I'd like to analyze the statement in that article beginning with this one:

Let's isolate the first important conclusion:
So it is true after all, right?

Then he comes up with a surprising exception to this rule.

To prove this exception he uses a single hand from a HU-scenario and draws the following conclusion:

Unfortunately the blinds do rotate in the real world! If that wasn't the downside of all of these fancy theoretical models, then scientists would rule the world, throw 105 mp/h fastballs and win all the money. The fact that they don't, should make you at least somewhat suspicious.

You can't draw any conclusions from breaking up the game in half and analyzing the asymmetrical disadvantage of the player in the BB, if that very same player gets the button in the very next hand.*

In fact GTO is the way to realize the exact EV of any given position regardless what the other guy does. If the EV on the BB is negative, then the GTO-player will lose money. If the EV on the BU is positive, then the GTO-player will win money. Overall, since the game is zero-sum and played in a symmetrical way, he is supposed to break even (if there wasn't a rake).

The real way to make money from this is to mention "GTO" in the title of your next book though. That's an instant winner, because every donkey believes that he can finally buy the secret miracle method how to turn xxxx into gold. The overall concept isn't new: http://en.wikipedia.org/wiki/Alchemy



* Well, that is not completely true. If this is the last hand of Poker you will ever play, your seat does indeed matter, because there will be no symmetry after all.

Two players playing equilibrium strategies in a no-rake heads up game will both have an expectation of 0 if they play an even number of hands and rotate seats. That's not the same as saying that either of these players would have an expectation of 0 playing against a different strategy. That latter point is what many people misunderstand, and what this article is arguing against.

There are plenty of other exceptions, but it only takes one to disprove a rule, and this is a very easy one to illustrate. Imagine again the example of the two equilibrium players. If they play an even number of hands, each has an expectation of 0.

Now let's assume that one player is replaced by the world's tightest poker player. This player folds every hand except Aces. Even if other player does not change his strategy at all, the expectation of his equilibrium strategy will be greater than 0, and thus greater against this player than it was against his first opponent. Of course he could do even better if he adopted an exploitive strategy, but my point is that he doesn't have an expectation of 0 against any opponent strategy simply because he himself is playing an equilibrium strategy.

This is an extreme example to prove a point. Even against many more reasonable but unbalanced strategies, such as those employed by many real world players, the equilibrium player would still have an expectation higher than 0.

My argument does not depend on the players not rotating positions.

The "thing" is the indifference criteria. When computing GTO-strategies you are trying to make you opponent indifferent, so it doesn't matter if he bets, calls or folds. In the end you will realize your EV regardless what he does.

The best example is Rock-Paper-Scissors. The GTO-strategy is simply to randomize your decision. It doesn't matter if your opponent starts with scissors or comes up with the most sophisticated move-order possible, he will always have an expectation to win exactly 50% and so do you. By making sure you won't lose more than half of the time, you also made sure that you won't be winning more than half of the tims. Notice, it is not a freeroll where you win at least 50% of the time. You win exactly 50% of the time, not more and not less.

The difference between Rock-Paper-Scissors and Poker is that in Poker players act after each other. One player has more information than the other. This positional advantage translates into positive EV. The positional disadvantage obviously translates into negative EV. So what exactly is the correct strategy for Poker? Well, the correct strategy for the player with negative EV is not to play at all.

So there you have it: Poker shouldn't be played at all! From a GTO-standpoint playing Rock-Paper-Scissors is just a complete waste of time, playing Poker is actually a stupid idea for the guy in the BB. Now I am sure that neither you nor Pokerstars love to read this, but it's true.

Now here is something about your argument that GTO wins money against a guy who folds almost everything. My feeling is that this is already accounted for because of the indifference criteria. To get the full picture we need to analyze the mirror situation. What is the opposite of folding almost everything? Betting almost everything or pushing all-in with almost everything? We know from the Sklansky-Chubukov-Rankings that depending on the stacksize, pushing all-in can actually be a winning strategy, so even defending GTO will actually lose money against it. How about that?

So we found a situation where playing GTO is guaranteed to win something and we found a situation where playing GTO is guaranteed to lose something. What does it tell us? It tells us that we can prove anything if we just construct the model in the way that it supports the argument we want to make.

Btw, my argument has a practical safety net: Even if I am wrong, it doesn't matter since the GTO-approximation for LHE is about 8 TB in size. That's basically a full public library of data and since Rainman is dead, nobody on Earth is capable of memorizing it.

