Equilibria, yes, Nash Equilibria, no. The confounding factor with multiplayer games is that there usually exists some way to improve on the equilibrium strategy via collusion, which could happen without actual communication between players, either implicitly because they each have faith that the other will do it, or by chance when they both happen to 'fall into' the collusive strategy.

I've linked a pdf that goes through explaining GTO in 3 cards heads up poker (https://www.dropbox.com/sh/v8tbv4ejfzgw00b/J7SFCp0RtW).

The paper would say that both Durrrr and Sauce123 are right and wrong. It's been discussed ITT but I think there's confusion on what is GTO and what is exploitative. I believe Durrrr is saying that GTO will be -EV if he incorporates and exploitative strategy. Sauce and others are saying that GTO is the same EV against any strategy, however the paper would argue that it may not necessarily be a +EV (or profitable).

From the article -
"We see, then, that there is no contradiction to common sense after all. The best play can in fact be determined from our cards, the betting, and our opponent’s tendencies, without having to consider what we will do in other, unrelated situations. But to come to this realization, we must acknowledge the difference between what is “optimal” and what is “best.” When your opponent plays optimally, then it doesn’t matter what you do. You will always have the same EV, so all plays are “best.” But when your opponent does not play optimally, the best play for you – the play which maximizes your expectation - will not be optimal either."

He also discusses "equilibrium" and eventually "how to win (against GTO)".

Let me know if the pdf doesn't link.

Enjoy!

It might hurt high stakes head up poker. But everyone has to remember that bots that try to use game theory will beat medicre players at a slower rate than good players will.

Imagine a chess tournament with a lot of weak players. Imagine further that winning players get extra points in relation to how few moves they need to checkmate. Great players will choose non optimum moves against bad players to win quickly to pile up those extra points. If the tournament was 62 1400's, one 2400, (not even a grandmaster), and Deep Blue, the computer would have no chance to outscore the master.

Do you keep qualifying your description of Nash's theory because you're not familiar with it, or because you doubt it?

Yeah, cause the average recreational player out there isn't going to stop at "Wow, there are bots that can beat legends!", he's going to go into GTO, chess analogies, Cognitive Science, probabilities etc...and eventually understand that bots aren't that bad since they won't crush as hard as good players do.

This thread is a super-polarised mix of interesting discussion/concepts and total crap.

uh? where does sauce (or any non-troll) say this? its obv not true.

Obv GTO posting style.

GTO by definition as proved in the paper is a 0 EV strategy. I should rephrase in saying that it's not the same EV but the decision made will be 0 EV overall.

I suppose the argument kinda teeters between whether GTO is +EV and the answer is NO. However, GTO can "solve" hu poker to be 0EV but would need a large amount of computing power.

And he is just wrong. It would be better (or equal) to just play exactly GTO than any modification of it against GTO.

What is true is that you can adjust GTO (sometimes drastically) against other non GTO strategies (ie a shovebot) and do much, much better than GTO would, but I don't think that is what he is saying.

Is anybody saying that? I don't think there is any strategy in say 100 BB NL that is the same EV against any strategy except maybe if you allow folding out of turn and the other player can't.

Actually I may be overstating it by saying there is usually a collusive strategy. I confess I'm not sure how often this comes up, it may be only in special cases such as high ICM situations. What is usually the case, I'm pretty sure, is that there exists a 'kamikaze strategy' whereby another player can hurt you and themselves, and there's nothing you can do about it.

I didn't read the paper, but showing an example of a GTO that is 0 EV (like RPC) mean all GTOs are 0 EV.

This idea of a GTO strat using the information of draw cards in draw games is interesting. Hopefully someone will flesh it out a bit more. Conceptually it seems like a GTO strategy must incorporate draw information, but in a way that is not tied to opponent's range or strategy. That is, it's not that opponent drawing 1 tells you something about his hand, but rather drawing 1 in 5cd is going to be a strong play in some situations, and so the GTO frequencies must reflect that to "defend" against it.

So just to clarify, you are saying that a GTO style would be one that is 0EV, therefore whether playing against a fish or another GTO style the EV would still be 0?

He also quotes an article which apparently talks about "how to win (against GTO)". Take that how you will.

well that was my other question, how reliable or good is that article haha

They use a Best Response Algorithm which determines the maximally exploitative strategy versus the bot as well as the expected winrate if that strategy is applied. If they ever arrived at a bot where the BRA resulted in a 0 EV counter-strategy, it would mean the bot was playing GTO. As a result, it is considered a measure of the distance of the current bot's strategy from the actual solution.

