Coaching: Fixed Limit [Advanced] with Boomer2k10

Yeah! Sorry Boomer, I didn't mean to piss you off, just frustrated that's all.

When I joined PS it was to learn LHE and it was great. I attended the beginner & intermediate coaching sessions with TerrorBlade then Datsmahname and when I hit gold I sat in on Byron's and had my (metaphorical) hair blown back, not to mention your videos.

Unfortunately life got in the way of things and I had to give poker a rest for a year. Now when I'm ready to get back into things there's nothing on my level.

No I don't want to watch you play .25/.50 every week, I don't see the point either. I find most of the advanced topics fascinating but there is a huge gap between my current level and being able to put any of it to use.

Let's say that I'm not ready for the red pill right now and that I'm too dependant on the matrix.

So do I continue with LHE or take the blue NLH/PLO pill? Fucked if I know.

The material that is currently on the site available at silver level is more than sufficient for you to beat micro stakes

Do you need to know GTO to beat 0.5/1? No and no-one is implying that. I use concepts from GTO because it's part of my game and has been for years, however these concepts often clash with pre-conceived ABC-style play and I find it very inefficient to teach one style of poker only to have to undo it the moment you come across somebody half competent.

With all the information that's available you don't need me to tell you how to beat fish. That information has been round for years. What I'm needed for is when you need to understand the how's and why's of the game and to be honest some of what I do links directly back into how to beat fish for the maximum.

The problem is I come at poker from a completely different angle than has been taught in the past except by a select few.

Most coachings have been "Here's how you beat bad players" or in some case "Here's how you beat a TAG with really big holes in his game".

Well that only gets you so far and it teaches you a very rigid, inflexible and predictable form of poker which is not suited for taking on good players and will cap your ability to learn.

What I try to do is show the underlying reasons and maths behind the game which do confirm some of the assumptions made in the past but also tear some of them apart. It teaches you better exploitation but not before you've learned how to balance your play.

The classic way has been "exploitation first and figure out balance later" whereas my arguement is "how can you know you're exploiting your opponent as best you can when you don't even know why or how far his play is from a balanced position?". My method may be more difficult but it allows for more accurate and devastating exploitation in the long run. I'm not turning my back on the exploitative game just because I base my on GTO and balance initially. You make money by exploiting your opponent's mistakes, that's always been true. I just think my way, while more difficult initially, just produces better results in the end.

I decided to get stubborn and stick with LHE. I just "obtained" a copy of "The Intelligent Poker Player" and was just reading the same thing.

I have also in the past had the viewpoint that FL at the micro's need not and should not be treated from the advanced perspective (balance and beyond). From a pedagogical point of view, I still believe so, but I'm very happy that Boomer has firmly refused to abandon his line (GTO, or at least balance, first, only then exploitation). We'll have to enjoy the situation while it lasts.

The problem is how to attract newcomers to the LHE commuity. Seriously, a beginner will not be able to understand more than two consecutive words in a theoretical coaching (which is still available to him being "basic" or "silver"). Add to this the fact that much of the written beginners material is deprecated and will not be made up-to-date.

Hey Boomer,

I watched the recording of yesterdays coaching and I still dont get this model.

You mentioned that we should bluff 31% on the turn instead of 20% that one street balance would dictate, but we have to bluff 14% on the river.

Say on a board like A222 you bet the turn with 31% bluffs and the river is a 2, how are you gonna end up bluffing 14% ? Your value range remains almost the same so you have to x with bluffs otherwise you would bluff too much. And this can be exploited for ex by overcalling the turn with a wider bluffcathing range and simply folding the river with these additional hands when you bet.

So if this is true your play can be exploited which means its not GTO play.
Whereas if someone bets the turn with 20% bluffs and the river with 14% bluffs I dont know how it could be exploited ?

Take your 20% model and have cards on the river that improve your draws...that's where it can be exploited because you're not bluffing enough.

That means you're probably going to end up either missing value (not value betting correctly) or having your turn betting range = your river betting range.

On the turn there are future streets and actions to consider which is why you can't use a 1-street model.

