worldzmine is absolutely correct. and doodoo i explained like 30 posts ago that in our pm i was trying to simplify it for you and you were basically close enough to on point that i wasn't gonna open the can of worms again. and later went on to state that perhaps that is responsible for some of the confusion, and that i regretted doing so.

i was honestly just trying to help you, and you msging the pio guy something that, not only did i not actually say, but, something i had told you on 3 different occasions was being taken out of context/misworded, was incredibly academically dishonest on your part.

on literally 3-4 different occasions i typed up very concise paragraphs of what GTO is, what pio does, and what the ev is a measure of, and, you instead chose to send him something that you wrote. something that i half heatedly agreed in a pm to keep from confusing you more, (which i later clarified in depth)? yeah, ok pall, well done.

it also seems coincidental that the two people arguing so adamantly against what im saying are seemingly the only two people who never used a solver. dunning kroeger effect much?

Academically dishonest?

I would say pretending to know something and then back pedaling when it is proven that in fact you didn't know the concept you were talking about is much more academically dishonest, but that's not really the point.

Oh Worldzmine agrees with you? You mean the guy that thinks 3betting 98s OTB to a CO open is better than 54s? That guy?

Congratulations. I'll use my own discretion when deciding who I am going to take advice from on this forum, because there is a lot smoke being blown around from people that have no clue what they are talking about.

And you are one of them.

Btw - I use solvers all the time. And monker ranges. And every other piece of software to help me get better at this game. So I have no clue what you are talking about with regards to that.

Run along to class now, we don't want you to be late. Also - when the adults are talking, it would be in your best interest to listen more and talk less.

TIL. So the mixed frequencies come from the fact simply that vs a villain who bluffs at, and calls at gto frequencies (ie. to remain unexploitable range vs range on a specific board) the ev of checking a hand instead of betting it is the same EV

Lost a few brain cells reading this thread

would love to see any example of "solver taking a non-max ev line for a particular hand to maximize ev for the whole range"

Can you expand on that? Are we playing against the clueless or the non clueless?

This is the problem:

Everyone is confusing the terms "line" with "action", which isn't your fault because even though these applications of game theory to poker have been theoretically tractable for 60+ years it seems--rather shockingly to me--that no efforts whatsoever have been made by anyone (not ITT because I wouldn't expect here to have done) to actually formalize these PURELY MATHEMATICAL definitions (not in Miriam Webster way) I mean in a FORMAL MATHEMATICAL way

Almost everyone ITT has attempted to speak a language they have never heard before and also, they don't even know what the language is supposed to be.

In fact I'm pretty sure most people ITT aren't even aware that, indeed, they're actually supposed to be attempting to engage in a PURELY MATHEMATICAL (not ARITHMETIC, which is they level 0 understanding of "poker math" most people have, posters ITT included) discussion, and they're supposed to be doing that because it CANNOT be any other way: there is NO OTHER FORMAL which the discussion can take. If you disagree, you are wrong.

OPs question and the "debate" that ensued is the domain of pure POKER THEORY which means poker, being unimpeachably at it's core a purely mathematical game, when we discuss pure poker theory there is literally no room for anything other than rigid mathematical formality. This is why poker will never die because as shown this is impossible for 99% of the population.

What going on in here is a mathematical felony: spewing off CONCLUSIONS before even formalizing definitions (and I do mean FORMAL, if you do not know roughly what a FORMAL math definition is then you honestly have no business posting ITT.) You should be lurking and learning but you simply can't further the discussion. It's actually rather arrogant (is it arrogant to call out arrogance??) to post here without at least trying 100% hemming within pure, formal mathematical bounds. Nobody would ever look down on someone who earnesty tries to do that--I have about 3-5% knowledge of this stuff, I'm no expert.

All these terms you're using are MATHEMATICAL OBJECTS: strategy, line, action EV, et al. I at least attempted to throw some loose starting-point definitions out there but no one engaged. Very sad, but makes me happy because, again, it reinforces just how juicy these games still are.

Here's a practical example of what happens: if that book that was posted (which I hear is highly regarded so wtf do I know) had used the term "action" to refer to "bet" and "check" instead of using "line", things would have been a lot clearer because really just replace "line" with "action" and suddenly instead of talking about "Max EV Lines(s)" (has to be plural--this is mixed strategy) we're discussing "Max EV Action(s)" which is a pretty damn good description of what's going on.

This is (I think) literally nothing more than a semantic problem.

