In the Ex. 4.2:
if SB range is too strong (that means 33rd percentile hand has EQ>5/12) he will profitably shove more than 2/3 of the time. To find SB frequency of jamming we look for the weakest hand in SB range that EQ is at least 5/12, right?
When BB is not bet/calling enough as he should in equlibrium, SB again will shove more than 2/3 of the time, right?
If SB range is too weak (2nd inequility is not satisifed) what % of the time SB will be shoving, less than 2/3?

I tried to calculate minimum equity for SB shoving range (when his range is too weak compared to BB) and I got 0.24, could be this correct?

Hi Will,

Thank you for the fantastic book. After several weeks of study my head is spinning. I'm hopeful that things will start to click and that I will be able to incorporate these theories into my poker strategies in an effective way.

This is not the book for those that don't want to put the work in. The game has been evolving, and the days of success employing unrefined simplistic strategies are gone...at least for me.

I've been playing with your Equity distribution visualization utility, and have found situations where the Hero's equity distribution graph somehow includes hands that aren't in hero's range. IE Hero's range is top 13% but sets of deuces are being included in the graph? Is this a bug?

I would also be interested in forming a study group if anybody is interested? I guess the challenge will be to find people that are in the same ballpark in terms of level of understanding/application so that it can be mutually beneficial.

The three books that I am working on are, EHUNL, MOP, and Applications of NLHE. I will be adding volume II of Will's book ASAP as well. If anyone is interested please PM me and we can see if there might be a fit.

Thanks again for the fascinating book Will, I highly recommend it.

Are you in the Toronto area? Having read MOP, I originally bought Will's book because I was ending up heads-up a lot when I WAS grinding at PartyPoker, but unfortunately PartyPoker has become a terrible site to play so I am now trying to implement the book's principles in live tournaments and cash games in the GTA.

Sorry for the dumb question, but would someone be kind enough to point out what I'm not getting here?

I've kinda hit the wall in chapter 7 of Expert HUNLHE, seems like the more I reread it the less clarity I get.

There's something about the way he's graphing the solution structure for asymmetric BB bet or check game(p.239) that seems somehow backwards to me? I'm sure I'm just missing something obvious?

For example I can't understand why on p.241 he states that...

"Where EQBB(Hb vs a call) is the BB's equity with hb versus the SB's calling range, that is, the chance he wins the hand when he gets called. This is simply 0 since, as mentioned previously, the SB always has a better hand than hb when he calls."?

Now while I understand that SB's calling range is much stronger than BB's betting range, but isn't there an overlap in the ranges where some of BB's strongest betting hands will beat some of SB's calling range? Am I just misreading the graph somehow and that the entire SB calling range beats all of BB's betting range?

On static boards, what makes you think that bcb line is higher EV than cbb line?

There's no example 4.2, but I guess you mean section 4.2? Yea, so the situation again is that we start with 30 BB effective. SB minraises, BB 3-bets to 5bb, SB calls. On the flop, BB can c/f, b/f, or b/c, and if he c-bets, his sizing is 1/2 pot. SB plays jam or fold vs the c-bet.

If SB range is too strong so that he's jamming > 2/3 of the time, then the BB never plays b/f on the flop since c/f is better than b/f with all his weak hands. Since BB is always b/c'ing, SB should jam when

EV(jam) > EV(fold)
60*(sb's equity) > 25
sb's equity > 25/60 = 5/12

So yes, you are correct. (And keep in mind that (sb's equity) here is equity versus BB's bet-calling range, not his whole turn starting range.)

I don't think this is necessarily true. If BB's range is very strong, SB might be shoving over the c-bet very rarely at equilibrium. Then if BB began bet/calling a bit too little, SB might start shoving a bit more but still less than 2/3 of the time. Did you have a different situation in mind?

Yes, it will be less than 2/3. That's essentially what the inequality means.

Big picture -- we'd like it to be the case that SB is shoving over the c-bet just enough to make BB indifferent between c/f and b/c (and this turns out to be 2/3 of time time). But there are two ways this doesn't work out
- if SBs range is strong enough so that even if BB never b/f's, SB can still profitably jam more than 2/3 of the time
- if SBs range is so weak that BB can c-bet 100% and his cbetting range still isnt weak enough that SB can jam over it 2/3 of the time
In either of these cases, the indifference breaks down, and we can't use the convenient tricks to find the equilibrium.

