Cool, looking forward to seeing what you come up with

I have bought the book and I must say it is not easy to read for me. (English is only my second language). Overall I think the book is very helpfull for players who have the time to work apart from the tables. I think its suited for 6max and FR because you will often encounter only a single opponent postflop.

Will, really helpful to have you responding to questions in this way.

On page 149, describing equity distribution graphs, you say (last paragraph):

"Hero's distribution is just a plot which shows, for each hand, the number of hands in Hero's range which it is ahead of ..."

Shouldn't that be 'the number of hands in Hero's range which it's behind?

For example, if Hero's curve passes through the point (0.25,0.6), that means there's a hand in his range which is behind 25% of his own range (ie. it's his 25-th percentile hand) but ahead of 60% of villain's range (i.e. has 60% equity).

If I've got that right, I'm also trying to get my head around the precise way that Hero's distribution is 'flipped' to get Villain's distribution.

If (0.25, 0.6) lies on Hero's distribution, then the same hand, if it was in villain's range, would beat 60% of villain's range, and beat 75% of Hero's range (would have 75% equity) so we'd see Villain's distribution curve passing through the point (0.4,0.75).

More generally, if Hero's distribution curve passes through (x,y) we should see Villain's curve passing through (1-y, 1-x).

Is that correct?

Hero's top is on the left, so it really plots the percentage Hero is ahead of. So (.25,.6) means a 75-percentile hand has 60% equity. If Hero's curve passes through (x,y), then Villain's passes through (y,x) (if we neglect the possibility of split pots).

Thanks for the reply, Cangurino.

My use of the word 'percentile' was incorrect.
As you say, (0.25, 0.6) represents a 75-percentile hand (not 25-percentile). It's stronger than 75% of Hero's range. Thanks for clarifying.

I disagree with you, though, when you say that if (x,y) is on Hero's graph, then (y,x) appears on Villain's graph.

I went back and looked again at p150 where WT explains that the 'flipping' of Hero's graph that he's talking about is reflection in the diagonal from top-left to bottom-right.

When we normalise the scale on each axis to [0,1], that diagonal is the line x+y=1.

When the point (x,y) is reflected in that line, it goes to (1-y,1-x).

The reflection that takes (x,y) to (y,x) would be reflection in the line y=x, surely?

Again, maybe I'm missing something fundamental. (Wouldn't be the first time. )

Yes, you are right. (x,y) goes to (1-y, 1-x).

Thanks, Cangurino, that's useful, having that confirmed.

Can now safely turn the page and await my next [headexplodez.jpg] moment.

(@WT - that's meant as a compliment )

Well, the distribution graph certainly gives both pieces of information -- the number (or fraction) of hands which it's ahead of and the number it's behind. But yes, we actually plot hands with the x-axis indicating the number of hands it's behind, so I agree it would have been better put the way you suggest.

Yup, this all looks correct to me, except for the percentile thing.

In hindsight, I think I should have gone with the opposite convention and drawn equity distributions increasing from left to right. That would have simplified a few things...

My book order finally arrived today! Yay! Can't wait to finally dig into it!

I'm glad to hear it. I think it addresses a lot of the questions you've posted in Poker Theory lately, so hopefully you'll get a lot out of it.

Cool, I've also been reading Tendler's Mental Game lately. However, I think the biggest (and only significant) form of tilt I experience lately is what I call: "ignorance tilt", kind of related to "mistake tilt" mentioned in the book. Essentially, I lose confidence and focus in my game after losing a big pot in a tricky spot I'm not used to. For example, I'm grinding well into a session and comfortably playing a lot of common spots I'm used to, then this aggro/tricky player shows up and wins a big pot with a line I'm not very used to (ie. raise flop, check turn, overbet river). My game drastically plummets from there for the rest of the session and my concentration is spend ruminating about what went wrong in that hand and what I should have done instead. In essence, I am tilted not because I won or lost but because if the same spot came up again, I still wouldn't know what to do.

Hopefully this book will provide more insight as how to best defend against less common spots/lines that fewer players utilize with some general frameworks/methodologies.

Off topic question. I can try.

Did you ever explored approximate equity distributions?
With this I mean a bucketing of combo's in groups like AKo AKs 22.
But previous streets can still remove ie AcKc AhKh. So AKs still resembles AsKs for the FD and AdKd for the backdoor FD ie but their equity is grouped together beneath AKs.

