Card removal effects. That range is 536 hands, and there are 1326 hands total, and 536/1326=0.40422.

But in this example, we hold 7h4c, so Villain can't have any hands with those cards in them, so his shoving range is actually

33,55,66,88+,A2s,A3s,A5s,A6s,A8s+,K2s,K3s,K5s,K6s, K8s+,Q5s,Q6s,Q8s+,J8s+,T8s+,A2o,A3o,A5o,A6o,A8o+,K 5o,K6o,K8o+,Q8o+,J9o+,Ac7c,Kc7c,Qc7c,Ac4d,Ac7d,Kc4 d,Kc7d,Ac4h,Kc4h,Ac4s,Ac7s,Kc4s,Kc7s,Ad7c,Kd7c,Ad4 d,Ad7d,Kd4d,Kd7d,Qd7d,Ad4h,Kd4h,Ad4s,Ad7s,Kd4s,Kd7 s,Ah7c,Kh7c,Ah4d,Ah7d,Kh4d,Kh7d,Ah4h,Kh4h,Ah4s,Ah7 s,Kh4s,Kh7s,As7c,Ks7c,As4d,As7d,Ks4d,Ks7d,As4h,Ks4 h,As4s,As7s,Ks4s,Ks7s,Qs7s,4d4s,4h4s,4h4d,7s7c,7d7 c,7d7s

which is 513 combos. And if we remove two cards, there are a total of nchoosek(50,2)=1225 total combos remaining. And 513/1225=0.41878.

The bolded number looks incorrect. If you get it all-in and win, your stack will be 13.2+9.7 BB.

Got it, thanks Will

Here's my corrected version of the problem I initially posted above. I apologize if I'm misusing this thread, I just want to make sure I can do one calculation correctly before doing several



I changed the ranges a little since I saw that it makes no sense to have QTs+ and Q8o+

Asked in Science, Math, and Philosophy as well:

Is there a way for us to rearrange the procedure above in such a way that we know what line we want to take and crunch out what's the max or min range V has to have in order for us to take that line?

For example in the post above, I find that raise/folding is the better option. How do I find the required range of V (and consequently shoving and folding frequency) in order to raise/call profitably? What about folding?

How did you get Villain's shoving and folding freqs here?

It's not quite clear what "max or min range" means. There's no unique mapping between your all-in equity and the size of Villain's jamming range, since ofc, Villain could jam lots of different ranges that are the same size that you have different equity against...

But, if, say Villain has to jam the top X% of hands where that's defined by equity vs ATC (you could do any parameterization of Villain's strategy here). You could solve this and get numbers.

So if you want to know when one line is the best, then start with inequalities
EV(that line) > EV(other line #1)
and
EV(that line) > EV(other line #2)
and solve for the variables of interest in those EVs.

I see, makes sense. In regards to the initial problem...I got shoving freq by plugging the range and my hand into equilab. So V's range with my cards T7hh accounts for 38.37% of hands. And 1 - .3837 = .6163 for the folding frequency. If I did that wrong then fml....hopefully it's just my notation of A2o+ that caused some confusion. I imply it to myself that when A2o+ is part of someone's range, I intuitively assume they take the same line with A2s+. I guess the proper notation of that would be A2+ though since that isn't necessarily true

I bought this book on my kindle when it was first released over 2 years ago. My main game is 6-max, so I never really got round to reading it much...

Now I'm onto chapter 5, really really engrossing book, although sometimes easy to get lost in the wording, especially when repicking up the book from the night before. Anyway, I just ordered the 2nd book, hats off to Will for this great book, will be looking to ask some questions further down the line!

I just started reading the book an I need to be sure I understand from the start.

This is a hand I've discussed with a friend:

Stack Hero - 139.878 (47 bb)
Villain - 74.684 (88 bb)

PREFLOP:
Hero has Ah6c

Villain (BTN) opens 2BB and Hero 3bet to 4.5 BB, Villain then 4Bet to 9 BB.

We have 3 options:
1. If we FOLD to 4Bet = 83.42BB

2. EV Stack 5Bet/FOLD = 83.42BB-18BB= 65.42BB

3. EV Stack 5Bet/CALL SHOVE:

Equity Ah6c vs {AA, QQ-99, KhKs, ATs+, KQs, ATo+, KQo}
MP3 31.77%
MP2 68.23%

EV Stack:
41.25 BB + 0.3177 x 95.64bb= 41.25 BB + 30.384 BB = 71.63 BB

Can someone tell me if this calculation is correct?

Thanks

*
Stack Hero - 139.878 (88 bb)
Villain - 74.684 (47 bb)

Let's pretend we bet t90 into a t150 pot. To be perfectly balanced in this scenario, we need to have 90/(2x90+160)=26.5% bluffing hands and 73,5% value hands. Does a flush draw count as a bluff without restrictions?
The problem I see here is that if I make a flush otr, BB is no longer indifferent between calling and folding because my range is so strong.

You'll get your answer in vol. 2.

Ah ok, yea leaving out the suited aces was the issue I guess. Calc looks right now.

Glad you're enjoying the book and finding it applicable to non-HU spots!

If you start with 88 BB, put 4.5 BB in the pot, then fold, you end up with 83.5.

I guess the small difference is because you rounded when you reported your initial stack as 88 BB? What exactly is the big blind here?

This is right if your 5bet is to 22.5 total. Not if you mean to make it 18 total.

Looks close. I don't really know what stacks are, so it's likely right.

So now we know that folding to the 4bet is best, but if you 5bet, you have to call a shove. Obv A6o is a pretty bad hand to have vs the range you give y, but you're kind of committed after you 4bet to 22.5bb with effective stacks of 47bb.

