The “thinking about your entire range” philosophy from the beginning seemed off to me. You cannot just be okay with folding the hand you currently have because “you have these other hands 'in your range' to call down with”. Because your opponent can destroy you by leveraging removal by only taking a particular line with one single type of holding.
Let's say you open the BTN, BB 3bets, you call. The board comes KsQh2c and villain jams for 5x pot effective.
This is your flat to 3b range:




According to the Minimum Defence Frequency formula, you only need to defend around 16.67% in this spot. That is roughly 29 combos out of this range of 173 combos (this is accounted for the removal inflicted by the board cards):




So you pick your top 29 combos to call: KQs, KJs, KTs, K9s, AQs, AQo, QJs, QTs:




You're thinking "Thank you Doug Polk, I am now unexploitable because I chose which combinations I am calling down!"
But the surprise is that your opponent is only taking this line with two combinations: QdJd and QcJc.
Well if you apply your opponent’s removal power, the combos you actually end up calling down are:




So if you apply MDF here, or really any “which hands am I calling here” approach, you are bleeding money since a huge part of the combinations you think you are calling down, you never have.



So you cannot ever know your range because it is dependent on your opponent's hole cards. Really this range stuff is a world of twisted mirrors, the only thing you can ever know about a hand is what your hole cards are.

1-MDF formula is not applicable here
2-second pair can certainly have + jam does not mean we are exploitable
3-You can consider blocker effects when constructing a defensive range. Solver does this.

Why is MDF not applicable here? Is it only applicable on rivers when your hands can no longer draw?

The whole point of defending at a certain frequency is so that villain can't have a profitable bluff with every single low equity hand, not so that they can't have a profitable shove with second pair. If he is doing this with KK are we also "bleeding money"? Of course not.

No, the entire point is that he only takes this line with the blockers. With KK he bets for example 100% pot. My point is that you can't construct your range to be unexploitable because you don't know what your range is, it is dependent on your opponent

and if villain is only 5x jamming sets he's printing every time you call too...

Villain taking +EV lines doesn't mean you're getting exploited. Some hands just are entitled to a share of the pot by virtue of their strength/blockers/equity...

Also yes, this is how blockers work, you weight villains range towards hands that you want him to have in order for his plays that "supposedly" try to make your range indifferent to end up benefiting the exact combo you hold.

also yeah, MDF is not how defense range would be constructed but that's besides the point. MDF makes villains hands 0 EV, not indifferent.

Your overall idea is correct, villain's hand determines your range. It's precisely because of this that some combos result in slightly profitable bluffs on the river for example, even against an optimal defending strategy, because having those blockers decrease the probability of getting called so much that the precise MDF is not met and thus that exact combo yields a small profit. But unexploitability does not mean defending MDF at every node, sometimes defending exactly MDF is very bad and can be the most exploitable thing in the world. Being unexploitable means capturing the maximum amount of EV that you can while minimizing the risk of capturing less than that.
Regarding your example:
There is an exact frequency and range you should defend with vs this size such that you capture the maximum EV possible and at the same time there isn't a strategy your opponent can take that lowers your own EV. You arrive to this range and frequency by using an algorithm that iterates over and over, going through the different flop, turn and river nodes. This is what solvers do, with the output being the optimal defending strategy.
This optimal defending strategy isn't necessarily your top 29 combinations like you mentioned, it's a result that takes into account all possible threats, blocker effects and other factors, although in general the output tends to be something similar like the range you came up with.
Let's suppose that indeed that is how the optimal defense looks like (it likely isn't, MDF serves as a rough estimate on early streets, it's mostly useful on rivers where your opponent arrives with a lot of bluff candidates):
When your opponent shoves with QJ/QT, he will face more folds than he should, thus he would have a +EV shove. There is nothing you can do about it. But QJ/QT is middle pair in this scenario, which are hands that have a pretty substantial share of the pot already. By going all in with them... yes, your opponent is making money, but chances are he can make way more by betting small and getting some value from worse. It's ok to let villain capture EV with QJ/QT by getting us to fold at a very high frequency, QJ/QT are +EV hands either way. If he's doing that it's actually benefitial for us since shoving is likely the least +EV decision in his shoes. The same logic could be applied to when he has KK, he has THE best blockers to go all in with and make us fold way more than MDF, and sure he makes money by doing that, but chances are it's not the best option.