How did you do it? I don't think there's any one number that's the minimum equity (do you mean equity vs BB's flop starting range or his c-betting range or his jam-calling range?) for hands in SB's shoving range. It depends on how often BB is folding to the jam, which depends on the details of the situation.

Hey, yea, that behavior is intentional. You can think of an equity distribution graph as including all hands, and if 0 of a hand is in the respective range, that just means it gets 0 horizontal space on the distribution graph. These hands will generally be on a vertical portion of the graph, and you can always just consult the range views to see if a hand is in range. A few people have mentioned this, and I guess I could change the program's behavior if it confuses people, but doing so would really only reduce the amount of information you can get from the display, so I think I'll leave it. If I otherwise update that utility, I'll try to remember to add a note about this to the README file.

It isn't that the entire SB calling range beats all of the BB's betting range. It's that the entire SB calling range beats the BB's particular hand Hb. Hb is the BB's weakest betting hand. That is, it's a bluff, and never gets called by worse.

This is illustrated on Figure 7.9. The way SB splits his ranges is shown on the vertical axis. When facing a bet, he calls with hands between 1 and Hc and folds hands from Hc down to 0. And you can see that the BB's hand Hb (which has percentile EQBB(Hb) with respect to the SB's range) falls below Hc, SB's worst calling hand.

Maybe some general comments on the equity distribution graphs would be helpful since they are in some sense drawn backwards, and I agree it can be confusing.

Maybe think of it this way -- the function itself EQ(h) is what we're actually interested in. We give it our hand percentile and it tells us our equity. Of course this function is increasing from left to right, since as our hand gets stronger (that is, h gets larger), our equity gets higher. In the symmetric distributions case, for example, EQ(h)=h. In this case, if h=0 (i.e. we have the 0th percentile hand or the worst hand in our range) then we have EQ(0) = 0, i.e 0 equity. Or if we have the nuts, h=1 and we have equity EQ(1) = 100%. So the equity function EQ(h) looks like the straight line y=x.

It's only that in the book, we plot it backwards. Instead of drawing EQ(h), we draw EQ(1-h), so that the nut hand 1 with equity 1 shows up at the point (1-1,1) = (0,1) and the air hand 0 with equity 0 shows up at (1-0,0) = (1,0). So in effect, the function we care about is EQ(h), but we plot EQ(1-h), and the graph of one of these is essentially the mirror image or the other, obtained by flipping it around the vertical axis.

I did it this way for historical reasons -- to draw them the same way as they were drawn by other people previously. However, I don't think they were used as quantitatively as in EHUNL, so these problems never arose. I imagine I'd switch things up if I was doing it all over again, but I don't think the convention's too bad once you get used to it.

Are you asking me? I wouldn't say I think that. Why do you ask?

Oh yeah, sorry will, I was asking you. I just reread the section on static boards and it says "focus on the more common case where the SB c-bets". I overlooked this assumption in my post. The only reason I ask is that every static board that I've looked at, there was never an incentive to value bet unless you were planning on betting on future streets.

But here's a different analysis then. Without thinking about it much, it seems that the SB would want to bet the turn and check back the river with the top of her middling hands which can get value from the bottom of Villain's turn calling range. And then he'd want to check-back on the turn and bet the river, her weak middling hands which aren't strong enough until the BB checks the river and signals his range is even weaker than it was on the turn (because he value bets with the top of his range on the river).

A tangental question, in the book are you assuming the SB c-bets with his entire range in this section?

First off, thans for answering my last question personally, Will.

Reading further, another question occured. It's starts at p. 46, where we apply the equilibration exercise. I got the example you made here but had some troubles setting one up myself.
So what I would like to know is whether we can create an equilibrium for the following situations (unfortunately, I don't know whether this still occurs in the book since I didn't read much further yet):

1. Unexploitable calling range vs let's say 30% 3bet preflop
2. Equilibrium for cbetting flops after minraising pre vs an opponent that calls very loose (K high on almost any flop etc)

I would be thankful if anybody could help me out.

No one is even close to solving this complex of an equilibrium. The further you move away from the river, exponentially larger the game tree becomes. We can solve river situations with limited bet sizing. But no one is even close to solving turn situations. And river situations aren't exactly very helpful, since the ranges you get to the river with are not GTO ranges, so your not coming up with the actual correct GTO solution, since the ranges you begin with are off.

hope that answered your question.

thanks for your answer. now i know why i wasn't able to solve this problem

Got it!