If you post a HH for a review and say "review my line" LITERALLY EVERYONE means "review the ENTIRE HAND" i.e. "review every ACTION" I took in the hand because, again, it makes perfect mathematical sense to define "line" as a unique collection of actions that gets you from a root node (if you don't know what "root node", you may not post here) to terminal node such that we can create a set of all possible unique lines which, together, spans the entire tree, which is literally what a strategy is (an optimal strategy is a SPECIAL CASE of a strategy but really in a general sense a strategy is just a map to get us to the finish line for the set of all possible finish lines (i.e. terminal nods) and whether the strategy is optimal is irrelevant for the definition.

Which is why the concept of a "Max EV line" (singluar) is particularly ******ed because:

1. It's ******ed on its surface. You can only ascertain an optimal "line" looking backwards through the tree because that's how trees are solved--in other words, you need to build the line (via actions, because that what lines are--a unique map of actions through the nodes of the tree) first before you know it's optimal, which means you need to assign frequencies to each action for each combo first (i.e. solve the tree), and then you need to split those combos into the ranges assigned to each Max EV ACTION (i.e. has positive frequency) and then having taken that MAX EV ACTION WITH EACH INDIVIDUAL HAND (like punter said) you can move onto the next action until, finally a terminal node is reached and the game ends and only then do you actually have a line. Saying you always play the "Max EV line" is mathematically putting the cart before the horse.

2. This is a mixed strategy game and there is no reason to think that there even exists a singular max EV line. In fact, I don't believe the term modifier "Max EV" is good because that implies the single highest EV line within the tree-actually at equilibrium that's impossible because all lines have the same EV of 0.


I do believe it would be better to say "no matter what node of the tree we're in, we always take the highest EV ACTION(s) with all combos in our range" and in doing so you would ensure you are taking an OPTIMAL LINE.

There's absolutely no guarantee betting 77 on that texture is highest point-EV because 77 have the nut-low blocker effects and also need absolutely 0 protection and V folding is a complete disaster such that we have an extremely high incentive to x 77 here just based on pure EV considerations, no balance necessary. Checking 77 absolutely, positively may be the highest EV play, even at equilibrium--I've seen spots where the nuts are checks back IP in pure or almost pure fashion.

77 in particular on this texture benefit so so so much from just making sure V isn't allowed to leave the hand. There is absolutely 0 a priori reason to think betting 77 other than some hand wavey "but I have a set"-type explanation. Same for that FD example in the book posted.

In fact, I'm not sure it even makes sense to even try to think of changes to EV in that manner AT EQUILIBRIUM (not talking about exploitation) because as long as you're not in a terminal node, you have subsequent actions to be taken with your current line meaning your decision at every point in time is based on your knowledge about what's going (i.e. how many chips won/lost) in those terminal nodes subsequent to your current spot in the tree because that's literally where money is made--that's where the chips are shipped. I think by definition in fact you can't calculate EV anywhere except the terminal nodes.

Assuming this pretty much ignores how EV is actually calculated, because it ignores that there are subsequent nodes in which the decision to x ends up better than the decision to bet such that in this hypothetical spot where you ran a tree and saw 77 mix flop b or x the EVs came out to being the same. Slowplaying is absolutely a GTO concept.

If you'd like to go ahead and prove that betting (btw it's silly to say "betting has to be higher EV" without specifying sizings) 77 there is unilaterally higher EV without having to consider unexploitability, please do so.

In fact, I don't believe it makes sense to think of changes in EV in this manner by definition of EV (which, again, hasn't been presented in a formal manner), but that definition has to include that it's average chips won/lose IN THE TERMINAL NODES.

So saying "betting has to be higher EV but we're sacrificing that EV for the good of our range" isn't trivial--what about the terminal nodes if you were to always bet? If you always bet you give V the optimal conditionals to perfectly shed his air then you've also ****ed up his range to where his range has, from our perspective, suboptimally few 0% equity hands that may improve to a worse hand than yours and pay off turn and river bets but, these same hands are otherwise folding flop--in other words, betting 77 purely, even without exploitative considerations is not necessarily higher EV because about what about those terminal nodes?

Also:

Fixed. Did not mean to say all lines have EV=0.

^ tldr : he makes some very good points

BTW, there are two propositions (mathematical propositions, formal ones) being bandied about in here:

1. An equilibrium strategy may (the proposition should be "May", not "does", not "will", but "may", because "does" and "will" implies all the time and may means "it's possible" and no formal proof has been given for that in a general case) sacrifice a particular COMBO's EV now to enhance EV later (assuming it even makes sense to think of EV "in a vacuum" in this manner--which I don't think it does in theory because this ignores the entire rest of the tree--ok so 77 on [742] calls a cbet now what? We maxed our EV for that COMBO, are you sure? We x 77, that means in a vacuum we valuecut ourselves, are you sure?)

