Hi Will,

Think I've found a typo. Chapter 7 p.251:

"If
EQsb(Hv) = 1.1Hv (ie the SB range is moderately strong)
we have that he value bets 31% of his range and bluffs with 16%"

1.1 must be a typo because the whole range is plotted from 0-1
So then I guessed it must be 0.1 because that would be a strong SB range but that would mean he value bets 90% of his range and not 31% of his range as is stated in the sentence that followed.
If he value bets 31% then EQsb(Hv) = 0.69

So should 1.1Hv actually be 0.69Hv ?

Or am I completely lost? :-)

Great book by the way! Looking forward to Volume 2.

Thanks for your time.

Hi deanster,

Glad you're enjoying the book.

This is in the discussion about the SB bet-or-check river game, and Hv represents the SB's weakest value-betting hand. We describe a player's hands with numbers between 0 and 1, where 0 represents the weakest hand in his range, and 1 is the strongest. For example, if Hv is 0.70, it means his weakest value-bet is his 70th percentile hand.

If the players have the same ranges on the river, then a hand's numerical value and its equity will be the same (up to card removal and chopping). This is because its percentile (the percentage of Hero's hands it's ahead of) is the same as its equity (the percentage of Villain's hands it's ahead of) if Hero and Villain have the same range.

On the other hand, if a hand's equity is below its percentile, it's indicative of a weak range, since it means that particular hand combination is ahead of more of Hero's range than Villain's. The equation EQSB(Hv) = 0.9Hv represents this case -- the equity of this cutoff hand is 90% of its percentile.

Similarly, EQSB(Hv) = 1.1Hv indicates a strong range, since the equity of the hand is higher than its percentile. Also, it turns out that the equity of this particular cut-off hand is an important descriptor of the strength of the SB's range in this toy game, which is why we focus on it in particular.

Hope this helps,

Will

I see.

So with pot size river bet:

In the case of symetrical distributions:
EQsb(Hv)=1Hv the SB bets 33% of his range of which 22% is value so Hv=0.78

But when distributions are slightly asymetric it makes a big difference.
It works out that when:
EQsb(Hv)=0.9Hv SB bets only 15% of his range of which 10%is value so Hv=0.9
And when:
EQsb(Hv)=1.1Hv SB bets 47% of his range of which 31% is value so Hv=0.69

Thanks Will

A bit late, but thank-you very much for your answer. Really cool, that you can ask questions to the writer of a book you just read (and still reading off-course). Have a good holiday!

Greetings,

Laurents

Np guys, happy to help.

Hey Will,

Sorry if this has been discussed already. On Pg. 251 We find that that BB's GTO calling frequency is 4/9 hands. I am trying to figure out why SB wouldn't expand their Bluffing range if the BB isn't calling 50% of the time. Is it because checking produces a higher EV for the SB after the hand H(b) then betting would produce because of the BB folding just a little too much?

Thanks!

Hi Uraby,

Yes, this is a very important lesson from the SB bet-or-check game.

The BB calls a bet less than 50% of the time, so indeed, the SB's complete air is not made indifferent to bluffing -- it strictly prefers to bluff. However, the SB doesn't actually have any complete air in his range here.

His bluffs are hands that have a bit of showdown value if they check (but which never win at all if they bluff or get called). So, the BB has to fold a bit more than 1/2 the time to incentivize them to call.

If the BB did in fact fold exactly 50% of the time, then the SB's weak hands' EV(bluffing) would just be S, but their EV(checking back) would be S+PE where E is their equity, so the SB wouldn't want to ever bluff. BB has to fold a bit more to inrease these hands' EV(bluffing) to make them indifferent.

This effect has some important consequences (and gets some further discussion) in the examples at the end of the chapter...

Great, thanks! Just reading along here, on Pg. 254 you refer back to the symmetric case saying that Villain is Calling 55.6% of the time. Just wondering if maybe thats a typo because you got 4/9 when you solved that example at the beginning of the SB-Bet or check game. Couldn't find it in the Errata so maybe I'm just referencing the wrong example. I also noticed in that particular figure title on Pg. 249, figure 7.11 it says solution structure for asymmetric SB bet or check game. So maybe I just have the ideas of asymmetric, and symmetric distributions mixed up. Could you clarify what those 2 things are for me?

Thank you

fmp

You're correct... typo. Villain's calling threshold is 0.556 here, so his calling frequency is 1-0.556 = 4/9. Thanks.