Geeez....so many typos. Well, I assume you know what I meant even if I spelled it wrong.


Btw, here is something I have to add for the record: If you are in the BB and the other guy folds everything but Aces, you are not winning because you are playing GTO. You are winning because of the rules. In this case your strategy doesn't matter at all!

In other words, since you didn't have to take a single decision yet, you didn't get the chance to play a strategy up to that point. The strategy only matters when you can make a decision based on it. For that purpose the only relevant hands against such a player occur, when you are on the BU and that already cuts the possible scenarios down to half.

Now if only the hands on the BU matter (in a higher sense) to judge the quality of your strategy against such a rock and since you have a positive expectation from the BU anyways, it doesn't make me wonder, that you show profit against him.

Very interesting discussion guys! It's a complicated topic, but one that I think Andrew did a nice job with at a basic level. I agree with Shandrax that the "world's nittiest player" example is a flawed one, because Hero's strategy will make very little difference to his win rate in the BB. A better example might be the "world's biggest calling station," aka an opponent who limps every hand but aces on the BU. (and raises with aces obv). Now Hero can use an equilibrium strategy to take advantage of that mistake and lock in a minimum amount of EV (it still may not be positive EV since Villain has position in those hands, but coupled with an equilibrium strategy when Hero is on the BU it will yield a net +EV). If Hero believes that Villain will make more mistakes on later streets (most likely by calling too light) he can take an exploitative line and perhaps do even better, but the GTO strategy will guarantee that he won't give back what he already gained.

I'm sure I said this badly, but maybe an older (albeit typo addled) article by someone who knows what he's talking about will shed a little more light on the subject.

Yes, I think the RPS analogy is at the root of the misunderstanding about GTO. You're right that no matter how exploitable your opponent's strategy, a GTO strategy can only break even in RPS. This is not the case in poker, as I've demonstrated.

Yes, it's awfully difficult to overcome a positional disadvantage, and if you were going to play just a single hand knowing you'd be OOP, you'd have to have quite a huge skill edge to want to accept that proposition. I'm not sure what your point is here.

A player employing this shoving strategy against another player employing an equilibrium strategy will not have a positive expectation. I don't know where you're getting that idea from, but it's incorrect.

To be honest I'm not clear on what your argument is. If you are claiming that it's often desirable to employ a maximally exploitive rather than an equilibrium strategy, I agree. If your claim is that an equilibrium strategy in poker can only break even (or lose money to rake), as is the case in RPS, then you're wrong.

I agree that you don't have to worry about anyone memorizing a perfect GTO strategy and employing it against you. I would lay long odds, however, that there are elements of your game where attempting to bet or call in a balanced way, rather than pursuing whatever strategy you currently pursue, would produce better results for you.

Sorry, but I don't think you have demonstrated it with your example. You were trying to show that you make profit against a very bad player, because he folded too many hands. The question is if that is because of the strategy of the other player or because of the rules of the game.

If we play chess and you are white and resign before making a single move, I certainly didn't win because my strategy was better. If I am white and play 1.d4 and you resign, it's a totally different story, although not much different.

We are talking NL Hold'em here, not [0,1] or any other theoretical model, right? In Hold'em you only have 169 hands to cover an infinite amount of possible bets. I would be surprised if there is a perfect solution that always guarantees 0 EV for the second player. It may certainly be close to zero, but I doubt you can ever reach exactly zero.

My claim was based on the indifference criteria. GTO is supposed to produce the same result, regardless what the opponent is doing. I only used RPS as an example and I also pointed out that there is a difference between RPS and Poker. In Poker the player who acts last has more information that allows him to make better decisions.

Since everyone came up with his favorite player model to prove this or that, let me just introduce mine. It is called "Busto":

There are two players (A, B) with a stacksize of $1.000.000 each. There is $1 in the pot. Player A is never bluffing when he is betting big, but in contrast to any other theoretical model or card-face-up problem, he doesn't tell anyone. Player A moves all-in. Player B is playing GTO and he has a bluffcatcher. I think we can stop here, because everyone knows where this is heading. So much about laying long odds....

Now let me say this: Equity given up by one player because of his mistakes has to go somewhere. It may very well end up with his opponent. In other words, if one player is desperately trying to throw away his money, the GTO-player will accept at least some of it. That is what we can agree upon.

The biggest moneymaker in NLHE poker apart from ranges and hand reading, is still Human Psychology and a good understanding of how the brain handles the very simple game of rock/scissors/paper.