Yeah of course. What bothered me is the conception that you can just not worry about folding what you have because "you have those and those hands to call down with in that spot" kind of Doug Polk school of poker thought. My point isn't that you shouldn't be aware of your range, but that it isn't fixed, or pure, or absolute - your opponent can take it and do things with it. Especially if he takes lines where he bets huge and only does so with blockers. There's a lot of leverage in that imho. I think that's something to be aware of but I'm very new to this game so maybe it isn't



Can you expand on this? Why is MDF not how a defense range would be constructed? What else is it for? And if your opponent's hands are 0 EV, how are they not indifferent (if the hand ends right then and there)?

Yeah, thanks, this is exactly what I was getting at. But given a huge leverage of something like a 2x or 3x pot bet, and a line that is only taken with blocker combos, can't this become like a huge, huge exploit?

Can you please provide an example of this? I'm very curious since I'm only getting familiar with MDF

Something I'll write down for sure


Yeah, the example I picked was imperfect. A much better one would have been where villain has strong blockers but no made hand. Like AJ on KQ6. In that case there is something you can do about it, you just have to call more than a "properly constructed" range would deem enough: like adding 76s, 86s, 65s to your calling range

The "Doug Polk school of thought" is fine, since you will never be able to apply it to any accuracy that makes blocker effects matter.
It's more like: "I have top pair here but I'm actually at the bottom of my range because I called 2 overbets and my draws turned into flushes, if I were to call with top pairs i would be stationing my whole range."
Whether this leads you to folding 50% or 40% is not the important thing, the important thing is that you don't station/fold everything.

For 0 EV to be indifferent, other actions would have to be 0 EV too. Obviously QJ on KQx doesn't have 0 EV on any action

Does it really not matter? Doesn't this game work on small edges? The difference between AKo and AKs is only a few percentages of equity but these hands are meaningfully different. If I can make my opponent call 20% instead of 30% I feel like there's place there to make myself some money.

So what you mean is that MDF has no use?

It would be important if your brain could calculate it with any accuracy, the truth is it can't. If someone made only these kind of frequency mistakes, they would be a high stakes crusher no doubt.

MDF is not really all that useful to design strategies, it's still a cool concept to know at which % folding threshold your bluffs start making money

Got it, thanks a lot mate

"like adding 76s, 86s, 65s to your calling range" yes, something like that ends up happening in some situations quite a bit, but more so on the river where villain employs a polarized betting range and we have bluff-catchers. It's not unusual to see that an optimal defense on the river consists of mixing it up and calling in an MDF fashion with different types of hands.
In practice mixing it up in such a way is not very important really, and probably too complex to implement correctly... though it's still pretty interesting to discuss the concept.