Symmetric distributions means the players hold the same ranges. In this case, if we have some particular hand combination, the fraction of Hero's hands it's ahead of (its percentile) is the same as the fraction of Villain's hands it's ahead of (its equity). So, we get EQ(h)=h. In other words, the equity distribution is a straight line. This makes the math easy to do.

Asymmetric distribution is anything else -- the players hold different ranges. So, in some sense, symmetric distributions is a specific, easy case, and asymmetric distributions encompasses everything else.

Will,
This is Donovan Doust aka Navonod.
I have been hyping your book up to no end and because, buddy, I flippin mean it.
GOOD SHOW man. Ive read about everything out there, well..a lot. Now that I have ur EHUNLHE volume 1 I truly feel like every time I read any other books I am wasting time I could be using to reread your book for the 5th, or 6th time.

Im gonna be making soms vidz for Splitsuit, and have already done some for pokertube and soon pokervip as well and feel like reading ur book has given me some of the tools I needed not just to beat up on small stakes regs but to handle complex situations and youve given me a great arsenal of tools for coaching players as well.

I feel like my poker education, rather than a gradual progression, has been a string of plateaus and the occasional light bulb moment. Your book was a minimum of three such light bulbs and itll likely be more than that eventually. I get more out of it each time I read it.

Probably already been asked and answered but is volume 2 coming soon or no? I ordered it months ago from a store that said it would be in next month every month gor 4 months. They promised theyd have it feb1. Is that possible?
Oh, and I just wanted to ask you if you wanted to play heads up for bankrolls?
JOKING
thanks for the best poker book ive read period without a close 2nd.
Navonod

Hi Donovan,

Thanks for the kind words, I do appreciate it .

Vol 2 is on my editor's desk now, and I think we expect to have it in print mid Feb. and available not too long thereafer. Sorry it's been delayed a bit. On the upside, I'm happy with the content, and there's a lot of it. I'll try to get a substantial excerpt posted asap.

Cheers

This came via PM, but I thought it was interesting enough to respond in public, since it brings up something I didn't really realize before that could confuse people. Hopefully the author doesn't mind -- name removed just in case.

So, I often use "distribution" to mean "range" essentially. The distribution I'm referring to is the probability distribution that gives the probability a player has any particular hand. (For example, 2% chance he has AdKh, 3% chance he has 7s4d, etc.)

I think this could be confusing, since the only "distribution" I really defined was the equity distribution. Adding to the confusion, the player's ranges and the equity distribution essentially contain the same information (up to card removal and chopping effects) on the river, so it's easy to conflate the two concepts. However, the concepts need to be made distinct on earlier streets where the equity distributions don't give the whole story (due to draws).

When I say we need to keep the distributions in mind when we play, I'm referring to the ranges. We need to keep our opponent's range in mind, even to just play exploitatively, and I've argued elsewhere that strategic play requires us to keep our own ranges in mind as well.

I do think that equity distribution graphs give us a good way to do this on the river. They let us forget about the particular cards and board involved, and just tell us exactly how various parts of our range stacks up versus Villain's. Also, I disagree that they can't be approximated or visualized, or that small changes in the ranges make large, hard-to-understand changes in the equity distributions. All an equity distribution is is a careful examination of how much equity all the various parts of one player's range have versus the other's. And estimating equities is a very important part of playing poker. However, if the graphical approach doesn't resonate with you, that's fine. All that's crucial is that you keep the player's distributions (i.e. ranges) in mind, one way or another.

Also, FWIW, Vol 2 deals primarily with preflop, flop, and turn play, where draws are important. Since equity distribution graphs don't capture all the essentials when draws are possible, they will not be nearly as prominent in Vol 2.

So to recap,

the equilibrium exercise as of right now could only be solved by hand making lots of calculations? Some help from CREV, I guess, could be useful, but the only way to play with that exercise is to do it manually on a piece of paper?

Not sure what you mean, to be honest. Are you referring to the part where you actually need to solve for a maximally exploitative strategy?

If so, then if your spot is simple enough, that might be something you can do by hand, but otherwise, that's pretty much what CREV was made for?

No, I meant solving a situation for the equilibrium strategies for both villain and hero. CREV is unable to do that, it's only able to give you a maximally exploitative strategy, but as someone has mentioned before, it just works in an infinite loop of counter adjustments.

