Math checks out. Also makes sense intuitively if GTO is supposed to be unexploitable that it sacrifices small amounts of EV in some parts of the game tree to achieve that. If you take the example from Janda's book, what if the turn brings in the flush after BTN checks flop with a strategy of betting all flushdraws on the flop? You cant remain indifferent to big blinds turn overbet donk shove anymore because you cant profitably call it with any hand.

To me, it looks like you quoted the wrong post because they don’t seem relevant. It doesn’t even seem like you understood what i was talking about at all, not that I expect most people to. It’s not about people being dumb, well maybe . Math is a language of itself, and logic is something most humans have to train for and arent born with

One of my main points was your second sentence as well.

Your example also has leaks in it. You can profitably defend vs the BB’s turn overbet donk shove if BB is overbluffing. If he’s range donk shoving ott then any pair or A high is a snap call.... and obviously we are snap calling any traps otf

What you said in mathy language is that the infinite set of irrational numbers between range 0,1 is smaller than the infinite set of irrational numbers between the values of EV of different lines no? Thus the propability that the EV of different lines is the same is zero.

Edit: Isnt the dEV of different lines much smaller than 1 usually though? Not that it matters to the proof in any way since it would still be an uncountable infinite set

No..... believe it or not they are the same size. Trust me, it sounds really stupid but if you’re interested look up if [0,1] set has the same cardinality, ie # of elements, as the set containing every single number: (- inf, inf). Infinity is not a real number so when you play with it, weird things happen. Possibly a clearer example is that the set of all even integers has the same number of elements as the set of all integers. Counterintuitive? Very much so yes. True? Absolutely. It’s been proven

Ima be dipping so peace out everyone

To your edit, yeah that’s exactly what i was trying to get at. The differences in EV is basically impossible to be equal to zero. Maybe you could conduct a hypothetical situation under the right circumstances and twist definitions/arbitrations, and make assumptions such that in one case or a few it’s true, etc but if your EV is ever different from 0, then by OP’s and other’s logic of taking the highest EV line at 100% frequency always, then you should never mix. Which is a logical fallacy, since solvers obviously mix, and not for the reason that it yields the highest immediate EV. Even hands that have higher EV difference than 1 get mixed (hence for all epsilon > 0, epsilon is usually a small number but we can arbitrarily choose epsilon to be 1, 2, 3, 4, etc)

Interesting, I remember reading that there are different sizes of infinity and intuitively thought that for example the continuous range [0,2] would be a bigger infinity than [0,1].

Btw I obviously meant real numbers, english isn't my first language so got mixed there.

Ok guys, this has been a funny read so far (all you guys arguing and stuff ) but I'm gonna throw my perspective out there and hope, that since I am somewhat new, my views haven't been completely warped by the hurricane of varying ideas and theories out there.

The way I look at GTO is like this ....

Heads up, we have 2 ranges. All a solver does is take the bet sizes and ranges and start competing against itself over and over and over, until it finds the point where nether player can do anything to make more money.

To me, this means it is playing the highest EV strategy against a perfect opponent, who is also playing a perfect strategy.

Assuming both players play the same preflop ranges (GTO solved ranges or otherwise) then nobody every makes any money and we just lose to rake.

If we do this for every spot, against every player, in every situation (mw, 3bps, 4bps etc) then nobody is ever winning, because we are all using the same strategies with the same EV's.

Now, if we play the GTO solution in a given spot, and someone else doesn't, any mistake they make is EV in our pocket and we would be winning. By how much, who the heck knows, but we would definitely profit since we are playing a way that wins the most vs someone perfect and they are playing less than perfect.


Ok so, the other side of things for me is, when someone plays less that perfect, there will be a way to maximally exploit villain. (if they fold too much to turn barrels or something we can barrel loads etc)

Obviously in game, we aren't gonna barrel every hand and make people aware of their leaks, but better instead to just have a semi exploitative strategy and continue to exploit that leak and hope they never notice because we are not doing it 100% of the time. That is, unless we think villains counter adjustment to our exploit will be even worse than their original leak, but that's gonna be hard to quantify and comes down to the art side of poker I think.