The human brain is
a) greedy
b) can switch to a primitive emotional state within seconds
c) can only take so many rock/Paper combo's before assuming scissors will never be played

In order to master playing unexploitive you should first master your Mental Game.
Then you have to think about ways to exploit the unexploitable player, which because of the Human Psyche and the fact a human has time constraints and Physical needs is not impossible.

Paul "Vulcans" Ratchford for example is promoting unexploitative play, as a way to increase your winrate. He even wrote a book about it, Exploitive NLHE.

Offtopic: The biggest moneymaker in Poker is collecting the rake.

for sure, but as a player, that's out of control ;-)

Getting the most rake back possible should of course be a goal as such.

Nope, in my opinion people overestimate GTO and it has becoming a selling-argument for book. GTO is some sort of perfection that nobody can reach and even if he did, he would simply lose money playing this style in raked Poker.

I also have trouble with applying Blackjack-kind-of-analysis to beat certain player models where you make exact assumptions on how this person plays, where in reality this just turns out to be a complete waste of time. On the other hand I have trouble with doing complex calculations with the only purpose to make complex calculations anyways and I do not solve problems just for the fun of it. Needless to say, I don't care much about theoretical math in n-dimensional space.

What I actually need is a textbook with a complete strategy very much in the style of Nesmith Ankeny's "Poker Strategy: Winning with Game Theory".

Furthermore I don't need someone to tell me that I should polarize my play. I know that already. I don't need specific examples either. I need a formula. I need something that works always and that works instantly and not something that I have to compute first, because I don't have time for that. I need some sort of pointcount with cutoffs whatever about the strength of my distribution. It doesn't even matter if it is only an approximation.

As long as nobody provides this or stuff like that, we are all wasting our time. But in the meantime, I don't have a problem with you doing some math about AKQ-games or [0,1] or anything else that you want to explore. I can guarantee you, it won't get you anywhere. You will feel a lot smarter though.

Well, obviously one posting got deleted after I answered it, so this explains that.

What I can't explain though is why the first sentence doesn't read "it has become a selling-argument for books". It looks like there is too much editing going on.

Don't come to the conclusion that I am ignorant, just because I don't really believe in the Holy Grail of Game Theory. I read pretty much everything about the subject. It is just that I ask a simple question in the end: Does this information help me in a concrete way or is it just "interesting"? Unfortunately 99% of the stuff in print is just "interesting" and doesn't have any concrete practical application at all.

If B is playing a GTO strategy, he never calls without AA in this situation. That's not based on any assumption about how often his opponent is bluffing. When you say Player A is "never bluffing", what does that mean? Is he ever shoving without AA? If so, he will lose money on this shove, because he often wins $1 but very occasionally puts in $1,000,000 very badly. Meanwhile, what happens when he doesn't have a hand to shove? Player B collects the blinds.

The entire point of my article was that a GTO strategy is capable of collecting the equity forfeited by an opponent's mistakes. If you are playing against an opponent who makes no mistakes, then you aren't going to win money no matter what your strategy is, and you will probably lose money unless your own mistakes are very small (a GTO strategy can't capitalize on all mistakes, but it can capitalize on many).

If you know what your opponent's mistakes are, you can do better than a GTO strategy would by adapting to exploit those mistakes. But if you know only that your opponent is making mistakes, and not what those mistakes are, a GTO strategy will enable you to benefit from those mistakes.

Without knowing the equilibrium strategy for the game, can we truly know what a "mistake" is?

Yes. We can't make a list of every mistake, and we can't calculate the exact cost of a given mistake, but we can prove that certain things are non-optimal and therefore mistakes against a GTO strategy (or an appropriately exploitive strategy). I gave a simple and obvious example in my article, but there are many others. If you check and call your entire range in a given spot (ie you never fold to a bet), that is an exploitable mistake. If you check and fold your entire range in a given spot, that is also an exploitable mistake. We may not be able to say with certainty what is the exact correct frequency or which hands are the best for check-folding and check-calling, but we can demonstrate that certain frequencies are incorrect, and also demonstrate that some strategies dominate others.

My scenario obviously describes a river situation, because in contrast to common belief you can neither have the nuts preflop, nor complete air. For the sake of the argument I will answer this nevertheless.

If B plays GTO, he has to call with a "bluffcatcher" at a certain percentage. He doesn't even have to win exactly half of the time to break even, because he gets $1.000.001 for $1.000.000.

Since flipping coins, rolling dice or using any other sort of randomizer seems rather weird at a poker table, some brilliant professor named Nesmith Ankeny suggested as early as 1981 to call with the top x% of your hands in such situations. I could actually quote the exact page, but I am too lazy to check my bookshelf.

