Absolutely.

Ranges will be different of course but the general ideas are very applicable.

Hey, Will,

Say you were watching these two equity distributions (ie a 3betting and a calling range on these two flops) -






I am wondering, what does your eye notice there? What should I be looking at?

How would you compare the two, from Hero's and from Villain's POV?


And which specific strategies do you think these two distn lines merit?


In short, I am interested in ideas on how to conclude something practical from EV distribution lines on the flop, right after I draw them with that piece of software you recommended. On these two examples as a start.
There is one example in your book, but I couldn't infer too much from that one that I would apply here...

Can't wait for the 2nd.

There is no practical application for the raise/shove game is there?

Also I think you mentioned somewhere in this thread spamzor getting the theory wrong in his famous Nash Thread. What part did it get wrong at the time?

Grunching, so apologies if this has been asked before. What program did you use to create th range vs range visualisations?

Great book

you can download one from the publisher's web site I believe

Yea I guess I should have thought more about his comment much before I posted.

Suffice it to say that, regardless of whether it can give us a bound in any very specific situations, making complete air indifferent between bluffing once and giving up will not give accurate approximations of equilibrium frequencies in most spots, and the technique is often misused. However, if we actualy understand why indifference relationships arise, and we're a little more careful w/ some details, we can do a lot better.

It's coming along well. Actually, I'm planning to finish the manuscript this week, and then there's editing, etc, but hopefully that won't take too long.

Yea, the book is pretty theory-heavy, and I think it's fair to say that most of it isn't specific to any particular game, although all of the examples are drawn from HU play. There's a number of free excerpts about, and reading those is probably the best way to see if the content is interesting/useful for you.

So, equity distributions are most useful a tool for river play. (With a couple exceptions) they more or less completely describe the players' ranges there, and so (combined with the SPR) they provide enough information to say a lot about what equilibrium play will look like.

This is not the case on earlier streets because of the possibility of draws. There's a huge strategic difference between a 30% equity made hand and a 30% equity draw, but they show up the same on those graphs. On early streets, EDs give one piece of the puzzle, but don't tell the whole story like OTR.

Haha, yea... hm. I guess what you're getting at is that, at equilibrium, SB is going to have a limping range at the stack sizes where we might otherwise see a raise/shove dynamic. So, raise/shove isn't a great representation of the actual equilibrium at any stacks (in contrast to shove/fold which is pretty good at v short stacks). That said, we'll see a lot of similarities between the raise/shove equilibriums and those of (an approximation to) the full game.

And certainly there's a practical application from the BB when facing a SB who on playing raise-or-fold from his button at certain stacks (idk, say 12 BB dep). Then, playing shove-or-fold in the BB vs the raise is a v reasonable approach.

Yea, that thread had good advice, but it was a bit confused about some details. For example, I believe spamz made the (then-common) mistake regarding whether all hands are played "+EV" individually at equilibrium, and this conceptual error spilled into the practical advice a bit. IIRC, this was all hashed out in the thread, and Nichlemn made a number of good corrections.

Thanks . Are you referring to equity distributions? Yea, as genher mentions the EDVis utility is available on the book's webpage. The plotting tool xmgrace was used to actually create the graphs in the book.

Thanks for the reply and for the ideas, Will.

Feel free to ship a manuscript copy of vol2 my way for proofreading and typos and stuff if you happen to find an extra one laying around!

1) title for vol.2?
2) published early 2014?
3) is the last chapter of this saga or a vol.3 is needed?

Gl for your future projects and thx for taking the time to write down these.
In my humble opinion is a revolutionary approach for hu nlhe!

1) Same title, different subtitle: "Strategies for multiple streets". Not too exciting a name maybe, but it's a pretty accurate description of the contents. Vol 1 focused on single-street situations, but things become quite a bit more complex and interesting in the multi-street case. (zomg draws!)

2) Yup, looking like it.

3) Vol 2 should be it for now. It's currently quite a bit longer than Vol 1, but hopefully we'll be able to tighten up a few sections during the editing.

Glad you've enjoyed it!

Please don't tighten up by kicking out content, raise price instead.

Just give us the Cliff's...
A 40 page book is fine, 2 or 3 chapters.
Time is money etc.

^ haha

If anything of substance doesn't make it, I'll put it up online or something.

Dos anyone know of a software like EVdis which can store ranges and boards as scenarios which can later on be loaded without reinputing them?

If you use the Edit buttons in EDVis, you can copy/paste ranges to save yourself some clicking.

Otherwise, the holdemresources.net calculation might do what you want, not sure.

Hey Will,

You mentioned how equity distributions tell us the whole story on the river. I agree, but I'm having a bit of trouble reading that story from the distn itself.

I am playing with a simple river spot; "simple" because stack sizes left are just a bit under the pot size (it's a 3bet pot).
However, I am having trouble trying to come up with optimal strategies from the distn itself. I have toyed with these for over an hour today, browsing your book and the examples, the theory, yet still I am unsure on how to proceed.


BB can bet, xC, or xF.
SB can, facing a bet, call or fold. Facing a check, he can bet or check back. So simple enough.