About MDF: it's most relevant when having bluff-catchers on the river. The EV of your bluff-catchers depends directly on how often villain gives up with his bluffs. In simple terms (it's a bit more complex than this), at equilibrium when villain checks back with a weak hand you win the pot, and when you face a big bet you gain 0 chips in the long run (calling=folding=0). So the EV of your bluff-catchers is:
EV bc = (% V gives up) * Pot + ((% he bets) * (% you call) * 0) + ((% he bets) * (% you fold) * 0)
The last part of the equation is 0 at equilibrium, basically whenever he bets big and you call or fold the EV of your bluffcatcher is 0.
What you want to get ahold of is that "(% of time he gives up) * Pot ", which is your share of the pot. To make sure you get those chips, you call in such a way that he doesn't have an incentive to change his "(% he gives up)" frequency. If he all of a sudden is having profitable bluffs, because you fold let's say 100%, then the "(% he gives up)" drops to 0%, and so you start constantly facing bets and never get to realize the equity of ANY of your marginal hands, which can result in a massive EV loss. Thinking "he always has it" on the river when holding a bluff-catcher can be disastrous if he in fact never has it. Thats in part why it's most useful on the river. Using MDF on the flop however can be a bad idea in a lot of scenarios.
One example would be when defending MDF on the flop in a Button vs SB 3bet pot on the AAK texture. The Button in this situation is folding like 50% of the time on the flop vs a 33% SB c-bet, more than what MDF would dictate, even with hands that have 30% equity. This is because calling the flop only allows you to see a turn, where the probability of facing a larger bet is not 0%, and so this 30% equity cannot get realized in a way that isn't exploitable. Another situation would be in a BU vs SB 4bet pot, where the SB is folding like 60% of the time, even when facing a tiny c-bet of 10% pot. Optimal flop defense takes into account future street scenarios among other factors.

what in the world lol
i might be dumb but i don't think this is it

EV of bc = %bluffs*(pot+bet) - %value*bet

How often he gives up is irrelevant, or rather, it's only useful when we also know his overal value/bluff ratio in range, from which we could then derive his value/bluff ratio in his betting range

MDF has nothing to do with the EV of the bluff catcher, but with the EV of bluffs:

MDF = 1/(%betsize/100 + 1)

Reason you don't defend MDF is beacuse villains bluffs have checking +EV on early streets

"EV of bc = %bluffs*(pot+bet) - %value*bet" That's the EV of the bluffcatcher when facing a bet, which is 0 at equilibrium. In my simplified example Villain has a range of 0% equity bluffs and 100% equity value bets, so the overall EV of hero's bc is "%give up * Pot" + %bet * 0

"MDF has nothing to do with the EV of the bluff catcher, but with the EV of bluffs:" Yes it has to do with the EV of bluffs, it literally means the % you should defend such that Villain's bluffs have 0 EV.
What I meant was that if you fold more than MDF, for example if you always fold, you give Villain an incentive to deviate and start bluffing every 0 equity combo, in which case you would never get to showdown and lose a lot of EV.

Let's say villain has a medium hand that they can check for 1bb EV, or turn into a bluff with 0.5bb EV. Which action should they take?

You can't blindly apply MDF in a vacuum. As with most things in poker, It's always more complicated than the beginner explanation.

So am I the only one that’s still trying to figure out how OP thinks we can never know our own range??

Notice when I explicitly said "he only takes this line with these two combinations"? Well I meant it, meaning the rest of his range is in another part of the game tree, maybe it's betting 75% pot, maybe checking, whatever. So let me simplify even further, even though there's no need, let's say villain only ever jams 5x pot with literally one combination, QdJd. Well these are the hands you will literally never ever have here - no matter what - KdQd, KdJd, QdJd, QdTd, AdQd, AsQd, AhQd, AcQd - they are in his hand. You think you have them in your range, but you never ever do, and you don't know that you don't have them - thus you don't truly know your range.

Wow that is condescending.

We see your point- there are always gonna be card removal effects and other noise that are hard to account for. Hell most players completely ignore the bunching effect too.

Let me ask you a question PB97, how do you constuct a strategy without thinking about ranges? How do you justify a play in a vacuum without context?

Let me give you an example. I open with 54s, get called. Board comes A85r no BDFD, we act first. What's your play? You're not allowed to ask what position I opened, since that's only required for thinking about your range. You're not allowed to ask who called you, as that's only required for thinking about their range. Can you provide a meaningful answer without this context?

Wasn't trying to be condescending at all lol

You misunderstand me as well I'd say, in one of the replies I said that you inevitably will have to understand what your range is. My point is your range is an imperfect abstraction, and not something out of the reach of your opponent. He can affect it without you knowing it. Further I was trying to draw attention to this fact: your range is a function of your opponent; what you hold in your hand never is.