Which one happens more often? Improving on a draw or both players playing the board where there's no point doing anything except checking for both players unless they want to pay more rake? There will always be fringe situations where models fail becasue we don't know the exact mathematical model of GTO LHE play but if the point where it fails is when the board = the nuts that's more than a good enough model to beat any human being. (And the model doesn't even fail because that particular board is completely solvable and in fact every single line is exactly the same EV in a rakeless environment as long as you don't fold)

And no it's can't be exploited by calling the turn super wide and then folding because you'll end up over-folding the river when your opponent bets so how is that exploiting anything? Exploiting the smaller part of your opponent's range (bluffs which have equity) and getting exploited by the larger part (i.e. value bet to death) isn't exploiting someone it's being exploited.

Take your 20% model and have cards on the river that improve your draws...that's where it can be exploited because you're not bluffing enough.

I can still bluff enough with the draws completing on the river by valuebetting less on the river (which you should do anyway because your opponent has to fold 16% on the river). Also if one card completes all your draws then your bluffing range is not set up correctly, I think its better to create a range with a mix of various draws and pure bluffs than to overbluff with the hope that you hit.

Which one happens more often?

Even with a strong draw you are going to improve only 20% of the time from turn to river. So 80% of the time you end up checking bluffhands. Which happens more often then ?
Not to mention that you wont have a draw every time, nay most of the time you wont. I dont remember exactly but the chance of floping a straight or a flushdraw is something like 10%.

And also I dont understand the calculation why should you bluff 31%, where does that number come from ? If I count how many times draws complete that varies from board to board on a static board I pointed on previously the chance you improve is almost equal to nil, where as a T7J6 with two flushdraws is pretty high. Why on earth would you bluff 31% on each of one them ?

And no it's can't be exploited by calling the turn super wide and then folding because you'll end up over-folding the river

Even with the overfold it is +EV to call those hands. The profit is made when you overbluff the turn, I can simply prove that:

Say an overcall strictly regarding the turn under these circumstances is + 0,1 EV with A8. The EV of folding is 0. So calling the turn and folding the river is 0,1 + 0 = 0,1.

Originally posted by kavboj84

Even with a strong draw you are going to improve only 20% of the time from turn to river.

Wrong. Strong draws like FD + Overcards can be up to 30-33% if you include drawing to a hand that can value bet.

The way the model works is that you work backwards from the river arriving at the correct bluffing frequency.

Not to mention that you wont have a draw every time, nay most of the time you wont. I dont remember exactly but the chance of floping a straight or a flushdraw is something like 10%.

You're 16% to flop a flush draw with a suited hand preflop.

What you're assuming here is that I'm going to want to bet all my bluffs on the river and that is not correct under almost any circumstances you have to have some hands you are willing to give up with from turn to river and given many of your hands will improve from turn to river then having only a 6% decrease cannot be correct, especially when many value hands on the turn will cease to be value hands by the river meaning by # of combos 14% bluffs on river =/= 14% bluffs on turn.

By your logic we should only be continuation betting the flop with the absolute nuts because our opponent is immediately getting 5-1 even in a single raised pot (meaning we should only have about 16.67% bluffs in our flop betting range) and I'm pretty sure you know that's not correct so there must be a flaw in you using single street methodology when it comes to analysing your betting patterns.

And also I dont understand the calculation why should you bluff 31%

You work backwards from the river.

Again, this is a model not a wholly encompassing GTO solution (hint: your opponent can raise you, you can also bet/3/bet for value or as a bluff so there are complexities added).

It's a model calling Stacking Bluffs and you can read about it either in:

The Mathematics of Poker - Bill Chen & Jerod Ankenman
The Math of Holdem - Collin Moshman & Douglas Zare
Further Limit Holdem - Phil Newall

The idea is based on a couple of assumptions:

1) You will always bet your hands for value
2) You will give up on draws street by street
3) The mere presence of future betting rounds means that you can bluff more than your opponent's immediate pot odds would dictate.