Let me give 5 examples, for 5 different river cards, all from the same line yet we can end up with very much different spots on the river. I think that's where the biggest payoffs are, and OTOH I noticed that a lot of my decisions are based on "he's bluffy" or "he's not that bluffy" or "he likes to call", which is inadequate.

1)


2)


3)


4)


5)



How should I go on about solving these cases?

The first thing that pops into my mind is, I should determining what lines are to be used and which ones aren't -
is BB polar enough to just bet everything and virtually never xC;
and, if he isn't, does he have air that really wants to bet, along with the appropriate amount of value hands, and the desire to xC with the rest (or xF).
Is that a good framework, and if it is, would the graphs above be good examples for those lines?


The simplest example should be the 2nd one; and the solution for that one will work as a subsolution for the others.
In that polar distn where the river blanked, we will want to see what's the cutoff hand exactly at the half of the SB's range. (making BB indifferent to bluffing)
Then from that we see how much BB can vbet; and from there we determine how much he will bluff.
If I'm following everything correctly, we do that by making SB indifferent to calling with his median hand?
Ie, we bet everything better than that one, and add half as many bluffs to our betting range (half as many due to our bet size = pot, ofc)?

General thoughts
Example 1) - BB bets 10 from his air and top 20 which is strong. He can xC the rest, cannot bet more than 30%

Example 2) - mentioned in the last paragraph

Example 3) - similar to example 1, but weaker BB. So I am curious how that difference affects our solution

Example 4) - weaker still; I am curious will BB even bet anything, or just xC, due to too weak of a valuebetting range

Example 5) - ranges are a bit merged, yet it feels largely similar to example 3, with a smaller/weaker vbetting range than 1), and Villain (SB) being able to vbet quite a bit when checked to. Would you agree?



What else does your eye see in these?
What would your additional tips be on how to study them/work towards the solution?
I literally managed to crash crEV today while running sims on these and helped Scylla spot that crash in his software.
So any help is appreciated.

It had later on dawned upon me that the hand which we "value"bet has to be ahead of half of villain's range when he calls.
So the vbetting treshold in the example 2 should be easy to find.
If villain is calling with top50% of hands, our vbetting hands should simply be ahead of top 25% of his hands. We can take a look how many hands of ours are there.
And then we fill that up with the appropriate number of bluffs. So we can make him indifferent to calling in general, not only with his median hand.

(There is a slight discrepancy there because we're OOP, meaning the equity of some of ours medium-strength hands is going to be a bit harder to realize, but we can ignore that for now)


I guess something I saw at some crev video a while ago had me confused there for a bit.

Yea, along w/ SPR and up to card removal and chops...

Yea so generally speaking, we covered two ways in the book for solving games: fictitious play and solving systems of indifference equations. Either would work, altho the first choice probably requires computer software (does CREV rly not do this yet?), and the second can be tricky to get right for complicated games with asymmetric distributions. I definitely agree that things get a lot easier if we can narrow down the lines used until we just have a one-bet half-street game.

Do you mean that you think of Hero/BB (the black curve) as being polar here? Actually I think that's not quite right. It's true that he has a lot of air, but unfortunately I doubt he'll be able to bluff with it. His range is capped pretty low except for a small number of nut hands. SB has way more near-nut holdings.

BB has a ton of holdings with between 70 and 85 perc equity, and SB's range is pretty polar with respect to that chunk of hands. So, I think this looks like a polar-versus-bluffcatchers-plus-traps kind of situation where the SB is the polar one, and is betting to make BB's bluff-catchers indifferent. BB's total air, on the other hand, probably just has to give up.


Yea, I agree that the rest of the examples look like spots where BB will have a leading range.

So, I didn't solve the 1-bet-behind game in the book, but it's pretty similar to a couple that we did look at, and can be treated in much the same way. Actually I did solve it in a video, and looked a lot at how our max exploitative play changes for different Villain tendencies and a little at how the structure of the solutions change with asymmetric distributions.

Lot to be said there, but here's one way to start seeing how things work out:

First, the structure of the solutions:
- SB will jam polar when checked to
- SB will call some fraction of his best hands when facing a bet
- BB structure turns out to not be unique, but one thing that works is something like:
|--bluff--|-----c/f------|-------c/c-------|--value-jam--| (not to scale)

Then, start by supposing we know BB's value jamming hands.
- Then we can get BB's bluffing frequency/range necessary to make SB's bluff-catchers indifferent to calling.
- Then, we can get SB's calling range necessary to make BB's strongest bluff indifferent to bluffing.
- Then, we can get SB's overall jam-when-checked-to frequency. It needs to be the same as his call-jam frequency to make BB's strong and nut hands indifferent between checking and betting.
- Then, we can split his overall jam-when-checked-to freq into bluff-jams and value-jams in order to make BB's bluff-catchers indifferent to calling.
- Finally, we choose the BB's threshold between c/f and c/c to make the SB's strongest bluff-jamming hand indifferent to bluffing.

So -- first, everywhere in that logic where I mentioned making something indifferent includes an assumption about the player's distributions, and is a spot where the structures of the solutions can change if distributions are such that the indifference breaks down.