Anyway, that's my 2 cents on this

Of course I have a bunch of holes in poker theory. I play 10NL - but I am willing to learn to get better.

Also your first statement implies the second statement so it is a nonsensical sentence.

If I have a bunch of holes in poker theory--->nosebleed player is implausible.

On another note I asked Scylla (he made GTO+) this question in binary form and he said neither is true.

It's great that there's still a lot of confusion about this subject despite the fact that GTO talks have been around for 5+ years by now (e.g: poker is still alive).

Yikes.... I thought my definition was pretty clear.

So I have to ask, why is everyone sold on GTO and solver play, if most people don't even understand what it is?

This goes beyond poker. The most important thing you can do with any function in life, is have the understanding first. If you don't have the understanding, you're just imitating some function.

Don't do something just because someone else or some pro is doing it. Build your understanding first.

One thing you'll never see Piosolver do is eat a sandwich all loosey goosey.

look, the reason that A is "untrue" is because the wording
noone ever claimed that the solver takes the most profitable line with its range[getting to our pm in a sec], ( in my first post i specifically mentioned that it infact does not do this)
i realize in a PM you asked me " sometimes we lose EV with individual hands to make up for the EV of our range. So overall our range is taking the highest EV line but not every hand in our range. Is that right?" to which i said yes.

after all of the confusion, me saying well no not quite because... "its not the highest EV range line., its the highest NET EV comprehensive unexploitable strat" seemed like it would only be splitting hairs at that point, esp considering by then you seemed basically on track. so yes, in that sense maybe i am to blame.

however, if you again read my original post, i had already cleared this issue up.
the initial contentions are, the mistaken notion that: the solver always takes the most +EV line, (which minotar and myself conclusively disproved by demonstrating that it employs mixed strats)

and what i and several others have been saying all along which is that: the solver gives you a completely unexploitable strategy, which does sacrifice immediate EV for the sake of overall balance and range protection, but in the long run vs all opponents this will be the most +ev strat since it is the only strat that cannot be exploited ie the only strat that will never yeild -ev vs any opponent.

again, the alternative to what i am saying is that "the solver always takes the most +ev line" which, i hope is very clear by now to be untrue for many obvious reasons, and logically impossible based purely on the grounds of mixing.

Haha

I think this is just a semantics problem. Deviating from the correct frequencies means you become exploitable therefore it will lower your ev against an optimal opponent. If it is impossible to increase your ev once you have the correct frequencies, then it means the ev is maximum. If mixed strategies are used then it means the ev of each line is strictly the same, an optimal strategy never sacrifices ev for the sake of balance, this is a very common misconception and a very tenacious one as we can see here.

Read the text in the picture again : "then by definition the ev of both lines against an optimal opponent still has to be equal (otherwise we'd simply take the higher ev line)"

In other words, lower ev lines are never taken.

That's precisely why most players are all-in on GTO. Critical thinking is hard and most people would rather be spoonfed the golden answer to mimic.

You don't need to know how an engine works to drive a car.

And unfortunately you can't correctly apply gto concepts in real time without first understanding them. Critical thinking is needed still.

Man this guy writes like I do.

It would be nice to know what drives/incentivises the mix.

Anyway the mix is your opportunity as a poker player, this is where you get to get creative if you're about maximising (vs particular opponents), or blindly mimic frequencies you don't even understand otherwise.

I remember a thread where I was talking about the reasons I would or wouldn't x/r or probe the nut flush on the river and was told by OP I had to mix as this was his solver's output, so it's a wonder he now admits he doesn't understand why.

You not read the book Eggs? Hit me up for screens

I know you're just making a sweeping statement that's generally true, but to give a contradictory example we see plenty of cbet-with-range spots at equilibrium where we cbet 100% of the time.

I've literally never read a single poker book, or article, or watched a training video, and prior to joining the forum I had very little poker related discussion with anyone beside me, myself, and I.