Oh, well the EE isn't meant as an equilibrium-finding algorithm, although it's closely related to fictitious play, which is. I'm not aware of any publicly available equilibrium finding software except for the holdemresources.net calc which is unfortunately preflop-only.

Mhm, so if I do wanna analyze a situation, as you recommended, and as described in the book, I should do it manually correct?

Well, you can do it however you'd like?

The method shown in the book for finding max. exploitative strategies can be used by hand, but it's also the kind of repetitive, mechanical task that computers are good at. Again, although I don't use it myself, I believe a lot of people are happy with CREV for this purpose.

No general purpose max. exploit software is provided with the book, if that's your question.

Hi Will,

Just wanted to check my understanding of a couple of things with you if that's ok.

Firstly, regarding the question on p.248 about graph 7.10 d
The BB range is weaker so the SB is able to call with more than 50% of his hands.
Consequently the BB cut-off hand Hb is no longer getting enough folds to make it indifferent between betting and check folding. Check folding is now clearly the best option. So the BB will check fold more hands and make his Hb cut-off hand stronger so that it is getting enough folds to be indifferent.
Now that the BB Hb is stronger the SB weakest calling hands will no longer be indifferent and will prefer to fold so his Hc cut-off hand will return to 0.5
In the diagram the Mb horizontal dotted line effectively moves up a bit so that the Hc of 0.5 beats 33% of the BB bet range.
Looking closely at the diagram it seems to me that Mb was at 0.25 and would have to move up to 0.35 to keep Hc indifferent at 0.5

Secondly, regarding diagram 7.9 on p.239
In this case the SB Hc is 0.7 so he is only calling the top 30% of his range.
In the case of a pot sized bet Hc is 0.5
So is it the case that the SB must be facing an overbet here?
By my calculation Hc = 0.7 when the SB faces an overbet of 2.33*Pot.
Am I thinking about this correctly?

Thanks for your time Will, very much appreciated. Cheers!

Hi Will,

The questions below are about chapters 3&4. I know that's quite a few questions and it gets worse, especially with chapters 6&7... Hope that's OK.

Is it because the pay-offs depend on the stack sizes?

Figure 3.4, p78: I'm not sure I understand what we're looking at...
Could you explain to me what's going on here?

Because we need to know Vilain's range to do that?

How do you get these frequencies using tables 3.2 and 3.3?

Vilain's counter-strategy here would be to tighten up his calling range, probably to the point that all-in is no longer Hero’s best option with many of his strong hands, as you explain on the previous page?

Also, is there a reason why you picked AKo specifically?


Because if he's not 3betting enough, SB can profitably raise ATC?

Also, when you say "Those 2 ranges", we can use the word ranges here because EQ is not a factor? I mean, sometimes in this chapter, we make a specific hand indifferent between two actions, sometimes you talk about 2 ranges/plays having the same EV. Is it true to say that:

- we can talk about ranges/plays having the same EV when our EQ does not matter and only frequencies do?

- As soon as our EQ does matter, the Indifference Principle only applies to ONE hand (The borderline hand with which we play a mixed strategy at equilibrium?)

How do you know A6o is a good representative? I mean, you know that BB is shoving 20.6% of all hands but how did you pick A6o, speciffically? Experience? Common sense?


The way I understand it, it means that BB would construct his ranges so that:
- He still has a folding range and a 3 betting range
- He will get it in with 100% of his 3betting range

Is that correct?

Yea the thing to notice is that SB's calling freq can't really change from 0.5 (assuming a potsized bet), since if it was a bit too low, BB wouldn't bluff, but then SB's tight response would make him wanna bluff a lot and vice versa.

Starting from the symmetric distns case, suppose we obtained the equity distn 7.10d by keeping the SB's range fixed but removing some value hands and adding some air to BB's range. Start by supposing BB kept playing all individual holdings the same way (i.e. betting anything with equity Mb and checking otherwise). Then, his overall betting frequency is lower, and when he does bet, his range is weaker, so SB can call more.

But SB has to keep calling w/ freq 0.5, so to make this happen, BB has to tighten up his betting range. And I think that's pretty much what you said, so great.

