not even ever check back with the second nut draw. So when he bets into you, maybe you just don't have a raise range. Sometimes runouts just benefit one players range more than the other, and the person who has a range deficit ultimately has to play defensively.

It's like in NL when someone 3bets you out of position and the board comes 222. You generally just shouldn't be raising.


If you lump all nut draws together you can say that you have to have a mixed strategy, but A6 is not the same as A4 on that board. Your strategy with A4 would likely not be mixed. It's only the hands that are on the cusp that need to be mixed. Maybe you raise all combinations of A6 with the nut draw but a fraction of AQ with the nut draw. Maybe you raise all AQ with the nut draw but a small fraction of KQ with the K of diamonds. Where you draw the line depends on how many Tx hands you have, but the mixedness of your strategy of those particular hands is of very little consequence since it likely only applies to one or two specific hand combinations.

The idea that you need to have a KdQx as a raise with some frequency so that you'll have bluffs if the river comes an A I don't think has any merit. It's ok if in some scenarios the bottom of your range is top pair. You'd still bluff with some fraction of it, and he'd still have to look you up with some of his two pairs and fold others.

No, because I have other hands which are less showdownable(such as Q6s-Q2s) to bluff with.


I don't think this is true. I think many draws will have mixed strategies, at different frequencies depending on how profitable the options are(the higher the profit, the more frequently the draw is raised)(the lower the profit, the more likely we are to call).

I didn't mean to suggest this. My thought is that if I have a value raising range, then I also want a draw raising range, and that range should be included in the bet or raise range at frequencies depending on the profitability of calling and raising(the most profitable draws are included at up to 100% frequency, or down to 0% frequency).

I highly doubt you could fit Q6s-Q2s into your bluff raise range after checking the flop back. You don't have nearly enough straights to offset that large a bluff range. I guess maybe you could have J7 for value too but it's almost all straights and you should have a very small number of those.

Presumably you want to have some hope of getting paired hands to fold to the raise and for that to be realistic given that he'll be laid 4:1 and will still have equity against your straights you need a lot more value than bluffs. If you're dipping as low as Q2s you probably have twice as many bluffs as you do value.

You can justify it in that you're only doing Q2s as a raise some fraction of the time but why would you choose to raise Q2s and A6 with a flush draw half the time when you could just raise A6 with the flush draw all the time and just fold Q2? It seems like your answer is so you can occasionally have something that really sucks at showdown to bluff with which I don't understand at all.


Why make a less profitable play just so you can have more bad hands in your range? You similarly don't need to have nutted hands (or draws to the nuts) in your call range.

Right, that's pretty much why I've been arguing for mixed frequencies with draws the whole time.

Not if I'm raising at low frequency and calling at high frequency.
No, I believe that a well structured river range is the PRODUCT of good flop and turn play, and I've shown upthread why I think including or excluding draws at 100% is exploitable.



If a specific combination is ev neutral on the turn facing a bet (ev call = ev raise), then I'm not making a "less profitable play" by having a mixed strategy. All draws are not equal in profitability. The more realizable equity a draw has, the more profitable it will be. When I call the turn with a strong draw, I'm not missing any value because calling is assumed to have the same ev as raising. Likewise, when I raise a weak draw on the turn, I'm not doing it "just so I can have more bad hands in my range," I'm doing it because I think it's the most profitable play.

Taking this further, some draws have so much realizable equity that they may be able to bet or raise at 100% frequency(indicating that ev raise > ev call), and the weaker draws have so little realizable equity that they may call(or even fold) at 100% frequency.

If you're having trouble realizing why there can be draws in the raising range that earn less ev than other, stronger draws(which are also included in the raising range), without a contradiction of profitability(if A6 is more profitable than Q2s then why would we ever check A6 sometimes and bet Q2s sometimes?) then I think it's an indication that you're looking at the ev wrong. Draws become indifferent to the options given vs strong opponents.* You're not supposed to be trying to make your opponent indifferent on the flop nor turn by adding draws to your bet/raise ranges. You're trying to maximize the expectation of the betting/raising ranges. So if it doesn't matter which option we choose with a particular draw (if ev raise = ev call), then bet/raise range construction with draws will be a function of (which combos best add value to my bet/raise range?)

The result is a strategy that has maximized the ev of both the bet/raise ranges(to the point of draw saturation, adding more draws would result in diminishing profits) and the check/call ranges(to the point of bluffcatcher protection, adding draws benefits bluffcatchers by limiting the opponent's bet/raise frequencies).

*I called this "self imposed indifference" here:

https://forumserver.twoplustwo.com/1...12/?highlight=

but the thought never really became coherent.

The point of contention here is that having a little bit of everything is not a 'well structured range'. If you raise UTG you don't have to include 2s in your range with low frequency to be able to sometimes 'have it' when the flop comes 22x.

I don't disagree that there're some bluffs that will show less of a profit than others, but the others that will show a profit will probably include other draws that have much better equity when called than Q2 no flush draw. And more important conceptually, if you wanted to start raising A2 with the nut draw or KQ with the K of diamonds the argument im making (which is untested from both our perspectives) is that you would only do that if you were in a spot where A6 was such a clear raise that you'd want to do it with 100% frequency.


There're ways to test this. I'm not capable of doing it but we could make it happen. Would you be willing to bet on whether raising Q5-Q2s (no flush draw) in that spot occurs with any frequency in an optimal framework? And also whether A6 with the nut draw would be less than 100%.

I’m not side betting. But I think it’s entirely possible that the naked straight draws are occasionally raised at equilibrium, which is not exclusive to A6 nut flush draw being raised at 100%.