I understand where you are coming from but it just doesnt make sense to use it as an example. Just because folding is a legitimate option in poker doesnt mean it can be used like you have in your example. Like I said, your example of folding every hand preflop is not a strategy, it would be the same as not playing the game. There are some other things you should not use as an example, such as folding the nuts on the river, or checking back the nuts on the river. Both of those actions are the same as not playing the game.

@LEO

yes but the optimum frequencies are totally opponent-dependent. that's where i'm stuck.

i can see the RPS example being valid because at least it's a 0ev proposition.

however a "gto" bot which never adjusts could be beaten (-ev)

like, i understand that there is a set of tendencies that exists that are going to be, in a vacuum, the best general plays as per each individual situation.

but a player with even a general idea of what those plays are or what they are perceived to be could exploit what is, on a more macro scale, gto.

and dude btw i appreciate the clarification so if i'm wrong just lmk

i don't really want to regurgitate stuff already said by smarter dudes earlier in this thread but as im one of the dumb fellas just along for the ride ill try to put it in simple terms.

"optimal" is confusing everyone because to play optimally against an opponent who is exploitable in whatever way means you adjust to what they are doing. the bot isn't playing to exploit and be as much +ev as possible, its playing so that its impossible for anyone to be +ev against it.

in the RPS if some dude is throwing rock like 90% just for kicks... the bot still throws 1/3 all because that strategy cannot be beaten. the point is that it doesn't matter if the opponent knows exactly what the bots betting strategy is because there is no counter to it.

yes in the RPS example the bot is never going to have an edge because the human can just as easily play perfectly, but the point is that no human is ever going to play gto at hunl.

How about he folds every hand preflop except for QQ+/AK which he always shoves. There, this hypothetical super nit is 'playing' the game.

If anyone wants to make a prop bet if this goes forward for 5-10k, I'll take Game Theory Inc. side. Doubt this will actually happen though.

so the bolded part is wrong right? also don't you immediately contradict yourself in the next sentence?

There are very, very few opponents for which we can calculate the EV of vs GTO and this is one of them. So opponents like this are the easiest way to prove (admittedly trivial) statements like "there exist opponents against whom GTO is +EV." We'd love to talk about the EV of GTO against more realistic opponents (ie non foldbots, shovebots etc) but doing that is too hard for us.

No source is necessary; this can be logically shown.

Imagine a game similar to RPS but where there is an additional 4th option called "Auto-win" that beats the other three options every time (and ties with itself). The problem is, for whatever reason, throwing auto-win is exceedingly difficult and it is thus infeasible for a human to do it 100% of the time. A gto-playing computer would do nothing but play auto-win, and its human opponent would always lose, even if he was able to throw auto-win 99% of the time.

This altered RPS game is still zero sum... You win what I lose, and if we both play gto, we breakeven (assuming no rake), yet it has rules (or other properties) which clearly make all non-gto strategies lose to it, whereas in regular RPS, they tie no matter what they do.

So it is not necessarily true that a gto strategy for a zero sum game must breakeven against all strategies; clearly zero sum games can be devised where a gto strategy has positive expectation over other strategies.

As it happens, gto for my altered RPS game also happens to maximally exploit all other strategies, but I could add further rules or other properties (like payout amounts for winning with each option, with auto-win paying the lowest) to the game to make it so that the gto strategy was not always the highest EV play in every situation, but (obviously) still beat every non-gto strategy in the long run over the aggregate course of all possible actions.

For example, if auto-win paid 1 unit when it won, and winning with any other option paid 10 units, I could make much more money by "outplaying" you with the normal RPS options, but I still couldn't beat gto auto-win playing computer. And if you were predictable at RPS, I could crush you for much more than the gto computer could, but I would still not be able to beat gto.

Though more complex, this is the same situation gto has in poker. It beats all other strategies, even if it does not always make the most possible money.

The 200k hand hulhe competition is obv never going to happen.

Instead, durrrr should take an hour or so to clearly write out why he thinks he can beat a GTO bot due to the game having incomplete information. The game theorists on this forum can then show him why he is wrong, and that one hour he spent can potentially save him millons of dollars from losing to the gto lhe bots in a soul crushing and ridiculous 200k hand HU LHE challenge (who the hell wants to play 200k hands of HU LHE??)

he isn't saying he can beat a gto bot he's saying one doesn't and can't exist. theres a difference. there are no gto hulhe bots and certainly none for hunl... the bet is that he can beat the bots that do exist.

Suppose you are playing 6-max NLHE. You are the BB with 20 big blinds, and a GTO machine in playing the BU.

