My secret is a carefully-managed caffeine dependence.

Not sure, but my best guess is: not easily.

The theory pretty much all applies very straightforwardly to PLO, but I imagine a lot of practical difficulties arise from the number of hole card possiblities in PLO. How do you even visualize or write down a PLO range? Also, a lot of operations involve doing something (e.g. finding the most profitable way to play) for every one of our possible holdings. And that takes a lot more time if there are a lot more hole card possibilities. Some sort of hole card bucketing is probably the right way to deal with these issues, but that's not something I'll be talking about.

Ah, I had the first option in mind, but yea you could do either/both.

The idea here is that we want to change the SB's strategy (i.e. make him play suboptimally) and try to see how the BB's strategy changes to exploit it. For example, in the equilibrium strategies on pg 268, the way SB splits his river range when he's checked to on the river looks something like:

(worst hands) |--b/f--|-----------check----------|--b/f--|-b/c-| (best hands)

We called the threshold hand between b/f and check hsbbf (hand SB bluff bet-fold). If we move that threshold hand to the left, it means SB is bluffing less when checked to on the river. How does that affect each of the BB's holdings' max expl play? Stuff that used to be indifferent between c/c and c/f is probably now all c/fs. (But not all of his previous c/c range was indiff.) What about stuff that was planning to c/r bluff? Should it now c/f? Should he blufflead more or less than before? What about the top of his range? Should he value bet more since he gets less money in the pot by checking?

Anyhow, if you consider each of SB's threshold hands and each type of BB hand, you can make a table, and fill it out with the proper max expl adjustment..

You're right, the last paragraph on p 174 doesn't seem to be backed up by the graphs it refers to.

Yup, your games look right, well done.

There's no advantage to being in position in the first game (the low nut frequency regime) because both players have the same ranges and nobody ever folds. They have the same amount of nuts, they both get paid when they have the nuts, both pay off when their opponent has the nuts, and when they do show down, it's a chopped pot.

Solutions to these spots are a bit weird/subtle to interpret, since many strategies are co-optimal. For example, you note the SB is doing something weird in a spot that never comes up because BB's play means we never get to that spot. Well, the thing is that SB can play any way he wants there and still be at equilibrium, so long as he doesn't play badly enough to make BB want to start showing up there with some hands.

Anyhow, I think I talk a lot more about this stuff in the vid. If you still have more questions now that you've seen it, please let me know.

Glad you're enjoying the book . Also I appreciate the corrections. We're currently preparing for a second printing of Vol1, and these (and the rest of the currently know errata) should all be fixed there.

Well, fwiw, that is 45 combos. I've bolded the term you might have miscounted. The symbol Q♠-6♠ - Q♠-10♠ is meant to represent 5 combos: Q6ss, Q7ss, Q8ss, Q9ss, QTss.

I agree that it's hard to list long exact ranges. It gets much worse when some combos are played with non-0%, non-100% frequencies. In vol2, I often turned to figures (like Fig 7.1) which should hopefully make reading, if not copy&pasting, easier.

Ah, well I discuss the short stack case in the next section. So, you're right, but it's not what I was thinking about there. I was actually thinking about the case where Villain is playing badly. As long as we can bet large enough, there's no advantage to always having the nuts over always having the nuts, at equilibrium, since we never get called in either case. But if Villain sometimes incorrectly calls, ofc it's better if we always have the nuts.


So, whenever I say effective stacks, I mean the shorter of the two stacks. Any extra money that one player has behind isn't usually strategically relevant. This was defined/explained on pg 16.

Well, as it says, B* is the GTO betsize, the special value of B that maximizes EV_Hero(bet B).

I'm referring to the river starting ranges.

I don't understand the question. Did you use the right quote here?

yes

This is the BB bet-or-check game.

If BB is going to be both betting and c/f with hands that never win when called (equivalent hands for practical purposes), SB must be calling just enough to make all those indiff to bluffing. (e.g. 1/2 the time for a PSB).

So, (if we're using PSBs), BB's betting range has to make SB's 50th percentile hand indifferent to calling (b/c then all his stronger hands will call and weaker hands will fold and overall he'll have the correct 1/2 calling freq). OK, so SB's 50th percentile hand is indifferent to calling if it has 33.3% equity vs BBs betting range.

So, I guess I'd say that what matters is that the equity of SB's cutoff hand vs our betting range is 33.3%. And that's equivalent to saying that the average equity of our entire betting range vs that one hand is 66.6%


The big idea in that quote/section is that the BB's potential bluffing hands all have the same EV of checking, even if some are slightly weaker or stronger in an absolute sense, since none of them ever get to showdown and win after checking. That is not so for the SB's potential bluffs, except for his v weak hands which are all equivalent since they all lose to all of BBs checking range, since BB would have led the river if he held one of his weakest river starting hands.