If we want particular numbers, we can just do the math. The eq distn there looks like:

EQBB(h) = max(0, h - 0.2)
or
EQSB(h) = min(1, h + 0.2)

So plugging into the solutions on pgs 241 and 242, we have
Hc = B/(B+P) = 1/2
Hb = (EQSB(hc)(P+2B) - B) / (B+P)
= ( (0.5 + 0.2)(3) - 1) / (2)
= 0.55

So, BB bets the top 1 - 0.55 = 45% of his range, aka, all hands with equity at least EQBB(0.55) = 0.55-0.2 = 0.35. So you were spot on.

Figure 7.9 is just meant to show the structure of the solutions -- it isn't drawn to scale for the equilibrium at any particular bet sizes.

However, at equilibrium in the symmetric distributions case, Hc = B/(B+P), so if Hc = 0.70 here, then I agree it must be the case that B = 7/3 P.

Yes, that's the root of it, but intuitively, we can't be getting it in as loosely when the stack-to-pot ratio is larger, since our risk-reward ratio is worse.

4-3s is sometimes in the SB's shoving range in the solutions to the shove-fold game, and sometimes it's not. Sometimes it's shoved some non-zero, non-100% amount of the time. This graph shows shoving frequency as a function of stack depth.

Villain's ranges are precisely defined, and the charts that quote refers to describe Hero's max expl strategy. In particular, they give the frequencies associated w/ Hero's max expl play and not Hero's complete max expl ranges. This is mostly because it takes a lot more space to print a range than a frequency.


Those figures are the well-known "Nash shove/fold charts". Look at all the hands that SB shoves at 12 BB deep. These total about 53.2% of all hands. Similarly for BB calling.

Villain could also play exploitatively vs a flat call, knowing Hero's range is lacking in certain hands, and this would likely make flatting those hands more profitable for Hero.

It's just the standard example of a high card hand that plays well all-in preflop against a wide range but can be hard to play OOP post.

And if he's 3-betting too much, SB would never raise-fold.

Yea, since we have no chance of seeing a showdown if we raise-fold or open-fold, the EV of doing so doesn't depend much on the particular cards we hold. (It does a little bit b/c of card removal effects, but I'm neglecting that for simplicity.)

When a line can get us to showdown, our particular hand often has a larger effect on our EV. It's not always the case that there's just one borderline hand that's made indifferent in those spots, although that'll usually be the case in Vol 1.

Well if he's playing jam-or-fold, he's gonna be jamming some number of the best hands (no reason to jam a worse hand and fold a better one). So we just need to find the best ~20.6% of hands (ranked by equity vs villain's calling range) and take the worst of those.

A decent estimate can be made by just taking the top 20.6% of hands vs ATC, and you can clean that up with an iterative process if you want -- start with that, use it to deduce villain's calling range, then go back and recompute the top 20.6% or w/e of hands vs that particular calling range. Tbh, though, I imagine I looked at the computationally-generated solution.

This is with regard to the minr/3-bet/4-bet/shove game. Tbh, the quote here isn't very well worded. How can SB keep his ranges at the bottom of the tree the same if he's not getting to those spots with the same set of hands in the first place? I think I meant that SB's 4-betting and shove-calling range stay the same. In this case, opening less must mean he folds less to the 3-bet.

In this case, yes, your strategy is correct. The same set of hands will be profitable to get all-in, but with hands that can't get it in, EV(fold) > EV(3-bet-fold), so BB only 3-bets hands that can get it in.

Hi Will,

I´m always impressed by your posts, so I´ve decided to order your book. I play about 4 or 5 small stakes online tourneys each week with between 50-100 runners. Could you please advise me as to how I can most benefit from your book, bearing in mind that I don´t play cash games?

Hey, thanks .

I can't think of much specific advice for MTTers about how to read the book. Much of the book isn't really format-specific.

One thing to keep in mind is that, at least in Vol 1, we focus on maximizing chip EV. This is generally a good approach in HU (tournies and cash) but not in 3+player tournies where effects due to the payout structure are significant. To account for that, you'll usually need to add an extra step at the end of each EV calculation to convert chips to $ using, say, the ICM.

Another suggestion is to spend a bit of time playing HU specifically. It's hard to get a lot of heads-up experience just playing MTTs, so you'll get a better feel for the nuances of the format and be better able to take advantage of final table situations if you practice HU on the side.

Cheers

Thanks a lot Will, really helpful comments.

The following point you made is especially interesting:

Hopefully I will learn more about it in vol2. In the meantime... Moving on to chapter 5!