It doesn't take into account the following:

1) Your opponent can raise you on post-flop streets making things "cost" more (reason to bluff less, especially with hands that can't stand a raise like a gutshot and 2 unders)
2) Your semi-bluffs actually have a value element and thus do not count as a "full" bluff whereas value hands are actually always value hands, they have no bluff component. (reason to bluff more)
3) You will give up some value hands between Flop - Turn - and River. (Leads to giving up on a few more bluffs but doesn't really change the ratio)

Here how the model works for a single raised HUHU pot:

You estimate/know the final river pot size and thus the ratio of value/bluffs you're working towards:

You act like all the bets on the river (full pot size) is for value on the turn. You then Scale Up the odds on the turn to find out how many "additional" bluff combos are needed on the turn and add them to the full pot size

You then repeat the process for the flop.

River: The final pot size will be 7BB. Our opponent will be getting 6-1 therefore we should bluff with 1 bluff combo for every 6 value combos

Now working backwards.

On the Turn our opponent is getting 4-1 to call our bet. So for every 4 value combos we have we should make 1 bluff. The final pot is 7BB so the equation is:

Bluff Combos To Add * 4 = 7

so bluff combos to add = 1.75.

This gives us a total of 2.75 bluff combos into a total number of bets of 8.75 (Adding the bluff combos to the river hands) = 31% of our range.

Again now on the flop all 8.75 combos are value on the turn so let's work out how much we need on the flop:

Well our opponent is getting 5-1 so for every 5 value combos we need 1 bluffing combo.

Scale up and the Equation is

Bluff Combos to Add * 5 = 8.75

And how lucky it's 1.75 again

This brings us to 4.5 bluff combos in a total number of combos of 10.5 = 43%

Now again, this model is not perfect by any means, but it is close to the previous model for sure and it also allows for something which the previous model did not which is giving up on bluffs. The previous model was very flat about what it bet on turn and river.

Now, don't forget things are different for the caller as well as the better here:

For example we often say that someone facing a bet off someone getting 4-1 should only fold the bottom 20% of their range. Well more often than not you're going to fold a little more often than that due to reverse implied odds, especially on the turn (again leading to, yes we should bluff more than pot odds dictate on the turn)

However this isn't easy to apply to the game even with the correct bluffing ratios known:

1) Draws complicate things because they improve your equity and will often turn into value hands
2) You won't be able to bet all value hands all the way but really all this means is simply giving up on an appropriate ratio of missed draws and often the reason you can't value bet is that draws have come in
3) Leverage - This is one of the main multi-street concepts. You're not just calling 1 bet on the turn, you are being threatened with another so the cost of calling a bet on the turn with a bluff catcher is actually more than 1 bet because it's so often followed up by another. (Reverse Implied Odds Concept leading to higher bluff frequency)

Now this model, as I've said is pretty simple but it cannot be exploited by calling. Can it be exploited by raising, yes possibly it can but that would still be very hard to do, especially if you do what is suggested in many concepts which is to bluff with hands with higher equity and accounting for future card distribution.

It becomes more complex as you introduce more options with regards to raising but still it is a far more accurate model and most like HUHU single raised pots than simply playing by the calling odds

Originally posted by kavboj84
Even with the overfold it is +EV to call those hands. The profit is made when you overbluff the turn, I can simply prove that:

Say an overcall strictly regarding the turn under these circumstances is + 0,1 EV with A8. The EV of folding is 0. So calling the turn and folding the river is 0,1 + 0 = 0,1.

And how would you know at the time that it was +EV to call on the turn unless I told you my exact hand in which case it turns into an exploitative game anyway? Or you knew my exact range on every single possible board?

And coincidentally your EV may be 0 (fold) but mine is WAY higher because you're over-folding and thus any bet I make on the river is automatically +EV, especailly if you're making an extreme adjustment.

You're talking about single hand examples which aren't realistic in poker, it is a range on range situation. You cannot make every single decision with every single hand your opponent has 0 EV, sometimes he has AA.

What I am playing against is your range not your exact hand.

The way you're describing it here your EV can never be below 0 if you fold all the time. Pretty sure you know how that turns out.

This looks way more interesting than 1-street balancing.