Not sure if such an approximation would deliver results still close enough or not.
Not sure if you have an opinion about this or explored this option in the past.

If approximate enough; some commercial progams can quickly be used in combination with excel to get close enough results instead of time-consuming bypass route for generating equity distributions by first getting equities per combo.

How hero's best response is found in 2.2.3 Equilibration Exercise given Villain's calling range? For first iteration:
Villain calls 205 of 218 hands he gets to the river
shoving EV > 48.5 and is fullfiled for EQ> 0.3
To find best combination of hands is about looking for hands that have such EQ?

For iteration 2:
Villain best reposonse is simple odds question?

Why Villain QsQc has not enough equity for interation 4? Is is because we block hands with spade combination and then hero shoving range has less possible bluffs?

Yea, although I don't have too much novel advice that directly addresses those mental issues, I actually think a studious, quantitative approach to poker can help a lot with them.

If you rely on your gut/instinct/feeling to guide your decision making process, then you're often not quite certain whether a play is profitable or not. So it's pretty easy for your game to get out of whack when things aren't going well, since it's pretty easy to convince yourself to make a (tilty) play if for all you know it could actually be good -- it's pretty easy to convince yourself to make a play if you suspect it might be bad, because hey, it might be good also.

On the other hand, many players won't make plays that they know for a fact are bad (at least not until they reeeallly tilted ). So, if you've done the math, and you know with certainty which action is the best in most spots, you'll likely be a lot less prone to make tilty plays.

Just my 2c... the mental game is not my area of expertise...

Hey Emus,

I guess I'm not exactly sure what you're trying to do.

If you're trying to generate equity distributions for away-from-the-tables study, it isn't necessary to do it yourself using Excel -- you can use the holdemresources.net calculator or the free ED visualization utility on EHUNL's website.

Cheers

Hero's best response is found here by considering each of his hands separately, finding the EV of each of his options with each hand, and going with the action with the biggest EV. His EV when he check-folds is easy to find, and we can find his EV of jamming a hand on the river once we know Villain's calling range.

I think you're right that it turns out here that he needs something like 30% equity versus Villain's calling range to make the jam here.

Yes, essentially. Again, Villain's just looking to make the play that maximizes his EV with each of his individual hands, but since he's facing an all-in on the river, he has a simple call-or-fold decision that he can make by just comparing his equity to his pot odds.

In more complicated situations, it will often be difficult or impossible to boil down a decision to a simple criterion based on an equity. For these, it's important to understand the general methodology where we make decisions by figuring out the EV of each action and going with the biggest. If it isn't clear, you may want to review the previous section (Section 2.2.1 - The Most Profitable Way to Play a Hand).

Yes, this is correct. The bluffing range we gave Hero actually includes a lot of missed Q-high flushdraws, so we see that Villain's QsQc actually has much less equity than his QdQc vs Hero's jamming range because of card removal:

Board: Js 6h 3s Jd 2d
Dead:

equity win tie pots won pots tied
Hand 0: 25.000% 25.00% 00.00% 9 0.00 { QcQs }
Hand 1: 75.000% 75.00% 00.00% 27 0.00 { AcAd, AcAh, AcAs, 6c6d, 6c6s, 6d6s, 3c3d, 3c3h, 3d3h, AcJc, AhJh, Ah8h, Ah5h, Ah2h, KcJc, KhJh, QcJc, QhJh, QsTs, Qs9s, Qs8s, Qs7s, Qs6s, JcTc, JhTh, Ts7s, 9s7s, 8s7s, 5c4c, 5d4d, AsTc, AsTd, AsTh, KcJh, KdJc, KdJh, KhJc, KsJc, KsJh, QcJh, QdJc, QdJh, QhJc, QsJc, QsJh }

Board: Js 6h 3s Jd 2d
Dead:

equity win tie pots won pots tied
Hand 0: 34.146% 34.15% 00.00% 14 0.00 { QcQd }
Hand 1: 65.854% 65.85% 00.00% 27 0.00 { AcAd, AcAh, AcAs, 6c6d, 6c6s, 6d6s, 3c3d, 3c3h, 3d3h, AcJc, AhJh, Ah8h, Ah5h, Ah2h, KcJc, KhJh, QcJc, QhJh, QsTs, Qs9s, Qs8s, Qs7s, Qs6s, JcTc, JhTh, Ts7s, 9s7s, 8s7s, 5c4c, 5d4d, AsTc, AsTd, AsTh, KcJh, KdJc, KdJh, KhJc, KsJc, KsJh, QcJh, QdJc, QdJh, QhJc, QsJc, QsJh }