Some more terminology

A HAND btw is a down, it's an inning, it's a pitch--it's an overall iteration of the game, a HAND is what happens before the button moves and another HAND happens, but COMBOS are dealt and played within a hand


2. An equilibrium strategy takes the highest EV action(s) FOR ALL COMBOS AT ALL TIMES i.e. in every node of the tree (meaning it takes into account V's responses bc V gets to drive too during the hand, proposition 1 does NOT take V's actions into account) the only action(s) in the strategy profile with positive frequency are the max EV action(s) at that point in time i.e no concept of "sacrificing"


If you wanted to disprove 2, you'd prob wanna do it by contradiction (and it would only take one contradiction, so all you'd have to do is post an example of a tree run down to a truly, truly minute degree of exploitability)

oh just realized "EV of a Line" is a nonsensical concept, a line ends in a single terminal node--the word you want is "Max chips won/lost line", not "Max EV line"

The EVs are explicitly associated with actions because its the EVs that drive freq assignment to combos which then drives the aasignment of those combos to each optimal action's associated range. So its always the EV of an action for a combo, not the EV of a line (which by definition doesnt exist).

Awesome post Eggs - I am going to have to re-read it a few times to digest (see what I did there?) everything you wrote.

Do you have a formal mathematical background? By the way you write and formulate your thoughts it sounds like you have studied at a university level.

I'm broke and ******ed

What causes mixed strategies on the river when we call some % < 100%? Calling is +ev and folding is 0 ev, yet the solver will still take the lower ev line some % of the time?

The solution is not at true equilibrium then (Nash Distance 0). It is probably at .5% or possibly .1% depending on what setting you have your solver at.

It is impossible to have different EV's for mixed frequencies when the solve has fully converged.

me n eggs had a little man on boy interactive chat n he showed me some things, most of which im not supposed to talk about kuz he said wed both get in trouble, but, heres the gist of it:

me to eggs:
what op said in first post that i was arguing was that, "so we know solver will always take most ev line at all times"

even if he had said action, i still think that wording is incorrect due to the fact that we would always be able to "extract beyond equilibrium" thus it wouldnt be the most +ev action chosen., to me, it was ambiguous enough to have caused confusion.


eggs reply:
" so we know the solver will always choose the most +ev action within the strategy profile" would be a flawless statement?
Technically no. It's a mixed strategy. Try:

***so we know the solver will always choose from among the set the most +ev action(s) within the strategy profile"***

the last sentence is pretty much the most concise way of wording what were all trying to say
youre all saying it chooses the most ev action, which it does within the strategy profile. however this is not the most absolute ev action, because as eggs said, we can extract beyond equilibrium with additional info.

Posters are also confusing the terms/concepts of "balance" and "(un)exploitability"

Posters are using these terms interchangeably, that's not quite right.

Unexploitability is a result of balance--its the symptom, not the cause.

Furthermore, understand it's RANGES ASSOCIATED WITH AN OPTIMAL ACTION that are balanced, not the overall strategy. An overall strategy is unexploitable when balanced rages have been deployed in an OPTIMAL (only max EV ACTION(s)) manner over the course of a LINE.

Furthermore, now define BALANCE (in a formal, mathematical way).

The definition is gonna have something to do with have a certain ratio of strong hands to weak ones.

Now define STRONG and WEAK, and while you're at it, define VALUE and BLUFF--again, in a formal mathematical way.

See how deep this goes?

As I've studied GTO, I recognized early on the need for a glossary. Truly amazing that poker authors have not done the same.

This thread is Exhibit A for why terminology is so important. It's not pedantry.

Really egregious.

And you might be the one poster who posted ITT who could've actually helped dispel some this confusion, but you'd rather not get into a "dick waving contest with GTO nerds". Talk about arrogance.

BTW I have no attachment to ^^^any of the very rough stuff I wrote ITT, if you wanna rip it apart go ahead, I'd consider it motivation for someone like you, who's clearly extremely mathematically inclined, to use those powers for good

You better believe the best drivers know the mechanical details of their car more intimately than they know their wife.

To be fair, I also said that what you are getting at (real word exploitative EV) has literally nothing to do with OPs question--it's totally irrelevant to a pure theory question like OPs. And what I didn't say, but heavily implied, that you contributed to seriously derailing the thread because of it.

Then I also told you to get your diaper changed.

but tell them how you also said i was RIGHT, tell them that part eggs, TELL EM

Right about exploitative play being better against exploitative people?