I'm wondering this:

• Is it worthwhile (is it GTO) to set up the whole hand with the sole aim of being able to bluff with the last bet on the river with the correct frequency? (Bets going into the pot on flop turn and river count too. Mistakes here conceivably cost more than gained o t r, it may be much like with the 3-bet or no 3-bet preflop issue, i.e. unresolved.)
• Does the board texture ever influence the bluffing frequencies? (Recall old discussion on BU 3-bet vs UTG open with a really juicy flop.)
• Are the "bluff-adding-formulas" rigorously derived in the references, or are they something that happens to give plausible results?

Perhaps these are topics that could be discussed during a theory coaching.

B t w, the last show wasn't added to the JustinTV archive last time I checked. (Couldn't really follow it live, kept losing connection and sound.)

http://www.justin.tv/pokerstrategy_english/b/507092892

Originally posted by YohanN7
This looks way more interesting than 1-street balancing.

I'm wondering this:

• Is it worthwhile (is it GTO) to set up the whole hand with the sole aim of being able to bluff with the last bet on the river with the correct frequency? (Bets going into the pot on flop turn and river count too. Mistakes here conceivably cost more than gained o t r, it may be much like with the 3-bet or no 3-bet preflop issue, i.e. unresolved.)
• Does the board texture ever influence the bluffing frequencies? (Recall old discussion on BU 3-bet vs UTG open with a really juicy flop.)
• Are the "bluff-adding-formulas" rigorously derived in the references, or are they something that happens to give plausible results?

Perhaps these are topics that could be discussed during a theory coaching.

B t w, the last show wasn't added to the JustinTV archive last time I checked. (Couldn't really follow it live, kept losing connection and sound.)

Well:

1) You have to consider the whole hand if you're going to have any pretentions at GTO play.
2) No the model's not super-accurate because you cannot predict the next card and it only gives you frequenices, not accounting for future card distribuion or clumping of a certain part of your range
3) Additionally given your opponent can take an aggressive line rather than just calling we have to assume that the "actual" frequenices are a little lower
4) However this could be compensated for by the fact bluffs can have a "value" aspect and this would increase our bluffing frequencies and, yes, board texturemay very well have an influence here, i.e. boards where bluffs natuarlly have high equity (i.e. wet boards)
5) An interesting feature of wet boards is that they may not change the %-ages too much but in a HUHU single raised pot situation you will often find the bots changing lines.

For example:

On a low, wet coordinated bought the 2011 version of Polaris, for example, could have a C-bet % as low as 75% on the flop. In fact this is actually a situation where donk-betting becomes a legitimate strategy.

However Ploaris and th other bots still c-bet close to 100% if not at 100% on board which favour preflop raisers so rang assymetry in their "eyes" overwhelmed the pure bluffing frequencies aspect

Posted the link to the coaching above.

Thanks for your answer. I'll have to digest it for a while, but in the meanwhile, I think a clarification of my first question of above is in order:

If we set up the strategy so that the river bluffing frequency is perfect (pretend river raises aren't allowed), then we play the river perfectly. However, it might be that in order to arrive to the river with the perfect mix of bluff and value, we have to make imperfect (exploitable) moves earlier on.

What I'm trying to say is that it might not be GTO overall to play the river and perhaps the river only in a way that is guaranteed to be GTO. At least, if it is indeed GTO overall, this fact remains to be proved.

To support the reasoning behind my question; it's abundantly clear that 1-street balance (bluffs in inverse proportion to pot odds opponent is getting, street by street) isn't the way to go on the flop and turn. But then, what guarantees that unconditionally retaining 1-street balance on the river is GTO? (It is here I suspect that the board plays a big rôle, but that's besides the point.)

Originally posted by YohanN7
To support the reasoning behind my question; it's abundantly clear that 1-street balance (bluffs in inverse proportion to pot odds opponent is getting, street by street) isn't the way to go on the flop and turn. But then, what guarantees that unconditionally retaining 1-street balance on the river is GTO? (It is here I suspect that the board plays a big rôle, but that's besides the point.)