Hi Will,

I'm coming back to poker after a long absence, and this book is pretty much the book I always wanted to write, but never had the dedication (and software capabilities) to do so. The emphasis on GTO strategies fits my mathematical background well, and even your definition of EV is the same as the one I used when I was searching equilibriums on my Excel Spreadsheets years ago

(doing stuff like this, with StoxEV and a lot patience for every stack size :
)


So anyway, thanks a LOT for this book. To me it's the most groundbreaking material since MoP, and it's also much more useful in a practical way, since you clearly make the effort of staying away from Toy games and anchoring your reflexion in real NLHU examples.

Now here's a couple of questions I have so far (I'm only halfway through, so if they're answered later in the book I apologize) :

1) In terms of preflop opening GTO strategy from SB, I think it's fair to say that under 15BB, min-raising is better than any other raise size, as to give the worst possible odds to a BB 3-bet shove (which is proven to be necessary 50% of the time).

When the stack sizes increase, the SB can expand its raising range because the risk/reward ratio becomes worse for BB and in consequence BB can't punish SB weakening his range.

There comes a point where SB is able to raise 100% of his hands, around 28BB deep. At this point you say in your book that SB can't expand his range more than that, and what happens when stacks get bigger is that BB has to abandon his 50% shoving frequency, since the odds he's getting on his shove aren't good enough to keep it.

But what if SB were to increase his open raise sizing?

For example we can easily see - it's one of the "do it yourself" paragraphs in the book - that for a 3BB raise from SB, BB only needs to shove 37.5% of the time to keep him indifferent between open-fold and raise/fold.

According to your graph, BB will shove 37.5% of his hands at around 40BB deep. This means that at 40BB deep, SB could raise any amount from 2 to 3BB, and BB wouldn't change shoving frequency. Now obviously in a raise/shove game, this isn't very interesting for SB, since he can give worse odds to BB and still force him to make a decision by min-raising.

But it got me thinking about "real-life" applications.

So here's my intuition : I feel like once the stack sizes become big enough, there comes a point where the GTO strategy for SB is to raise 100% of his opening hands. This probably occurs somewhat deeper than 28BB, because in real games, BB has the options of small 3-bets and flat calling, but I'm pretty sure it happens at some point. Let's call this stack size S.

Now here's where I'm less confident and would appreciate your imput : I think that once we get deeper than S, the GTO strategy for SB is still to raise 100% of his hands, and to gradually increase the raise sizing, so that it's as big as we can make it without having to open-fold anything. And I wouldn't be surprised at all if 1000 or 2000BB deep, the correct strategy for SB was to raise to 4BB with his entire range.

What do you think about this theory? And with your experience of HUNL, what would you say is the value of S (the stack size where we can min-raise everything)?

2) My other question is something I've wondered for a long time, and I think you might have some insightful thoughts about it.
I have this idea, that if a hand is strong enough to call an all in on a Street as part of a GTO strategy, then you can call with it (or shove with it) no matter how the betting went so far.
I don't have a formal proof, but it just feels right.
Let's say 16BB deep, A3s is good enough to be minraise/called according to the GTO strategy, then it's ok to do it against every possible opponent.

So if I'm playing a complete nit who will only shove TT+ and AQ+, of course it's not a good call (I need 43.75% equity, and I only have 32.4% vs this range) ; and of course it'd be better for me to play exploitatively and fold. But in the long run I will still be +EV because the money I lose here will be compensated by other hands that he folds against me, or lets me see flops.

Same thing if according to GTO strategy, KQ is part of my flop all-in calling range on Q84 30BB deep. It doesn't matter if the action goes "open shove from Villain" or "Min-bet / min-raise / min-raise / min-raise / min-raise / shove", or if Villain always shows AQ+ in this spot... I will be able to call and win money in the long run, because the lines he takes and/or the ranges he shows up with will make him lose money against other hands I could have in that spot.

Would you agree with all this?

Regarding your excel sheet - having only limp/fold without limp/call range is exploitable.

Increasing raise size leads to decreasing opening range - less profitbale play.

"Hero can play his equilibrium strategy and be guaranteed to at least break even on average."

I did not read whole book, but I think if villain's range in given spot is much much stronger than hero the equilibrium play is to fold top pair 30bb deep.