No, the sentence is correct. Rephrased -- if Villain never bluff-raises, then value betting is better than checking if Villain calls with a worse hand more often than he holds a better one (regardless of whether he calls or raises with those better ones).


Yes.

If even our bluffs are ahead of some component of Villain's range, he can't be thinking about calling with those hands, even if he thinks we're bluffing.

Well, we've certainly seen that if we get to the river and pretty much all of Villain's range is made up of hands with the same equity, then our distn is going to be relatively polar, and we know more or less what the equilibrium is going to look like there. And I think that is often at least approximately the case in river play. But I dunno that there's a general rule that applies when the situation is more complicated.

Ton of new stuff to learn, but I'm enjoying it so far, thanks .

Especially if you write good books...

I completely agree, I think this is one of the most important idea really.I often feel like I don't know what I'm doing and the main reason for this is that I don't know my own range.


I'm trying to understand how we get to this GTO betsize B*, mathematically that is.

Given what you say in the book, it looks like:
1. you created a function, say f(B) = EVHero (Bet B)
2. Calculated its derivative, f'(B)
3. Set f'(B)=0 and solved for B


I guess my question is, if the above is true, what's the outpout of f(B) here, do we have f(B) = FJ (S − B) + (FBC)(S + P + B) + (1 − FJ − FBC )(S + P)?

If so, how do you get its derivative, f'(B)? I have only worked with very simple functions, I'm not used to seeing so many variables.

I used the right quote, but maybe my question didn't make sense.
You say that even if a turn card improves a small fraction of our range significantly, these hands have reason to keep playing the same way as the weaker ones and I want to make sure I understand why.

What are these reasons?


Cocktail party level poker theory... Nice .

I agree that this is not the right approach to poker theory but look at it from an author's perspective. Well, OK, say an author who doesn't care too much: No questions from the readers! Sure, these cocktail party poker theorists may quickly go from no questions from the readers to no readers...

Point is, I really appreciate you taking the time to answer all my questions.

Thanks!

quick question:

on p48 9s6s is in villains range (at least i assume that is what 9s4s+ means), on p50 it's not anymore. why not?

It should be in the range. This has been mentioned in the errata available on this books webpage on the editor's site.

so it's not in the iteration/ev numbers in the book are slightly off?

Yes, this is right.

Yes, that's the EV of betting that we found a couple paragraphs earlier. So, I think derivation is beyond the scope of the thread, but taking derivatives of long expressions w/ lots of variables isn't to bad if you remember a couple rules:
- derivative of a sum of things is the sum of the derivatives of the individual things
- if X is something that doesn't depend on B, then the derivative (with respect to B) of X*B is just X
- derivative (with respect to B) of something that doesn't depend on B is just 0

Other than this, the thing you'll need to look into is known as the Quotient Rule, because FJ and FBC are fractions which involve B.

Well, this is one of the lessons we learned from the PvBC-plus-traps game. Suppose we're on the river, BB has mostly bluffcatchers but also a few slowplayed nuts, and SB has some value hands and some air. Then, BB's few nut trapping hands do make the most money by continuing to slowplay, checking the river.

Why? Well, since the traps are the nuts, they simply make the most money when they play in whatever way gets SB to put as much money in the pot as possible on average. And it turns out that SB puts a lot more money in the pot vs a check.

This is b/c most of BB's checking range is bluffcatchers, and we know SB will bet big with all his value and some air as bluffs when checked to w/ a range of mostly bluffcatchers. On the other hand, if BB leads himself with traps (which are then value hands) and bluffcatchers (which he turned into bluffs) then all of SB's air has to fold, and SB's value becomes effectively bluffcatchers and might sometimes fold.

So checking gets money in vs all of SB's value and some air, while betting gets money in vs no air and maybe not even all his value. So, BB gets a lot more money in w/ a check on avg.

no problem

Yup, the missing 96ss combo is already noted in the errata, but I don't believe there are any issues with the EV or iteration numbers. If you're getting conflicting results or something, be more specific and we can try to figure out what's going on.

Generally, I'd recommend to anyone starting to read the book to get the errata from the website and mark the edits in your copy.

Awesomeness... Thank you very much, Will!