1 street balance works on the river because there are no future cards

Therefore your equity is either 100% or 0%, you win the pot or you don't. Your hand is a bluff or it isn't, there's no semi-bluffs and there's no future bet threatening or any reverse-implied odds to consider.

it falls apart of the turn because of the future action and equities of the hands. i.e. a flush draw isn't a 100% bluff etc

Yes, what you write is obviously correct, but you haven't grasped what my question is. I don't know how to put it more clearly than in my last post. I'll try to exaggerate a bit. Suppose that you knew that putting in $1000 into the $1 pot would allow you to play the river perfectly, would you do it?

I'm trying to say that it might be too expensive to attain the isolated river GTO.

More formally, if the 1-street balances are X, Y and Z, what is there to say that GTO is
x,y, and z with x unequal to X, y unequal to Y, but z = Z?

A mathematician would probably make an ansatz of the form x = f(X, Y, Z, flop, OA(pf)), y = g(x, X, ..., flop, turn, OA(f)) and z = h(x, y, ..., flop, turn, river, OA(t)), where OA(street) is Opponent Actaion.

In other words, "no more cards are coming" is necessary for having z = Z, but it is not proven to be sufficient since previous bets potentially cost money. At the very least, it isn't obvious that z = Z in every situation.

That's the classic argument "if God dumped $1000 into the pot would you play the hand the same way" and the obvious answer is no but that's not a realistic scenario. We are playing LHE the cost of betting is fixed so therefore the only thing that matters is our frequencies.

I've already said the 31% figure is an oversimplification and for the most part backing that off may well be adviseable due to your opponent being able to raise you on the turn, however it also can't be far off becasue the model doesn't take into account that many bluffs on the turn have "value" elements (i.e. KsQs on a TsJs5x board has something like 35% "value" associated with it) and still 70% of your turn betting range even under this model is for value (and as we've stated more due to the value of your bluffs).

This combined with the fact that many of your weaker draws in position which can't stand a raise will not actually be bet leads to you actually strengthening our turn range over the previous model which was pretty much "28% and bluff from the bottom up" and therefore our turn range was weaker despite being 3% less "value" and more likely to have to bet/fold or bet/call with weak draws.

It works on the river because it's an equilibrium calculation, nothing more. If you bluff 1 in 7 times into what will be a 7BB pot on the river your opponent is completely neutral about calling or folding with a bluff catcher, his EV is 0 either way.

Now it's not 100% accurate because your opponent has the opportunity to do something other than call or fold, so the model gets more complex then but that's the point where we get into the Pareto Principle where we could make the calculation more accurate by putting years of work into it but it'd still be unusable in real time at a table and it wouldn't have THAT much benefit so it's simply not an efficient use of time

It's hard enough figuring out what your bluffing range is going to be when using an approximation (i.e. you have to know your range in real time and there's not many of us that can do that anyway) so adding onto that an extremely complex calculation to take into account every single fringe possibility is really a waste of time.

Thank you for your detailed reply. I'm not arguing against the model, not at all. By now you ought to be fed up with me for a good while ♦. Still not sure you get what my question really is here. Will be back on a rainy day with one more reformulation of it :f_biggrin:.

Eὕρηκα

Forgive me for beefing out on this here, but I think there is something of value to be found in this post for those who might not have grasped the rationale for model under discussion. (Who knows, I might still belong to that category, in which case I'm making a bigger fool than usual of myself. I'll take the risk.) The post uses a minimum of heuristics to motivate the model.

After viewing the coaching two more times, and after deciphering a couple of Boomer's posts of above, I think I get it. Barring possible raises from the problem, this must be pretty close to actual GTO frequencies for flop, turn and river. To me, it took two key points of understanding before temporarily reaching nirvana.

1. River bluff combos should be treated as value combos on the turn. (But why?)
2. Turn (and river) bluff combos should be treated as value combos on the flop. (But why?)
3. One street bluff/value ratios are to be used to calculate "additional" bluffs to append to them we already have on the later streets. (But why?)

First off, some hands are considered being "bluffs" at one stage and being "value" at other stages. This is initially confusing. Consider the river first. The river is "solved". We have 1 bluff for every 6 value bets. We can "normalize" the calculation and say that our entire river betting range is composed of 7 hands. [Boomers terminology above, and in the coaching, is (sry Boomer) unfortunate. He calls it "cumulative pot size".]