Obviously. It was made to exploit the population tendancies I was facing like ~4 years ago, I never claimed it has any value in today games.

Hi Will.. thanks for this book.. i wanna specialize into Hu Cash and i'm finding it very interesting..

Btw.. i think it would be nice if the anwers for the some exercises at chapter's end were published.. maybe at the end of the book..

Well, certainly if the SB is open-raising too large or too frequently in very short stacks, it makes it easy for the BB to turn the game into a mostly preflop-only sort of thing. And that's great for the BB since he's in position preflop, so the SB will want to avoid that. However, I don't think it's wrong to open raise larger (all-in or not) with a set of hands primarily composed of some of those that play pretty well all-in preflop but rather poorly postflop (say some Ax, 22). We don't have to use the same sizings with all of our hands.

Yea, this is of course within the context of the minraise/shove game where the BB must play shove-or-fold versus the SB's minraise. In the full game, the BB can also continue with a call or a non-committing 3-bet, and I would guess that at 28 BB effective, the BB's total continuing frequency versus a minraise at equilibrium is easily >50%, so that the SB isn't auto-profiting with his min-opens.

Not sure this last part is right. I agree that (in raise/shove games at 40 BB effective) when the SB is opening 2x, the BB is jamming something near 37.5%, and when SB is opening 3x, the BB is also jamming near 37.5%. But in between the SB's 2x and 3x open, the BB's jamming frequency actually goes up. Essentially, he wants to jam more than he does vs the 3x since he has to defend more to keep the SB's bluffs indifferent, and he can jam more than he does vs the 2x since he wins more when SB raise-folds.

Lot of interesting stuff here. Regarding your main point, though, I don't think the GTO strategy will be to raise as large as we can and still get away with opening 100%. The picture you paint is that for any particular stack size, we start out thinking about minraising, and then increase that raise sizing until we can no longer open 100%. That is to say, we focus on the worst hand in our range, and, assuming that the EV of this hand will decrease as we increase our open sizing, we keep increasing our open sizing until EV(opening that hand) exactly equals EV(open folding that hand).

Since our actual goal is to maximize our EV with each hand, I'm not really sure what raising large enough to make our worst hand unprofitable does for us. Of course, it may be the case that the EV of our better hands goes up at the same time as the EV of our worse hands is going down, but there's no reason that I can see that any particular EV will be maximized at the point where we can just barely get away with opening 100% -- it seems pretty arbitrary.

And your whole question sort of implicitly assumes that we open the same size with all of our opening hands at equilibrium, which is not necessarily the case either.

The guarantee that GTO play gets us in HUNL is essentially that our expected profit is at least 0 when averaged over both positions (but not in an individual hand and certainly not in every subgame). So making a play according to our GTO strategy would always be 'OK' in the sense that the play would be part of a strategy with this guarantee.

However, I don't think it's the case that, if a hand is strong enough to get all-in with on a particular street in one betting sequence, it necessarily means that we can also get it in given all other betting sequences on that street. In fact, if that were true, it would imply that a player's entire get-it-in ranges on a street are all exactly the same, regardless of betting sequence. (Is it obvious why?) And I think it's clear that this is not the case. For example, there are lots of spots with draws where it would be incorrect to call an overbet shove all-in, but we could play correctly and then find ourselves with the odds to call all-in after a couple re-raises.

I'm glad youre enjoying the book .

As far as the 'Test Yourself' questions, I feel like a lot of them don't necessarily have a cut-and-dry correct answer. In the cases where they do, the answer often directly follows the question blurb in the text, but most of them are supposed to be of the open-ended, thought-provoking sort, and the answer often depends on the particular games you play in or opponents you face. Some of the others are just very large questions that would take a lot of space to adequately explore. That said, if you're having trouble with something specific, post here and I'll do my best to help you out.

Warning: long post!


Hi Will/yaqh. First let me apologise for jumping in the thread earlier with some stuff that wasn’t really specifically to do with the book. I’m pretty bad at the internet sometimes :/

Anyway, having now spent quite some time studying the book I have a lot of (hopefully on point!) things to say, some of them critical I’m afraid since I’ve found some fairly serious errors in the text. I’m not one to nitpick typos but these are pretty major and it seems you’ve been let down a bit on the editing front. It’s a shame because, let me stress, this is absolutely the most advanced and important piece of poker literature out there, at least in the regular price range. There’s no filler and no throw-away material, and the text as a whole starts strong and just gets better. Some subject matter is prefaced ahead of time, while other topics are introduced and then repeatedly referred back to; as such, the holistic nature of the book perfectly epitomises the poker maxim that no street must be treated in isolation. Even the ‘try it yourself at home’ bits are really well conceived and calibrated.