"It's that the entire SB calling range beats the BB's particular hand Hb. Hb is the BB's weakest betting hand."

For some reason I was thinking in terms of range vs range when to calculate indifference we need to consider only each players threshold hand. Thanks for the detailed explanations Will, very helpful, as it can be so frustrating when I can't figure something out.

Much

Does anybody know here how (or with tools) to calculate chance to hit at least X% equity vs opponent's range on random flop?

Something like this you look for?

ProPokerTools Odds Oracle Results (2.22 Professional)
Holdem, Generic syntax
PLAYER_1 76s
PLAYER_2 20%
813 trials (randomized)


How often do(es)
PLAYER_1 have hand vs. range equity of at least 35% on the flop
44,8954% (365)


PQL

(just note that you want 7x6x for 76 suited in the odds oracle)

- the PPT fairy

Your questions basically boil down to solving the whole game of HUNL. A very direct approach to this problem is described in Chapter 2: draw the game tree, guess a strategy for one player, and solve for each player's maximally exploitative strategy in turn, with some mixing, until convergence. Unfortunately (or fortunately?) this approach is computationally intractible. However, there is a lot of progress to be made by studying simpler situations, and this is the focus of much of the rest of the book.

In fact, computers can solve quite large games these days -- for instance, we can certainly obtain larger solutions than could be reasonably be included in a single book. Solving turn situations at about the same level of abstraction as the computational solutions to the river situations in Vol 1 (e.g. a few bet sizes possible at each decision point, etc) is no challenge at all.

You are correct the GTO ranges found by solving river-only situations are not necessarily the GTO ranges of the full game. However, saying they're not helpful is a bit of a stretch, imo. In fact, there is lots of value in studying realistic approximate games.

Happy to help

+1 for PPT

The largest known solved poker game that I know is from Game Shrink paper, but it seems that it would be much smaller than solving the turn and river, if we had normal sized ranges, stack depth, and a few bet sizes. What size do you think the game tree would be starting from the turn?

Yeah, I've solved lots of river situations and multi street toy games and have found them all helpful, but I meant to say not directly practical in that you can apply them to a specific situation.

Yea, it depends strongly on things like the stack depth and how many bet sizes you allow, but for a few, very roughly --

Think about an individual river spot. For a bet and raise sizes, etc, one river subtree might have on the order of a hundred decision points. If the SPR is small, you can do a good job modeling river play with many fewer.

So, there's something like 50 of those river spots for every way we get to the river from the turn, and then there's the decision points on the turn itself. Maybe there's like 10 different lines on the turn after which we have to play a river, and another 100 decision points on a turn itself.

Then that comes out to 10*50*100+100 = 50100 decision points. Definitely manageable with a computer.

Well, I mean, I often find myself at the beginning of river play armed with little other than my knowledge of the SPR, the board, and an estimate of the players' ranges. Understanding how those factors affect the equilibrium of that subgame in a vacuum can definitely help me understand and plan my play in such real spots.

Page 83, the questions in bold.

1. SB's hands have same EV before shoving as after shoving. BB's hands have same EV after SB shoves as after BB calls. Why is this?

2. What does the comparison of the same hands tell us about the positional advantage.

Would anyone care to join me in trying to answer some of these questions in bold from time to time. I often think I have a decent guess, but I'd love to hash it out with others and see what I'm missing, etc.

Hey QTip, we've already chatted about this offline, but just wanted to point out here that the bolded specifications I added are important.

Also, in general, I've tried to stay out of requests for discussion like this, since I know there's a lot of value in trying to work things out on your own rather than just being told an answer. But if anyone wants my input, just say so!

Page 130, I created my own examples.

Scenario (just to make sure my calculations are right)

Eff. stack size = 25BB
SB raises to 2BB, BB 3bets to 4,5BB, SB calls
BB's Cbet size is 3,25BB

EV (check-fold) = EV (bet-fold)
x = 0,735

I plugged in some ranges and found out that on the 8h6c5d flop, as given in the book's example, the SB is folding 75s rather than jamming in the equilibrium which I feel he never does in reality.

Additionally I didn't understand how the 26.73% BB's betting percentage is calculated.

Page 126 bottom, it says that the "BB's bet-folding frequency is key. It goes to zero when the SB's range is too strong."
How can the SB's range strength become stronger or weaker if ranges are set to top 35%?