Now, the entire river betting range is to be considered as value on the turn. Why?

• The value part (or what turns out to be value) of it is, well, value.
• The part that we do fire on the river as a bluff is, by definition, at least breaking even since it is provably GTO.

For the last point, consider what happens when we don't bet it on the turn for value. This alters the pot size, and will shift us from river GTO, rendering us exploitable. This "proves" item 1 in the first list.

Thus, to our entire river betting range, which we from now on consider to be our turn value betting range, we need to add bluffs. How many? Well, the hands we add as bluffs are, on average, going to (don't think of particular boards or river cards here) be bet only once. Betting them more than once (in addition to the once we already bluff on the river (but value bet with on the turn)) will lead to true overbluffing on the river. This "proves" (the turn half of) item 3 in the first list.

Of course, at this stage, we do not know which part of our turn value betting range will turn out to be actual value. The important thing is that we know its size, so that we can calculate the number of hands that go into the turn bluff range using the one-street street bluff/value ratio.

Using this reasoning, we have and entire river betting range = turn value betting range of 7 hands, the opponent is getting 4:1 so the turn bluff range is 1.75 hands. Note that the river bluff range is a subset of the turn value betting range. The terminology is perhaps unfortunate, but I know of nothing better a t m. Our entire turn betting range = turn value betting range + turn bluff range. (This, again, Boomer calls cumulative pot size.)

How much are we actually bluffing on the turn? This is turn bluff range + river bluff range = 1.75 + 1 = 2.75 combos, making the entire turn betting range = turn value betting range + turn bluffing range = 7 + 2.75 = 8.75. This is 31% bluffing on the turn.

Going backwards one more time, the entire turn betting range generates money, so we must consider it as being the flop value betting range. Recycle the above reasoning once more to get the size of the flop bluffing range. This "proves" item 2 in the first list.

/*******/
The model can't tell what is what at any stage, but it probably gets the total sizes of the bluffing frequencies pretty damned right (in the absence of raising).

Consider playing in the BB versus SB. The flop comes Q 7 2 :spade:. Your opponent bets. You are supposed to raise as a bluff quite a lot here. I find this just as interesting as a whet flop. What's do you raise with on this flop?

Some comments from the sidelines.

Nice that Boomer brought the model of stacking bluffs into discussion. The model in deed manages to demonstrate the existence of this interesting phenomenon that in some situations it is possible to bluff surpringly much because you can give up some of your bluffs (and naturally not value hands) at will on later streets whereas your poor opponent must be prepared to call down all the way.

But to make it clear, the model builds on the strong assumption of the value bettor A knowing the cards of the bluff catching player B. Translated into ordinary poker without anyone seeing the opponent's cards, this corresponds to A having fully polarised range of nuts and total air, while B has bluff catchers that beat the air but lose to the nuts. That is, A always knows whether he is bluffing or value betting. This immediately makes the optimal play such that B has no incentive to perform any aggressive action (bet or raise), whether or not raising is expressly forbidden.

This assumption, in my opinion, makes this model to be very unrealistic in many practical LHE situations. The effect is systematic and not some minor inaccuracy that could be corrected by more complete calculations that would take into account some board development and raising, as Boomer seems to think. Some other models (e.g. the one that Boomer earlier used) may actually be closer to truth in the bluffing frequencies they suggest.

The assumption of complete polarisation is not the only thing that makes the model unrealistic. The model assumes hand values to be static. As stated already in some previous posts, in reality hands naturally develop as they interact with the board, and this effect may be quite important in reality. Flopped top set may well become less nutty on four-flush straighty board.

A peculiar consequence of the full polarisation assumption is that the aggressive actions of A (or lack of them) do not reshape B's river distribution at all: he always has 100% of bluff catchers. Normally this is not the case, of course. I think this explains some elegant properties of the optimal solution.

I think Yohan's original question, whether it is worth paying any price (or even any realistic price) for the possibility of playing river in a GTO way is a valid one. And to be very clear, the answer is a definite no.