Second, as an exercise, try working out everything starting with just knowing SB's value-betting range when checked to. IMO, it's a lot easier to actually have good intuition about this range, since SB's other option is to showdown, but it's a bit harder to work with. The hint is to remember that the bottom of SB's jamming range when checked to should be right in the middle of BB's c/c range.

Well, this is true when our other option is to go straight to showdown (and when there's no danger of getting check-raised). So, it's not quite right for the BB in these spots.

I am not sure what you mean by fictitious play so I don't know whether crEV does it or not.

CREV can give you a maximally exploitative strat given Villain's strat on a known board; and it can give you the info on which hand has what EV when played a certain way.
It is also pretty awesome in that it counts for blockers / card removal and knows how to get the best possible bluffing ranges.


I never would've imagined that, two things lead me to believe otherwise.

1) BB's hands with 70% equity are ahead of half of SB's calling range (ie ahead of top25% of SB's entire range), making them viable to valuebet despite not being the nuts, as this is the last street

2) The main issue with the un-polarized player betting, in theory, is that the (polarized) opponent could simply "call with what's better and fold with what's air" when facing a bet.

However, if in this case SB "folds what's air" (ie what's worse than BB's 70% equity hand) that is just fine for the BB, as SB is then folding what are decent bluffcatchers, hands with good equity.
And the BB has actual air in his range, which can then bluff thanks to that.


I ran this spot in crev, and even when maximally exploited (!), BB's solid betting strategy on the river - a bit over 2/3rds for value, 1/3rd air - nets him just 0,7 bbs below what his equity in the spot would entail him to. (He wins 28,9 bbs, pot is 74bbs at the start, and he's got 40% overall equity).

I think (but I'm not sure as I haven't been playing with that much so far) that's a very good result for the OOP player.

Could it be that we're simply "close enough", despite not being optimal?
I do notice that BB's checking range is very imbalanced in this line, he folds a ton when he checks (80%), but it isn't all that exploitable by the BB in this case...



OK, now we got the outline of "What"s covered... I have to ask a few "How"s. I'd expect they'll be quick(er).


How do we determine which ones are those?
I figured being ahead of over half of SB's calling range should do the trick..

Do you do that by checking for an individual cutoff holding, where the cutoff holding depends on the pot odds he is offered?

And checking whether that particular hand is indifferent?

Straightforward enough (I hope)

I have to digest this a bit more, the bit on calling frequency being the same as call-jam frequency eludes me a bit.. Freqs crEV spits out for me differ quite a bit, although it might be due to it taking a decent strat swing since SB's play isn't optimal, just solid.

I would assume the SB vbetting range when facing a check is slightly narrower than his calling range when facing a bet; his not-so-great hands will prefer the showdown; and he gets to the same frequencies by adding bluffs in his riverbetting range?

I like to digest before asking further questions, but let me know if this is the rough outline..


Edit - yes, on rereading I see that in the next point you state it works precisely like that

The bottom of SB's valuejamming range, I presume?


And let me just thank you for your help.. this is the kind of work your books lead to so now you can see what it looks like when they're put to use.

FP is defined on pg 56, but it's basically just alternately calculating maximally exploitative strategies, plus mixing.

Not sure how you figure this, but I don't think it's right. If we have 70% equity on the river, it means something like we're ahead of 70% of Villain's hands. But once Villain folds the bottom half of his range, we're only ahead of 20% of the remaining 50%. In other words, we have about 40% equity versus his calling range, which I doubt is enough for a vbet...

I'll take a closer look, tho, if you post the exact ranges.

Well, as I mentioned, I don't think that's quite right (see the discussion of river thin value betting criteria, pg 281), but even if it were, it just leads to the question -- how do you know SB's calling range? Anyway, that line of reasoning was just meant to demonstrate how all the parts of the players' strategies fit together here. If you want to actually solve it, write down the 6 indifference equations and solve them simultaneously.

Well, neglecting blockers and assuming the distributions are conducive to BB even having a betting range, this indifference should hold for the majority of SB's hands. But we can skip the step of working with any particular SB holding by just choosing the ratio of BB vbets to BB bluffs equal to pot odds the SB will be getting.

The BB's nuts will, of course, just take which-ever line gets the SB to put in a bet most often. And not only his nuts, but also anything strong enough that it "wants" to put a bet in and is guaranteed to get the chance to do so whenever Villain has a better hand. Such a hand will always take whichever line gets SB to put a bet in most often. And, for symmetric distns at least, this turns out to be a lot of hands.

So... if SB called even slightly more when facing a bet than he bets when checked to, BB would always bet w/ all these hands. And so BB's checking range would be capped, often pretty low. So then the SB would be able to bet a lot more, valuebet thinner etc. I.e. he'd bet more often when checked to. So basically, if SB calls even slightly less than he bets, BB's counter strategy incentivizes him to start betting more often. And vice versa.

Ofc this is all assuming certain things are true about the distributions... BB actually has strong hands for the SB to worry about, etc.

yes

npnp, I learn a lot from working these things out. I guess I have a tendency to pull out the big guns when I want to solve these sorts of situations myself .