i am suggesting that since we do not know what gto looks like. we dont know if a human is playing close to it or not.

hmmm how do you know this? (hint: you dont!)
yeah maybe


I highly recommend anyone reading this thread to take what they read with a grain of salt. You really have to question everything and only look at a few posters who understand the intricacies of whats being discussed before you take anything for fact.

C'mon man. I am disputing that no human plays perfect GTO, not that someone plays close to it or not, but that depends on what your definition of close is.

I don't know what perfect chess looks like but I know no human plays it because computers crush humans, even though those computers themselves aren't playing perfect chess. The best human chess players thus play closer to perfect than anyone else.

It may be theoretically possible for a human to play perfect GTO poker or chess. I believe the human mind is incapable of doing it thus making it practically impossible.

I mean you don't know with exactly 100% certainty that the sun is going to come up tomorrow but everyone still (correctly) plans as if it will.

Yeah....the phrase "it's pretty safe to say that [no human] plays GTO" is not really controversial.

Yes, we have some idea what GTO play looks like. And what we know is enough to decide that certain players do not play anywhere close to it, while others might.

Can we really just say that because NE exists for these games that they must exist for a different game with different decisions? Everything I've read indicates that NE is very specific to the information and decisions being made and that there a certain instances where NE does not exist.

One example is the Beer or Quiche game found in chapter 9 in this:

http://www.matthew-hoelle.com/1/75/r...ment_465_1.pdf

It seems that it is possible that HUNL may be solvable in certain situations but not in others.

We know that optimal solutions exist for more complex games like poker not because a few simple games have a solution, but because in 1928 John von Neumann proved they exist for all zero-sum 2-player games where the players have a finite range of actions. That's the minimax theorem.

He then wrote the book on Game Theory in 1944, and you can read it here:
http://archive.org/details/theoryofgamesand030098mbp

For those of you who are curious about the subject, I recommend at least browsing this book, von Neumann derives everything from first principles, defining even elementary concepts like functions, sets and vectors.

We analyze toy games because they are easy to solve and we hope they give us some insight into the solutions for more complex games.

Regarding the paper you linked to, the author finds NE for that game on page 162.

No....and I wasn't saying that (sorry if it seemed like I was). The "example" GTOs were given just to give people some intuitive feel for how it would work, how it could be +/neutral against any other strategy in a fair game and how it wouldn't have to adjust to it's opponent. Why NE must exist for these different games is because we can prove it must exist.....and that would be the case even if we couldn't construct ANY toy examples of a GTO.


EDIT: Jaytorr said the same thing.....but I'll leave mine up to give a different wording of the same concept.

Nice find.

Easy steps to understand that optimal strategy for NLHE with finite stacks exists:
1)realize every hand is independent of each other and so called metagame or history are just terms describing human psychological weaknesses
2)realize that full strategy for NLHE could be written down as series of numbers: for every hand for every flop/turn/river you do X where X is description of how often you do something (raise/fold/call) in given situation (where situation is defined by your hole cards, cards on the table and action so far)
3)realize that having two strategies described like in 2) available it's straightforward to calculate result of the game for every flop/ever holecards (you just follow the instructions and see who takes the pot)
4)draw a square shaped table with all possible strategies on X and Y axis, fill in results, calculate minimax
5)you either get it now or you have problems with one of the steps. Realizing which one makes asking specific questions easier and it makes it easier for other people to help you to get it.

If you get it, congratulations, you understand more about poker math than 99% of players.

Your GTO is okay but thats given perfect information. Even using card removal doesnt help much and with imperfect information as stacks increase it seems like GTO wouldnt be as useful. Knowing GTO alone is very useful but thats only part of what makes a good poker player.

I hope you've got a really big piece of paper for this.

The author found NE for that game in certain situations but not all of them.

Correct me if I'm wrong but it seems that the minimal theorem applies to perfect information games. It's hard for me to be certain in this because I've seen definitions of it where perfect vs imperfect information is not mentioned. But I have also seen definitions where it stares that it apples to zero sum perfect information games.

You're wrong.

I have a question... How helpful would some of the more high-level thinking posts in this thread be to someone trying to develop a GTO bot be? I'm assuming that many of wizz kids working on bots may not have the poker minds that many of the posters in this thread possess. Wouldn't this thread serve such people very well in terms of coming to a deeper understanding of how to adapt a bot for poker?

Idk, I haven't read all the thread and mb this is a silly question but I thought I'd put it out there.

all a programmer would have to do is translate a algorithm for calculating a gto-strategy into computer-language (incredibly easy task), no poker mind needed to do that.

the solution isn't out there because we do not have the computational power to calculate it yet.

Well, it's true that number of strategies is infinite (as frequencies for actions are given in real numbers) but you can assume arbitrary precision (say 10 decimal places), and then repeat the process for better precision getting as close as you want. This is just a technical detail which distracts from the main idea

0 help.
Nothing besides basic knowledge bright high school kid browsing basic wikipedia articles could arrive at had been posted yet here.
This thread is more about fighting with ignorance displayed by poker player population (including some very high profile players) rather than posting any ground breaking (or even mildly interesting) math info.

This is where I get stuck. How can the fact that the number of strategies is infinite be disregarded when the number strategies has to be finite? I understand you can make assumptions to make a model solution but that's not technically a full solution is it?

Well, if let's say we start with 10 decimal places for describing every frequency. Now the number of strategies is finite. Then you can go do 1000 decimal places and it's still finite. Then ask yourself how changing a frequency at 1001 decimal place can influence overall result - surely it almost doesn't matter right ?
There is formal mathematical way to prove it works but for intuition it's enough to realize you can go to some very deep precision where changing frequencies only so little don't change result of the game in any significant way (say you can prove that changing frequencies at 0.0000000000000000000000000x don't change result of the game by more than 0.000001$ at 5$/10$ or something similar which is as good as GTO in practice).

(Btw this is just one way to think about it, there are others. Here you can see that poker is the same as rock paper scissors just bigger (there are infinitely many strategies for rock papers scissors as well)). If you understand why there exists optimal strategy for RPS you can see the same thought process apply to poker, just strategies are a bit more complex than 3 real numbers like in RPS.

Nothing positive comes from posts like this except maybe personal satisfaction. I'm just trying to learn man.

Or maybe he is smart enough to reckognise that a Nobel prize is no guarantee a theory will not be proven wrong somewhere down the road.

Don't have to be Einstein to figure that out..

Ok makes sense. Thanks.

This is math, not physics. Theorems that are proven don't later get falsified by empirical data.

I fully agree with you, but your post made me realize that some posters/readers in/of this thread might have gotten impression that John Forbes Nash received the Nobel Prize in Mathematics. He didn't; there is no Nobel Prize in Mathematics and Nash did not receive a Fields Medal and he never will.

He was awarded the the Nobel Memorial Prize in Economic Sciences.

You're right. The Nash Equilibrium requires BOTH player to be playing 100% optimally.

This means that playing in a poker where the Nash Equilibrium has been reached would be unprofitable for both players (because of the rake).

In order to be profitable in poker (and in all zero sum games) you're going to want to implement a dominant strategy.