So, keep in mind that the structures of equilibrium strategies aren't necessarily unique. That is, you can potentially rearrange some of the action regions and still have an equilibrium. In these cases, you could also construct some mixed strategy equilibria, and it's not easy to draw those on a line containing distinct action regions. But yea, guess-and-check is a time honored approach.

We did actually set up and solve a variant of the SB bet-or-check game with multiple bet sizes on pgs 256-262, and there's some discussion of these issues there.

Well, this is the SB bet-or-check (half street) river game, and the inequality itself (h_c > h_b) is just saying that BB's weakest calling hand is stronger than SB's strongest bluff.

What would the equilibrium look like if this were not the case?

Well in the later examples the proposed solutions involve things like check raising with some % of our nut hands while also using some % to blockbet to prevent villian from bluffing us. I'm just trying to figure out how the EV equalities are set up for solutions like this. If we're not splitting our nut hand range by some hand value where both actions have the same EV then how do you solve for these types of strategies?

I tried to solve some on my own by making the assumption that our entire nut hand range had the same equity vs villian to simplify the problem. Then the breakpoint within that hand range would represent the % of one strategy over the other. The problem I ran into was that the EV of the actions were dependent on the bet sizing which in turn needed some % of bluffing hands but there are only some fixed % of bottom end range hands which makes me think the blocking bet size actually has to be a variable we're solving for and then my head exploded and I gave up

Can someone explain the Figure 3.7 on page 83? SB's shoving hands have the same EV at the beginning of the hand as they do after he shoves. Similarly, the EV's for the BB with hands with which he is going to call a shove are the same when he is facing the shove and after he calls it. Why is this?

SB's shoving hands have the same EV at the beginning of the hand as they do after he shoves only SB he is planning to Shove that hand, if he is going to fold the hand: the EV would be his initial stack minus 0.5BB. It is because no matter if we are after or before the decision point of open shove or open fold, Hero knows what he is going to do as soon as he sees his card. The EV in these hands only depends on BB's decision of call or fold and that is why the EV changes after BB calls or fold (supposing we can't see the BB's card, only his calling range).
EV's for the BB with hands with which he is going to call a shove are the same when he is facing the Shove and after he calls it because once SB has made his decision of Shove or Open-fold, the BB already knows if he is going to call, and supposing we can only see the SB's shoving range when BB makes the call (not the specific card): the EV doesn't change. Try to draw the tree with your approximate EVs and you will understand for sure.

On page 124 what is BB's flop betting range? I see the hands he sees a flop with and his bet-calling range but I can't find his betting range.

yaqh, I stopped the book on page 263, because i don't know how to use softwares to solve the game. Reading those results solved by you is meaningless, therefore i would like to buy your video pack and solve the game myself. However, i don't have paypal account, so may i purchase by transfering money to your pokerstars/fulltilt account? Also a small discount would be helpful because i am a 5NL player with bankroll just over $70

I've read ToP and TaP books. Is this a good enough prerequisite to tackle your 2 volumes?

I'm half way through volume 1 and I just bought volume 2. I was thinking about buying either volume 1 or volume 2 of the video packs. Which video pack should I buy if I can only buy one? Should I read the books before watching them or watch the vids after reading certain sections or topics?

States HUNL in the book title. If that means that HUNL does not have a blind increase and SNG does it has a big difference

Will,

Reading the book, the discussions about ranges and equity distributions are only possible if one knows on the fly the equity of his hand/range vs an opponents range. How would one know this at the table? Is their a quick tip to calculating it?

For someone that doesn't play higher than small stakes,mainly 6 players +,will these books and video series improve my play or is it more a mid or higher stakes players book?
Thanks

If you spend the time to really understand them they will help tons.

Will,

On page 174 you said, "We can see from the graphs that the BB's average hand strength increased slightly with the advent of the flop since the dotted line is on average higher than the solid one, and the opposite is true for the SB."
Is this a typo? It seems as tho BB's average hand strength decreased after the flop and SB's average hand strength increased.

This book is fantastic in every way. I can't recommend it enough.

About the "test yourself" segments, though. Is there any way to find out the correct answers? Some of them are discussed directly after, but most are not?

Or is it mainly just the process of trying and using the techniques that are beneficial even if i end up with the incorrect answer?

If anyone is interested I'm putting together an Expert Heads Up No Limit Hold'em study group? PM me on skype. My screen name is Ryan Foley. I want to start a small study group so that we can discuss concepts in the book and share our results to the exercises in the book.

a lot of Ryan Foley
http://snag.gy/QyMzN.jpg

My avatar is a pic of the MMA fighter Nate Diaz flipping off that camera. I'm from CA, USA if that helps.

Is the study group up and running?room for one more?

For anyone that's interested in the HU study group, my skype id is ryfoa6.

You can try to guess the structure of a strategy, then solve for the thresholds, then compute EVs to check if you've actually found an equilibrium. And if you haven't, then guess different different structures. But it isn't necessarily possible to describe mixed strategies as cleanly-separated action regions. Other methods like Fictitious Play (mentioned in chapter 2) work in general.

I'm not exactly sure what problem you set up here, but it sounds like you're not exactly sure what question you're trying to answer. If you're curious how the equilibrium changes as you change the betsize, then leave the betsize as a variable while you solve for the equilibrium, and then once you get expressions for the cutoffs (or other parameters describing the equilibrium), see how they change depending on the betsize.

Yup, I think mmowgli pretty much nailed it here. If it still doesn't make sense, try actually calculating the EVs yourself for some examples, using the methodology described in Ch 2, and it should become clear. Also, I think this stuff is pretty fundamental -- knowing what we mean by the EV at a decision point is v important for understanding the rest of the book, so try to get it before continuing.

His bet-folding frequency is found on pg 125 -- he's bet-folding a range composed of 7.5% of all hands. The particular hands that make up that range don't matter (up to card removal), since they can never get to showdown.

Well, I'm not sure why you mention pg 263, in the middle of the river play chapter. There's lots of paper-and-pencil (ie not-computational work) after that. And I think there are also lots of lessons from examining computational solutions, even if you didn't run the program yourself. If you don't feel that you understand where exactly the computational results came from, review chapter 2 -- the method is described there.

Anyway, yea if you want to solve situations yourself, the Solving Poker video series shows how. You don't need a PayPal account to pay using PayPal -- they process regular credit card payments. And as a US player, I don't have a realmoney Stars or FTP account. That said, as a 5NL player with a 70$ bankroll, more basic (and perhaps even free) educational resources might be more appropriate.

Yea, I'd think those books would provide plenty of "poker maturity" and comfort with a little math. Check out the sample excerpts for more about prereqs and to see if you like the style.