I reiterate my post. Prove HUNLH is a game.

In this context a game "consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies."

What makes you think HUNLH is not a game?

Ok well, David's example did not illustrate this at all.

Check calling with the 3 card draw, is going to show different profit then check raising it , is going to show different profit then check folding it. If you check raise the stand pat every time, how in the **** is that going to show the same profit as check folding every time? It won't even be close in any way. Check folding expectation is obv 0, and there is no way in hell that check raising the stand pat in that spot will also show an expectation of 0. GTO for the stand pat has to be to call the vast majority of the time, since if it was folding any significant amount would be inherently exploitable. Therefore if the GTO for the stand pat is to nearly always call, then check raising will not show the same profit as check calling, or check folding. It just won't. So my point stands that David's example was bull****, and the argument that it doesn't matter what strategy the opponent uses against GTO seems to be flawed.

Maybe I misread his message, maybe not. I will have another look at it later. Too much to follow on NVG at the moment...

You can consider it this way. However,

• GTO for the button and GTO for the big blind form an optimal pair (an equilibrium).
• To compute the equilibrium, you need to consider both strategies.
• Knowing just one of the strategies (say, for the button) would be pretty useless.
• We don't know what the expected result for either of those strategies is.
• It thus makes a lot of things easier if you combine them into one strategy for HUNL.

GTO is indifferent to strategies, not to actions. Each action taken helps defining our opponent's range, and we can use that. So I would expect position to be worth something.

One should probably consider the full-street AKQ game capped at one bet and compute both players' expectation.

pretty sure Durrrrrrrr didnt join twoplustwo until 2003/2004

And "golden years" typically the partypoker days, ie. the last few years before UIGEA.

Not true at all. For any given button,bb strategy pair, if your goal is to get closer to gto then playing just the button gto and the bb old strategy the new one will be better. Or we could get kind of unrealistic and say maybe we are playing someone who will play perfectly when we're on the BB but badly when we're on the button so we could play gto in the BB and exploitatively in the BB.

I wasn't referring to GTO strategies. Having some strategy for the button, but none for the BB doesn't make sense to me.

Well obviously you have to have SOME strategy for both, but we could do a ton of work/research into finding one of them and just do a basic/bad one for the other and it'd still be useful.

Obvious play is finding the strategy for the button. And then you button the gto-bot and win some nice cash (if the assumption that position does give you an edge is correct).

Wait what?

What are the moves that are available to the players.

Bet 1 unit.
Bet 2 units.
...
Bet n units for any natural number n.
...

That's a lot of moves. An infinite amount of moves available makes it an infinite game. Not really a problem from a mathematical point of view, Nash still applies, there is an equilibrium and we have a strategy (or a pair of strategies). Problem is that the strategy could be infinite and we want put that thing in a finite concept aka. a computer.

It's better to say a hand of HUNL is a game. We have a fixed effective stack size, therefore a finite set of moves and a finite game (each move either decreases the remaining stack or progresses the game in another way). Nash assures an equilibrium and we have our strategy pair.

Now we could try and say that HUNL is an iteration game of those games, but that would create quite a headache, if we try and model today's cash game rules. Is sitting out a strategy? Is it optimal? Can an optimal strategy ever accept being in the BB first?

So we just say HUNL is an infinite set of games {HUNL1, HUNL2, ...}, and we have an infinite set of strategy pairs that show has how to play it.

Sauce says he'll take the first million of these strategies, round the frequencies carefully, stuff them in a computer and attach a pseudo-random-generator to it. This is his gto-bot. While it's not perfect in a mathematical sense, it'll be a mean machine and the theoretical EV that could be gained against it is tiny at best at effective stack sizes that are practical.

Ignoring the technical problems like rounding and randomness it's obvious that there is a gto-bot for any finite subset of HUNL. But is there a gto-bot for HUNL itself? durrrr doesn't think so and his intuition lead him in a direction such things a frequently proven in mathematics.He looks at the bot to see the biggest effective stack it was designed for (by asking, since the bot doesn't exist yet) and tries to show that he could exploit it at much bigger effective stacks. The reply he gets: For the bigger stacks he would need to face a bigger bot. And he is left wondering how bringing an entirely new bot to the table is not adjusting the bot. I don't know if durrrr is correct or if there would be a gto-bot, if I would use a suitable model (probably a pfsm or sth. like that) for it. I'm not going to go into more detail because it would get a little technical and I don't think anybody would be interested, but his statement is far from trivial. People ridiculing his posts show their own poor understanding of infinity and not his poor understanding of game theory.