It folds to the GTO machine on the button. It opens to 2bb. You decide to 3-bet to 6.5bb.

Let's look at the EV of your bet. You are risking 5.5bb and stand to win 3.5bb if the GTO machine folds. Let x be the percent of the time the GTO machine folds. We get the equation:

3.5x - (1 - x)5.5 = 0

3.5x - 5.5 + 5.5x = 0

9x = 5.5

x = 5.5/9 = 61.1%

So if the GTO machine folds 61.1% of the time or more, your 3-bet exploits it. Therefore the GTO must defend its button open against your raise at least 38.89% of the time.

Game-Theory-Optimal: Whether you as the BB are only 3-betting aces, or if you are 3-betting all hands, the GTO machine must not fold more than 61.1% of the time (after it opens OTB and you 3-bet to 6.5bb). That is, regardless of your 3-betting frequencies, the GTO machine must not fold more than 61.1% of the time.

The GTO machine doesn't care if you 3-bet only aces, or every hand. On top of this, the GTO machine doesn't even care how often you defend your BB by calling. So it doesn't adjust its BU open frequency according to your big-blind defending frequency!

Exploitative-Optimal: However, if you are someone who only 3-bets aces, a more profitably strategy is for the button is to exploit you by folding everything except aces (you are not deep enough to set-mine against or bluff post-flop, ect.)

People posted toy games itt where they constructed GTOs that don't adjust and can beat any non GTO.

As for why this must exist in HUNL, given a very general set of criteria (which HUNL meets) you can prove a mixed strategy equilibrium must exist. There are alot of ways to prove this...the ones I understand involve translating games into well known and solved problems in topology. Unfortunately i don't think there is any good way to convince somebody of it's truth without learning how to prove the theorem.

ok mr literal. All he needs to do is quickly explain himself, and it will be pretty easy to prove why he is wrong. He posted the same stuff in hsnl months/years ago but again never explained himself. I think theres probably 100+people on this forum that could quickly explain to him why he is wrong. Although hes probably stubborn and wouldn't accept it. Either way a 200k HU LHE challenge against the world's best bots is ******ed. He will lose a lot of money whether his initial assertion about gto strategies is right or wrong, these bots are way too good

no, they aren't opponent dependent. they are derived assuming your opponent will always maximally exploit you, and you re-exploit until you are at equilibrium. that frequency (and bet size and hand range) at equilibrium is optimal, and it doesn't care what your opponent ACTUALLY does, only what your opponent COULD HAVE done.

That is why I have a problem with a "proof" using that example because it is wrong to use it

And I just wanted to add: So much disagreement/confusion would have been avoided if the convention was to use GTU (game theoretical unexploitalbe) rather than GTO (game theoretical optimal).

No it isn't. Not playing the game is 0 EV and the fold bot is very -EV.

Someone claiming the following:

"The GTO solution to poker is not neutral EV against all strategies."

provides a complete and perfect mathematical proof by finding at least one example where GTO is +EV. The folding bot is one such example. Raise/folding bots are another.

I think the isildur1 bot might be another (bazinga).

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Someone else claiming the following:

"The GTO solution to poker is +EV (more than neutral EV) against all strategies."

is claiming something entirely different.

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It's the internet so sometimes people think they disagree when they are only arguing different points.

No it isn't. It's a perfectly valid proof of what I stated. You are free to think the statement I am proving is trivial/uninteresting or there is a better way to prove it but that doesn't mean the proof i gave is wrong.

he said himself that he's too lazy to explain it, so instead he's just going to become the best lhe player in the world and play 200k hands vs. near perfect robots. what's so hard to get about that?

Haha.

Im not sure what the technical way to describe it is but using the fold every hand as a proof just does not make sense. There has to be some criteria for something to be considered a game and I would think one of those would be that neither player can fold the nuts, or check back the nuts, otherwise it really is not a game.

Prove that the GTO strategy is +EV against a strategy of folding everything but AA and that would work, except for the fact that noone knows what that preflop GTO strategy would be right?

Assume 1.5bb stacks and a player folding everything preflop except for AA, which he shoves. Are you really of the mind that this isn't -EV against a GTO strategy?

I am not trying to argue for one side or the other, simply that it is not proven (at least using the example of the fold everyhand pre thing).

I would assume that a GTO strategy would be +EV at normal HUNL stack depths vs a fold everything but AA strategy but the point is you cant say for sure, right?

so are you just going to counter every argument by saying that theres no way that we can know because we can't produce a gto bot? pretty stupid imo.

it is proven in theory. there are plenty of things out there that we can't physically produce/measure/whatever but we can still prove that they are real through math and logic and stuff.