Hello again Mr. Tipton. Thank you for your prompt reply and apologies for my tardy one. Its water under the bridge now but in answer to your question as to how I arrived at the frequencies I did (re: fig. 3.2/3), I am probably embarrassed to admit that I simply averaged out the stated values. Whether this places me into the league of the clueless imbecile I have absolutely no idea. In all honesty I started reading your book as a losing poker player. In the meantime this has changed and I now maintain a respectable ROI (at least so far so good for a while). In truth, I realise now that the level you write at and the audience you write for is not my level. However, through poker I have discovered a hitherto unknown maths/stat geek, and I truly enjoy reading and learning all this stuff, and so far the maths involved in your book has not yet defeated me ( I'm about half way through). And as poker is just an obsessional hobby and not a profession, who cares how long it takes me to get through it? I am not embarrassed to say that I literally might take hours understanding a paragraph. I actively enjoy it. But while we are here I may as well lay all my cards on the table. I wonder where we are going with all this maximally (minimally) exploitative business. I also wonder whether the days I am putting into understanding all this stuff will translate into any discernable practical benefit to me. Am I correct that the thrust of the idea is that I should play non-exploitatively vs villain until I establish he has exploitable tendencies which I might then exploit? Surely once I move to exploit said tendency I am no longer playing GTO. Ah, I'm light years behind here right? I could care less, truly I am really enjoying the challenge (to me at least) of your fine book.

Regarding Pokersnowie Im not so sure it isnt legit. As I understand it the authors simply told the bot the rules of the game. And then let it play against itself for years, allowing it to establish the most effective play vs itself in any number of possible situations. Being very far from a GTO expert myself as we have established, in my limited understanding would this system not produce maximally exploitative play? Anyway I bought the freakin' thing tonight based on a very simple criterion: it beats me far more often than I beat it. It is as simple as that.

Much respect, Mark

Hi Will!
I'm reading the book for the second time and realising that it should be treated like a Bilble and be read every weekend forever
I think my doubts probably will disappear very soon with Vol. 2, but just to be a Vol.1 expert:
-On page 88, at the end of "3.1 Shove/fold chapter" are explained the applications of the solutions in real play but you mention that at around 7 BB at equlibrium the SB can not raise-fold.
Do you mean it also in real play or only in preflop-only-games? I find the evidence in the next chapter for preflop-only-games, but don't know if at equilibrium in real play the BB shoving range (having the option to call) would give the odds to the SB to call with his entire raising range at arround 7 BB.
-On page 102 "3.3 Bet sizing considerations": In the last paragraph it is explained that in preflop-only-games the player putting in the last best was in disadvantage if he had a bad risk/reward ratio on his all-in bets, so it can sometimes to be worthwhile to make a bigger bet in a spot if doing so will cause a Villain to play shove-or-fold.
In real play it is only possible if Villain is playing badly, i.e he is shoving at deeper stacks than he should be? Or could even exist this possiblilty in a GTO strategy?
Thanks very much for your atention Will!

Hi i was doing 1 exercise and i found this place what i cant understand.

On page 115 book says that when we know that BB 5 bet shoving range is 20.6 % SB calling range will be hands with 38% equity and they are {A2o+,.....} (25.5% of hands)

(stacks are 50 bb and 4 bet size is 12 bb ) so yea i understand that we should have 38% equity not less coz we loose just 12 BB if we fold (50-12 = 38) and if call we win EQ*100 so equty should be greater than 38.

But when i check equity of A2o versus his 5 beting range its ~35%

As i understand 38% + is needed for each individual hand not all range together right? if this is true than A2o shouldnt be in his calling range right? When few hands come off this range it would mean its actually not 25.5%, but number 25.5 was used to calculate his total 4 beting range and also open raising range. Not a huge deal but than it means they would be a little different than in the book.

Hi Minotaurs!
I was on the same page and think I can help with that.
Maybe the mistake is in your ranking of hands. For simplicity, the book points out that the hands are ranking by their preflop all-in equity versus any-two-cards in these situations. So 20.6% would be something like:{55+, A3s+, K8s+, QTs+, A7o+, K9o+, QJo}. Versus that range A2o have near 40% equity.

Hi man tnx for answer. I did choose the top 20.5% of hands (used slider so i think software (PT4) takes top equity hands against any2 hand range) and as i told A2o had nearly 35 %

I sometimes use the Hand Range Selector from PT4, but it only allows two models to ranking hands: 'Sklansky-Karlson' and 'Hand vs 3 Randoms'. My easiest way to visualize a range vs ATC is in www.icmizer.com and selecting: Ranking Vs Random Hand.

Yea man ill check it out, i probably used Sklansky-Karlson so maybe there is this difference. Thanks man

I just realize that on p.116 the BB Shoving Range is written!!

Hey i have a noob question. If i play unexploitable i end up with 0 profit against another opponent who plays just like me right. But if i play against some crazy guy i would still profit by not changing my game at all (playing unexloitably) right? I would profit, just not as much as if i would be by maximaly exploting him right?

Hi,

just wanted to do the exercise on p. 125 and wanted to know how to calculate the top x% of Hero's range vs Villain's range on the 8h6c5d flop. Isn't it just all hands that have at least 33,3% equity? Can somebody pls recommend a tool which can do these kinds of calculations?
Thanks