As Boomer said too, the optimal strategy must consider all streets in combination. This time the full solution to the model happens to be such that on each street A plays as if all the hands he intends to bet on the next street were value, if the current street were considered in isolation. I guess this is more of a consequence of the very special assumptions of the model, not a general principle. At least I wouldn't dare to derive the solution from the principle, I'd have to solve the full model instead. Interestingly, the optimal strategy of B is to play every street just as it would be in isolation [e: took this from Chen-Ankenman, didn't have to solve for it myself].

Actually, I wouldn't pay anything for possibility to play the river according to GTO. No matter how you got to the river, it's always possible to play GTO from there on. In this light, Boomers argument of the model "work[ing] on the river because it's an equilibrium calculation, nothing more. If you bluff 1 in 7 times into what will be a 7BB pot on the river your opponent is completely neutral about calling or folding with a bluff catcher, his EV is 0 either way" is a bit beside the point, while being true naturally. It's the EV of the whole range of one player over all the possible lines (betting patterns) that determines his optimal strategy.

And for the record, I don't even try to play anything resembling GTO when I happen to play LHE. I just press buttons when I feel like it (and this shows!).

e: removed most hard line breaks that were left there

Hi Turuntor!

I'll write a partial comment for now. For the purposes of the discussion at hand, let's define a polarized range as being composed solely of nuts and air. (Some use the term for either nuts or air, but not both.)

Yes, the model, and in fact any model using "balancing" or GTO are defined in terms of polarized ranges. If we start with the much simpler scenario where we know the range of the opponent (not talking GTO now), then it is perfectly possible to unambiguously define value and bluff in any situation. In any one particular hand, it may turn out that value isn't true value, and that a bluff may be the best hand, but on average, it will be true.

This sort of averaging will never, per se, destroy any model - unless you do the averaging wrong. This should be clear in the realistic case of the opponent actually having a range. (The only problem for us is that we have to guess the opponents range.)

Now, this changes slightly in GTO play. Now we don't assume a range on part of our opponent. But now we employ the following trick: We assume that the opponent is playing GTO, and by appeal to the existence of Nash equilibria, the opponent now has a definite range (which in all likelihood isn't his true range). Again, the terms "bluff" and "value" are unambiguous. (The problem for us now is that we don't know GTO.)

These definitions of "value" and "bluff" are mathematically sound, but they may be tricky to interpret in terms of actual hole cards. In an actual hand played out, you would never be able to, on the flop, categorize a hand as a hand that you will bluff one, two, or three times with. This will be clear only after the event.

/**/

Interestingly, the optimal
strategy of B is to play every street just as it would be in isolation.

Actually, it is not. If player B did this he'd be folding too much because the total ratio of bluffs (by construction) in player As range is higher than the direct pot odds would dictate.

...bluff surprisingly much because you
can give up some of your bluffs (and naturally not value hands)
at will on later streets whereas your poor opponent must be prepared to
call down all the way...

Just as player A has hands he bluffs one, two or three times with, player B must have hands that he will bluff catch one, two or three times with. This is a sign of internal consistency of the model.

/**/

The model assumes hand values to be static.

No such assumption is being made. (Read my previous post.) Only the relative size of ranges is given. At any moment what is "value" (hence what is bluff) is well defined. (Remember, we are assuming we play versus ourselves and that we play true GTO. (Yes, that's obviously an approximation that can be questioned.)) Particular holdings are allowed to move between various categories. What the actual categories a particular holding belongs to on each street is revealed only after all cards are out. Very rarely particular hand is assigned to belong to a particular category (such as "value bet twice", or "bluff once").

/*****/

The biggest (by far) weakness of the model is that it doesn't "allow" for raises. Another weak spot is that it doesn't tell how much (per street) you should value bet (and hence bluff) in absolute terms, except for on the river. It also doesn't tell which combos to value bet or bluff on any street (except for possibly on the river, for which we don't need the model anyway). Only relative frequencies are explicit.

On balance, I think the model has some pretty solid theoretical foundation, while the "former model" does not. Its justification is mostly composed of handwaving arguments. I'll be thinking about how to incorporate (qualitatively at least) the possibility of raising. This isn't easy, not even on the river with the simplification that the river is capped to only two bets.