(http://forumserver.twoplustwo.com/sh...&postcount=80; also, someone linked to a paper on Best Response earlier itt)

To provide a better example than rock, paper, scissors and show that GTO is *not* 0 EV against all strategies in any game I figured providing an AKQ game example would be appropriate:
The rules are this:
There is a 3 card deck consisting of A, K and Q. The pot size is 1, the bet size is 1. Player 1 always checks, Player 2 can bet or check behind, Player 1 can then only call or fold.
To cut a long story short the GTO play for Player 2 is to always bet A, always check behind K and bet Q 1/3 of the time. The GTO play for Player 1 is to always call with A, always fold with Q and call with K 1/3 of the time.
Now, let us take this to 2 extremes: Player 1 folds K all the time and Player 1 calls K all the time.
If Player 1 folds K all the time then Player 2 actually gains EV as his bluffs with Q pick up the pot more often without losing EV anywhere else.
If Player 1 calls with K all the time then Player 2 gains EV. Player 2 is value betting A 3 times more often than bluffing with Q so when calling with K Player 1 will win 2 bets 1/4 of the time but lose 1 bet 3/4 of the time, again producing a -EV result even though he picks up every bluff Player 2 makes.
Notice how in both situations Player 2, playing the GTO strategy, gained EV through Player 1 deviating from GTO even though Player 2 never altered what he was doing.

GTO is GTO according to my understanding. I don't see how you can have multiple by definition.

Not all GTOs will result in the same EV, which depends on the details of the game. An obvious example is poker with negative rake, where the house puts in 10 BB into every pot before the button acts. Clearly all strategies (including GTO) will have a higher EV against all other strategies compared to standard 0 rake poker.

The paper does a fairly thorough job using this example and solves it mathematically.

If two bots were employing perfect GTO (I.e. bluff, c/c frequencies) then it would be 0 EV for both bots. In order to be +EV the author argues that one or both would have to bluff, c/c with non-optimal frequencies. If one or both bots can adjust and readjust to the
frequencies is where you get the idea of equilibrium (or approaching an).

I think the author presents a fairly solid mathematical proof. Perhaps Sklansky can weigh in his opinion on the paper. I'm not associated with it other than I found it on my computer.

Thanks nice post

If both bots play GTO then by definition one cannot change its strategy to become +EV. Any deviation by one bot will make the bot still playing GTO +EV and the bot deviating will become -EV.

Edit: and to provide a reference for that example: Mathematics of Poker by Chen & Ankerman

1000+ elo points is a huge skill difference. beating such weak opposition is usually a matter of quick tactics, not a long strategic game. computers are very good in tactics. against such weak opposition, someone playing "optimum" moves (basically, strongest possible moves in a given position) will crush the weaker player much faster than someone playing non-optimal

I only skimmed the article but here is what I've gathered:

The author says that GTO play is 0EV as long as the opponent doesn't make what he calls "Stupid Mistakes." This is true, with respect to the simple game he's discussing, in a sort of circular way where you call any mistake that loses to GTO play "Stupid," but it's not really a meaningful insight relative to full scale poker games. The mistakes you can make in the One Card Poker discussed in that article fall neatly into two categories:

1. Making a play you should never make. These are what he calls "Stupid Mistakes."

2. Making an acceptable play at a suboptimal frequency.

In One Card Poker, mistakes in the first category are plays that look legitimately stupid like folding the nuts or calling with the nut low. In 100bb HUNL we're not close enough to understanding optimal play to say with any certainty whether a play is optimal or in the first category or in the second category. For all we know, the first category might include things like raising with 32o on the button or folding 93o from the BB to a 2.5bb open or even 3betting a minraise to a size of 7bb with any hand. It is extremely likely, virtually certain, that even the best human players are making "Stupid Mistakes" almost every hand they play in HUNL and therefore that GTO play would not just break even but beat them badly.

edit: Also, you're mischaracterizing what both Sauce and Durr said. I have no idea where you got the idea that Sauce thinks GTO play has the same EV vs any response. I don't think Durr is saying that he can beat GTO play. That would be definitionally nonsense. He is saying that he doesn't believe a Nash Equilibrium exists for HUNL or HUPLO, that he would be very surprised if one existed for HULHE, and that he thinks he can beat a HULHE bot that attempts to approximate GTO/Nash Equilibrium play. He's wrong about these things but they aren't explicitly nonsensical.

edit2: The article doesn't discuss how to win vs GTO play. It discusses how to deviate from GTO play in order to beat an opponent who is making what I described above as "category 2" mistakes. e.g. folding exploitively often against a player who bluffs with the right hands but not often enough.