Sorry if this has been mentioned already, but the basic MDF calculation doesn’t apply to this scenario because there are still cards to come and nearly every hand in your opponent’s range will hold some amount of equity. The basic MDF calculation is modeled from a simple river situation in which all bluffs hold zero equity. You can use MDF in the scenario outlined, however you will need to modify the calculation slightly to account for the equity that the bluffs hold. In short, you can use the basic MDF calculation on early streets as a guide, but it’s important to note that you should always defend less than this value since all hands will hold some amount of equity.

This is something that is often overlooked, particularly around observations of overfolding to 3b’s or cbets.

Right. These facts don’t discount the importance of considering your range the best you can. After all, it’s a game of incomplete information.

The example provided doesn’t quite work. You’re applying MDF to a range of hands of differing equity. The basic MDF calculation assumes that all hands in the bluffcatching range hold equal equity. Therefore, once you know that your opponent is only jamming 5x pot with QJ, you wouldn’t now call with a hand lacking the necessary equity, such as QT to meet MDF. You would only call with hands that have enough equity vs the opponent’s range given the pot odds. Also, and surely someone will correct me if I’m wrong, but you would only call at MDF across the hands that hold enough equity to call. For example, if your range held 10 combos, 2 of which had 0% equity, and the other 8 held enough equity to call pot sized vet, you would call with only 4 of the 8 hands. Calling more than 4 leaves you over calling, and less than 4 as overfolding. The 2 0% are simply folded. Of course, in this model, a bluffcatching range can’t actually hold a 0% equity hand.

This is correct, but allow me to try to better explain it. When we reach an equilibrium on the river in the nuts or air scenario the goal for the polarized better is to make the other player essentially have 0ev with his bluff catchers. If a hand has 0 equity, then it isn't a bluff catcher and isn't really counted.

Let me provide an example.

IP range = AA, KK, QQ and 99 88

OOP range = JJ, TT, and 66,55

Board = 2 2 2 2 3

pot = 100
stack = 100

On the river IP will bet all 18 combos of value (AA, KK, QQ) and bet 9 combos of bluffs. If blocker effects don't matter then it doesn't matter how the 9 combos are distributed. IP could bet six combos of 99 and three combos of 88 or 75% of each (3/4)*6 = 4.5, so 4.5 + 4.5 = 9 also. The reason we arrive at 9 is because 9/(9+18) = 33% which in this case is the pot odds laid to the OOP player (pot/(pot + pot + pot). In this situation 66 and 55 are pure folds for OOP as they have 0 equity even vs the bluffs, so the defense frequency or "mdf" which would be pot/(pot+pot) = 50%. This means 50% of the combos of JJ/TT would be called or only 6 combos out of a total of 24 (so only a 25% call rate).

1. That is the equilibrium situation and would have IP winning 95 chips

2. Non-equilibrium where OOP folds everything IP wins 100 chips (because he always bets and there are 100 chips in the pot

3. Non-equilibrium where OOP calls everything IP wins >100 chips (because he only bets his value hands) this ends up being 109.45 chips

4. Non-equilibrium where IP over bluffs and always bets 88/77 OOP always calls JJ/TT and always folds 66/55, this would increase OOP EV because IP is unbalanced... JJ/TT gain ev here.

5. Non-equilibrium where IP never bluffs and only ever value bets AA/KK/QQ OOP gains EV because now when IP bets OOP can fold 100% and when IP checks behind his JJ/TT get to realize all that equity when in scenario (1.) JJ/TT is indifferent to calling/folding due to the odds laid.

You can attempt to calculate IP/OOP EV in the various scenarios as practice. (2.) is the easiest. I've included the EV for (3.) in case you would want to practice the calculation and reference it, although I cheated and used PIO :P