/rant

GTO is not going to adjust its strategy against stand pat hands whether you stand pat 100% or 0%.

The thing is they are totally independent. If you get a trained monkey to mash buttons when in the BB the GTO strategy for the BTN would be no different to if you were playing a GTO strat in the BB. So when developing the BB or BTN strategy you do not have to take into account what your strategy is in other positions.

I don't know if this is what durrr was saying but it isn't relevant to the question of whether a GTO solution to HUNL exists or not. There is no memory restriction on the bot...just like there is no restriction on look up tables, which is all the bot really is (given an RNG). The question of whether bots that exist could or could not beat durrr is a completely separate topic.

durrr posted about playing a limit holdem bot, first of all. Second, are you seriously arguing that his claims about beating a GTO bot in any form of poker are some kind of semantic argument about the general nature of "no limit holdem" as opposed to "no limit with stacks of N BB"?

I conjecture also that there exist finite stack sizes such that if you have an optimal strategy for that stack size, you can combine such strategies with rules for betting sequences exceeding those stacks and obtain strategies that are exploitable for less than arbitrary epsilon for any larger finite stack.

Also, I'm too lazy to look it up, but aren't the button strategies in academic bots exploitable for less than the button edge by now so we have a proof in probability that the button wins at HULHE?

jfc, i know.

never said it would.

i was saying that in the example that david gave, it seems that if the pat hand is playing GTO, that the available options to the 3 card draw hand of check raising, check calling, check folding, or donk betting will NOT have the same expectation against the strategy. I am calling into question the notion that if the opponent is playing GTO that all available options to the other player have the same expectation. The example he gave to illustrate that point, I think suggests the opposite.

obviously all the available options dont have the same expectations...

not sure what you're on about.

shaywh said "The idea is playing GTO in some spot means your'e indifferent to the opponent's action. As a result, the opponent can do what ever he likes to, since he will make the same average profit (If he could do more profit by one action, it wouldn't be GTO since GTO is balanced against any possible range)."

David Sklansky said "Anytime the river is a made hand aginst a drawing hand that has no value if it doesn't hit, any non gto strategy is as good as any other as long as the other side is using gto. A very slightly imperfect example would be in high draw when a three card draw checks to a pat hand and now considers check raising. If you know that your opponent will play this right, you don't have to."

I am going on about how David's example is wrong. The 3 card draw can't play it anyway he wants and expect the same profit if the Pat hand is playing GTO.

more likely GTO play would require that you check/raise/pat as a bluff with the hands that're just on the cusp of being too weak to check/call (and the check/calls will most definitely show a greater profit than bluffs).

and the reason is because some fraction of your bluffs are going to be checked post draw, and they need to show some non-zero showdown value in order for the initial bluff to show a profit.

a check/raise bluff pre draw with complete crap would probably show a very, very small loss.

the post draw bluff would break even every time though.

or at least that sounds right at the moment in my drunken state.

sorry didn't realize exactly what you were referencing. and i keep thinking this is always about a nlhu bot every time i read new poasts. my bad ignore whatever i said.

they're talking about average profit which is the same as EV, your talking about the immediate profit of a hand in a vacuum. Of course check folding has different immediate profit than bluffing.

Check-folding, check-raising or donk betting a missed draw will all show the same profit over the long run vs GTO

David's example is much clearer if you substitute "bluffcatcher" for "made hand" and "polarized range" for "drawing hand that has no value if it doesn't hit."

If I have a King and you have either an Ace or a Queen, with equal probability, and there's $1 in the pot and you can bet $1 or check, GTO is for you to bet all your Aces and half your Queens and for me to call half the time.

As long as you bluff with the right frequency, it doesn't matter whether or not I call with the right frequency. As long as I call with the right frequency, it doesn't matter whether or not you bluff with the right frequency.

It does matter if you fail to bet with the Ace, though, so in a nitty way he's a little wrong.

The more concrete example where he starts talking about a pat hand and a 3 card draw, I'm not really following either. He must be talking about the decision to checkraise bluff or just fold given you don't have it and have checked, but there's nothing in his post that makes that clear, I'm just guessing based on what would actually illustrate the concept he's talking about. It's still kinda weird because there's no reason the other guy would be betting when our range is perfectly polarized relative to his and the rest of what he's saying doesn't apply otherwise.

Is that what I said? If I did I didn't mean to. I meant that the three card draw plays GTO after he checks and the pat hand can do what he wants.