So let’s get the nasty errors out of the way. One recurring thing I noticed was that ‘SB’ and ‘BB’ were getting muddled up. Didn’t note down the locations at the time, sorry, but it happens several times in spots where genuine confusion is likely to result. Still, that’s relatively minor.

On the more serious front, a few of the river hand examples, so central to the book, are spoiled by basic inaccuracies. The range on page 198 is a mess... ‘Additionally, weaker queens, many nines, and the occasional stubborn six are in his range for getting to the river.’ But the range listed contains no (single pair of) nines! And the only pair of sixes is the Ks6s – not so stubborn considering it flopped a flush draw. Oh, there’s also the ‘two pair hand’ Qs6s… oops, the Qs is already on the board. Is this the range you actually worked with when analysing the hand? Looking at the equity distribution graph, I suspect so…

Staying with Example 2, I’m not certain that this is an error but I can’t make sense of it: on p.300 you list precisely three combos which are checkraised for value, then say ‘Bluff-raising Ks2s about 1/3 of the time he has it gives him the proper bluff frequency.’ But after the SB bets ~Pot his odds on calling a shove are 4 to 1, so to make him indifferent the BB should be bluffing 1/5, right? So wouldn’t that be 3/5 of the time he has Ks2s?

Moving on… probably not an error as such but reads confusingly: on p.307 you tell us that the BB’s EV is maximised with an overbet. Yet on p.305 you had said that the BB did not use any of several overbet options. That’s quite surprising – were the options you offered it very different that that optimal sizing? If so, why didn’t you go back and offer it that more profitable sizing once you’d learned that it existed?

On p.203 the Jd appears in both ranges despite being on the board. There is no KJ listed in the BB’s range but on p.308 you say that he sometimes checks that hand. Again, we just don’t know whether the typo is in the range or in the text, or which range you actually did use when calculating the solution to the hand.

On p.309 we have ‘The stacks are a bit awkward on this river. There are 15 BB in the pot and 22.5 BB behind.’ Sadly though, according to the hand history as presented on p.204, there are in fact 19 BB in the pot and 20.5 BB behind. Not so awkward then. Again, typo in text or you worked with the wrong numbers? We don’t know.

One last not-so-important thing I noted: on p.266, (c) should, I’m fairly certain, read ‘P4’. I’m less certain whether p.290 should read ‘P4’, but for the sake of consistency it would seem to make sense.

Again, despite these bloopers the quality of the book as a whole remains extremely high and I certainly recommend it to any serious player. I just hope it gets the second edition it deserves.


So with that out of the way, some other stuff I wanted to touch on:

Bet sizing

- Were you solving for bet sizes to any degree, or mostly using intuition and experience when choosing candidate bet sizes to feed to the solver?

- Is it actually possible to ‘solve’ bet sizing in general or in any particular spot, without solving the whole game? Or, hypothetically, if one were in the process of solving the game, would the bet sizes ‘pop out’ somewhere along the line, or is it a case of brute-force solving for every possible set of sizings and then seeing which has the highest EV? Are these questions inscrutable at the moment?

- You don’t mention geometric growth of pot at all. Is that concept only relevant to games with static hand values? Or would we expect to see sizings trend towards that pattern as ranges across streets get closer to optimal?

- There are several places in the text where it appears that the bet sizing ‘Pot’ has special significance. Perhaps that’s coincidence but it kind of chimes with my intuition… 2 to 1 being the lowest whole integer odds, I see it as being a sort of ‘knee of the curve’ point on the risk/reward graph: lower and the rate of loss of value outstrips the rate of risk reduction; higher and it’s the other way round. Does that sound about right, or am I seeing things that aren’t there?


Miscellany

- Would it be possible to get a look at the full results? I’d love to dig deeper into card removal effects and threshold hands, etc.

- Similar to the GGOP question, do you think that as ranges become closer to optimal the equity distribution graphs become a) more like a smooth diagonal line and b) less distinguishable between the two players?