BTW what are w eactually defining "line" as here--is it the action on a particualr street, or a particular path through entire the tree, or something else?

Definitely, definitely not true.

Also definitely, definitely not true, see Post #95

Do you know if solvers use the initial condition of assuming mixing occurs based on uniform distrbution (i.e. initial condition of equal freq for each action in the strategy profile) or did you just mention that interval as an example (I'm almost positive they use the discrete 1/n for an n-option profile btw)?

2nd bolded part, feel free to excoriate me if I'm wrong: isn't this a misrepresentation of what the ultimate computed strategy looks like (attempted explanation to follow):

My understanding is: the solver builds the tree, and as it solves it sees how each iteration in the sim performs, and makes adjustments based on the performance (i.e. the EV), and ditto for the 2nd player in the sim, and after many billions maybe even trillions of trials and adjustment the algorithm does its best to drive towards an "exploitable for" value of 0 which at equilibrium implies the highest EV strategy at equilibrium (not line, whatever line means here).

The EVs therefore are the average across all nodes in the tree. But by the time you get to that step of computing the overall EV for each action for each combo, the frequencies haven't been assigned at random anymore. They've been assigned such that each non-dominated strategy is unexploitable i.e. they've been assigned such that the overall strategy represents the maximal EV strategy at equilibrium.

So, my interpretation of what you said is the following IF-THEN clause:

IF a solver chooses frequencies completely at random (according to some distribution, I'd assume we don't even necessarily need U[0,1] in fact I assume it could be arbitrary and still converge to the same equilibrium--do you know if this is true BTW?) THEN it will not mix. Because it will be impossible to realize equality of EVs because of the infinitely uncountable nature of the set from which we draw the freqs at random meaning EV, being a function of the frequencies, therefore also exists as a continuous random variable on an infinitely uncountable interval --and literally any dEV>0 leads to a strategy option being dominated at equilibrium.

But by the time you reach an equilibrium those freqs have actually been computed to maximize the overall strategy's EV (which at equilibrium implies 0 exploitability--is it not the case that a strategy is unexploitable at equilbrium if and only if it is the maximal EV strategy at equilibrium?), they are not random.

The strategy itself is a distribution (frequencies summing to 1) but the frequencies composing that strategy are no longer governed by a distribution--they are no longer random (is it a contradiction to say "at equilibrium we play each non-dominated strategy at random according to the freqs" and "the equilibrium frequencies are not chosen at random?).

And if the frequencies aren't actually random at equilibrium, then why would it remain impossible to see true EV equality at equilibrium between certain actions? As Scylla said in DooDoo's PM--we can think of these freq as recommendations, and by the time we're actually ready to compute average EVs (which is ultimately the EV that feed freq assignment, no?) there's just a finite # of recommendations that are fixed and no longer random--there has to be otherwise we're not at equilibrium yet.


Do you know if solvers employ some sort of thresholding--meaning, if a computed EV1 is within some chosen distance of a computed EV2 we just assume EV1=EV2 to avoid exactly what you describe, where trivial EV differences don't **** up the sim?

btw I'm assuming "line" refers to a particular path through the tree starting at the root node and ending in a terminal node because when we define line like that we can define the overall strategy as the set of all possible lines, i.e. the set of all possible paths through the nodes of the tree.

Also when I said this:

What I meant to say was:


Also one final point/question:

Is it really correct to say a GTO strat vs a GTO strat is breakeven when you allow for both players to arrive OTF w/ different ranges meaning one player may arrive at flop having a range advantage meaning that player's range will extract more EV from the pot compared to his opponent.

Seems like when someone says "GTOvGTO is breakeven", what they really mean to say is, the difference in EV for each players' range is static over the course of the hand (i.e. neither player extracts any additional EV beyond the flop EVs)

^OK one more truly final, final thing:

I realize this statement:

Ignores the preflop strategies.

And it would make sense that if all lines are considered--stretching as far back as the 1st preflop action--then neither player will extract any EV at equilibrium i.e. both players strategies are in totality 0EV.