Yea, as lestro points out, this is kind of a Vol 2 question, but big picture -- I always find it helpful to go back to first principles to figure stuff like this out when I get confused.

So, what's fundamental here? That value-to-bluff ratio you calculated, 73.5-to-26.5, certainly isn't. It comes from applying the naive bluffcatching indifference (i.e. making bluffcatchers indifferent, with some assumptions).

But the bluffcatching indifference isn't fundamental either. Indifferences are a common-but-not-universal emergent property of the situation where both players are trying to play as profitably as possible vs each other. (It gets kind of repetitive in the books maybe, but I like to always go through the equilibration exercise ot convince myself that an indifference is likely to hold). Once you think an indifference holds, you need to think a/b what hands it applies to. (On the river, lots of bluffcatchers are mostly equivalent, but not so on earlier streets where hands have different outs.)

Then, you can write EV(call) = EV(fold) for those hands. Here, EV(call) will take into account the possibility of getting sucked out on, bluffed later, and having to play multiple streets in general. That is, EV(call) is usually lower than if calling just got us to showdown immediately. And of course, this depends on the exact composition of the bettor's bluffing range -- is he using draws or more airy stuff to bluff? (Btw, the 73.5-26.5 isn't right in a multi-street situation even without draws.)

So ya, for more details, vol 2.

That's interesting to know. Looking forward to v2, although I still got a bit of work to do with v1.
As always, thanks for the reply.

I tried to solve the 1st river example myself but failed at it. I could imagine two ways how to do this:
1) ipython: first step would be to draw the game tree and then assign ranges. But given a certain amount of value-betting hands ott, I find it hard to find the right fraction of bluffing hands to get the river starting ranges.
2) equity distributions: read the equity off the graph of the weakest value-betting hand hv -- but again no idea how many bluffs to add sothat we have a balanced turn barreling range if you say my 73.5-26.5 ratio isn't correct for multi-street play.

Thanks in advance

"We will sometimes find a bet to be our most profitable option when it is not clearly a value bet or a bluff but simply because it leads to a part of the decision tree where Villain employs a poor strategy or is otherwise unable to play well." (106)

I'm sure this will be touched on later as I continue in the book, but isn't this statement similar to the concept of "range merging" and discredited on 2p2? I'm not trying to argue against Will in any degree whatsoever, it's just that I used to think along the lines of the quote above upon "personal strategic analysis" since it gets different parts of V's range to incorrectly call or fold, but I remember reading a while ago that it's bad b/c it essentially means that we aren't sure what we are trying to accomplish by making such a bet and that reasoning stuck with me. Am I misunderstanding the quote or is 2p2 just wrong?

Well, you're right that the river-only examples in Vol 1 assume river starting ranges, and there's a lot of reason to want to look at models that cover more of the game. That's sort of the whole topic of Vol 2, so there's a bit much to say here, but maybe I'll just elaborate on why the 73.5-26.5 isn't right, even in the simple case that we ignore draws (i.e., even assuming that our hand values all stay the same from turn to river.)

Suppose, for the sake of contradiction, that our GTO strategy was to bet with that nuts-to-air ratio on the turn. Then on the river after we bet, we'd want to bet again with that ratio (we know this is the right answer for river play). To do that, we have to continue betting the whole range we did on the turn. So essentially, after we bet turn, we always bet river. Well then Villain's best counterstrategy will definitely not ever include calling turn/folding river since that'd just be throwing away money compared to just folding turn. So then our counterstrategy can't involve bluffing river, because Villain's never folding. This contradicts our original supposition, so it isn't right. Basically, we need to bluff more on the turn so that we can give up with some bluffs on the river and still be betting 73.5-26.5 on the end.

That quote is sort of vague, but I think "betting for protection" would be a better concrete example of what it's referring to. So for example, one common spot where protection can be appropriate is when we're on a pre-river street, and a lot of Villain's range is weak hands that just have, say, a couple outs vs us. (Maybe his range has some stronger stuff too.) The weak stuff all has to fold to a bet, but if he has to fold a lot of hands that have a little equity each, it can be a big win for us in total. That can make a bet the correct play, even if Villain just folds all worse and continues with all better (so I wouldn't call the bet either a value bet or a bluff).

I agree that range merging is almost always a bad idea. There's a section on this topic on Vol 2 actually. (Reading over the previous paragraph, protection sounds a bit like range merging, but range merging generally refers to a river bet, whereas the previous is an early-street thing because it requires draws.)

I see. As always, thanks Will

got volume 2 lying right beside me. cant wait to finally start

I recently purchased this book: heads up is not my game, holdem is not my game.

Reasons for purchase (in no particular order):

1. The author has spent much time in this thread responding to questions. He deserves a medal for all his input!

2. The book is an invaluable addition to the poker literature.

3. It is not going to get any cheaper (see point 2).

4. I fully expect (having read the Look Inside excerpt on Amazon) that it will help me in my games.

I'm building a simply model in order to gain some experience of your method to calculate EV.

I'm trying to find, how much I could invest preflop to setmine, facing an openraise.

80BBs eff stack
UTG opens with AA
Hero wants to flat with 44 to setmine

EV fold = 100
EV call = %miss * investiment pre + %hit * (avg eq * stack after we won)
so
http://www.wolframalpha.com/input/?i...80.81*181.5%29
100-X=6,5 (maximum investiment pre in order to BE)

What about if we are in the BB?
Instead of winning 181.5 BBs are we winning 180.5?
And in order to find out how much we can call what we should do?
99-X, or 100-X?

Ty in advance