- I was a bit surprised that in example 7 the BB wasn’t able to bet his whole range and take the whole pot. The small threat of trips is obviously what makes the difference. Is there a stack size at which the BB could lay claim to the whole pot, given the same ranges? More generally, for any ‘bluff-catcher plus traps’ spots, I imagine there must be an equation into which I could plug stack size, equity of trap hands and frequency of trap hands and that would spit out a yes/no answer to the question, ‘Can the other player capture the whole pot?’ Does that seem reasonable? (Feel free to leave the finding of said equation as an exercise for the reader )

- On a similar note, it was interesting that one of the examples included a non-all-in betting range that contained no nuts. Should there be a straightforward way of finding the min B and/or max S at which it’s ‘ok’ to have no nuts in range, or is that sort of thing contingent on the precise make-up of the equity distributions?


I think that’s all I’ve got for now. Thank you for taking the time to address this thread, and thank you for the book! I can’t wait for the next one

No worries, I think your questions were pretty relevant to the content of the book anyway.

Thanks for your kind words. That said, mistakes in the text are all mine, and my editors are responsible for correcting many issues that you didn't see. In a work with such a high density of info/details, I'm afraid some errors are almost certain to slip through the cracks. To be honest, though, I've been happy that the list isn't too long, although it seems like it might be about to get a bit longer.

As far as that goes, thanks very much for taking the time to write out errata you find. There's a list on the book's website to help readers, and everything listed there will be fixed in future printings.

Ah, ok. To be honest, this is the sort of thing I take especially seriously since it's likely to actually confuse a lot of people, whereas many issues are things people wouldn't even notice without the super-close reading it seems like you've made. Anyway, I really do think correctness is very important, so if you do happen to re-locate these spots in the future, please let me know.

Qs6s wasn't included in the calculation, and I'm not sure how it made it into the range listing, but the listing does seem to otherwise match up with the one I used. It looks like I forgot to include those bluff-catcher type hands in his range. Thanks for pointing out this oversight :/.

Yes, the BB should be bluffing 1/5 of the time here. The issue is that, in addition to Ks2s, he is also using very small (less than 5%) fractions of a variety of other hands as well.

Of course the exact solutions to a lot of these can be quite complicated, and listing exact numerical frequencies of all holdem hands in all spots isn't going to help anybody, so I did my best to describe the important features of the solutions in relatively-short, human-understandable form. Here, only Ks2s was specified since since its the only one he uses with particularly high frequency. But I agree that the wording here makes it sound like 1/3 of Ks2s is actually his entire bluff-raising range which is incorrect and should be improved.

The overbet sizings he was originally given but didn't use were 3x pot and all-in. The B* sizing I mention later was about 1.35x pot.

As far as why the analysis didn't include this smaller overbet sizing -- to be honest, historically, I think it wasn't until after I'd done the analysis that it occured to me that the distributions looked a lot like the PVBCPT game, and that we might be able to gain some insight by refering to the solution to the model game. And then I didn't think to go back to my earlier analysis.

But if you'll excuse the post hoc rationalization -- very few players, I believe, will make the overbet their standard bet sizing in this spot. That is, some people might overbet sometimes, but I think very few make it the standard sizing they use with most of their river betting range. And there's actually good reason to focus on analyzing decision trees with the standard sizings. If your opponent constrains himself to using one particular sizing on the river, then in effect, the game you're playing is the one where that's his only sizing choice, so that's the one whose solution is most useful to know. Players often, through habit or whatever, effectively constrain their choices of bet sizing before even starting to construct their ranges.

Does that make sense? Like, if you're playing someone who only opens to 3x preflop, then to think about playing against him, it makes sense to solve a decision tree where 3x is the only open sizing choice. You wouldn't want to solve a game that allows lots of different open sizings, because the solutions to that would probably include a SB who uses 2x for most of his opens, leaving him with a weird, small 3xing range that bears little resemblance to the one he actually uses. I mean, the guarantees of the equilibrium still mean you won't do too bad if you were able to just play a solution to the full game, but if Villain constrains himself to only use a 3x sizing on his button, then effectively, the game you're playing is the one where that's his only option.

Hands that conflict with the board are automatically excluded from the calculations even if I dumbly included them in the input ranges which were transcribed to the text. Embarrassing mistake. And also it looks like the 'SB' and 'BB' were switched here. The second range on pg. 203 is the BB's (and includes KJ).

Yea, I changed this example after studying it a bit, but it looks like I forgot to update the hand history on p 204 . Thanks for catching this.

I'm not quite sure what you mean here. Do you mean that you think (c) should say "B=P, C=4P"?

I think I need a break -- I'll get back to your questions in a bit!

Cheers