Back to the example: Since A never "bluffs", he will have Aces in that spot. I leave out the case where B has Aces also, because that's trivial, but it would be also correct for B to call with Kings at a certain percentage, because Kings have more than 50% equity against any worse hand.

The misunderstanding occurs because theory, definitions, practice and different games get mixed up all the time.

If GTO means making your opponent indifferent, then it doesn't matter what he does. If we can construct a scenario where it does matter, then GTO must mean something else. In this case the definition must be wider: GTO also includes situations where your opponent has the choice, how much money he wants to lose.

Since we got that straight after all, I think it is time for you now to finally reveal the GTO-strategy to Poker, because it would be very sad if we went through all of this for nothing more than hot air.

Scratch that last sentence, as it wasn't nice

I went through the original article again and I think by now, that I can nail down the exact root of the misunderstanding.

The GTO-part of the overall strategy is to make the opponent indifferent to calling with a King. That is only one part of the spectrum of possible situations though. The other parts of the spectrum are outside the scope of the GTO-solution. If there is nothing you can do in those cases, any strategy is no strategy after all.

If the opponent has a King, the GTO-solution will deal with it in a way, so that his choice doesn't matter. If the opponent has something other than a King, GTO doesn't cover it, so his choice does matter and that is where the mistakes add up. In such cases the "GTO-player" makes additonal money, not because of GTO, but for reasons that he cannot control.

A GTO strategy will play in such a way that it will break even no matter how the opponent plays his Ks. However, if the opponent makes a mistake with a Q or an A, the GTO strategy will profit from that mistake. So the GTO strategy is not indifferent to any action the opponent takes.

I don't understand what it means to win "because of GTO". The only way any strategy makes money is because of a mistake an opponent makes. No strategy will make any money against a perfectly balanced opponent. If you don't believe this, I'd be happy to find a way to bet on Cepheus playing against you heads up.

If you know what mistakes your opponent is making, you can often do better than a GTO strategy would. When you don't have a specific exploitive plan, but your opponent is making mistakes, then a GTO strategy will do better than break even.

Your "shoving the nuts" example is making a wrong assumption about how a GTO strategy is derived. In that situation, the GTO strategy will always call with the nuts, and then only call with non-nut hands if it has to do so in order to make the opponent indifferent to bluffing. When you risk $1M to win $1, it's not going to have to call with anything other than the nuts to make bluffing unprofitable for you. If you ever shove without the nuts, you will lose money.

Anyway, your penchant for misreading my article, misunderstanding game theory, and then mixing those together into snide comments is making me regret even trying to explain this to you. I appreciate the redaction, but there have been rude comments like that in every single post you've made, so this is going to be my last interaction with you.

I can fully understand your desire to bail out at this point. It took me a while to find the path through the labyrinth and I may have made a few mistakes along the way, but it finally became crystal clear that my initial assumption was correct.

The GTO-player can only control his own decisions, he cannot force mistakes from his opponent. Decisions that happen outside the strategy cannot be part of the strategy. This is the only to explain why a strategy that is based on indifference can profit from bad play.

The nice thing about GTO is that it keeps the cost of opportunity low, so you can afford sitting at the table, waiting for weak players to show up and make their mistakes. This is only true of course if you knew the GTO-strategy to Hold'em, which you don't and unfortunately never will.


Since this statement is totally ridiculous, I can only assume that you didn't understand the scenario. If not, then you are indeed totally clueless and should stop writing about stuff like that.

This was my point exactly. Proving that an action is a "mistake" means showing that it is not the optimal play according to the GTO strategy and that it is not exploiting your opponent enough to risk being exploited back. In other words, unless we know the GTO strategy, it's hard to say for sure what is a mistake and what isn't.

To put it in terms of RPS, without knowing the GTO strategy how can you say that a distribution of 50% rock, 25% paper, 25% scissors is a mistake? If you're playing a million games vs. an opponent who plays 25% rock, 50% paper, 25% scissors then it should become clear pretty quickly that it's a mistake but vs. an opponent who plays 50% scissors you will seem like Phil Ivey.

Poker is obviously a more complex game, and because of the rules a player can make mistakes that cost him/her money regardless of how the opponent plays (including a line like open folding every hand but AA). Or you can make a mistake that costs you vs. 99% of opponents, so that everyone with any poker knowledge agrees that it's a mistake. But mathematically speaking, you would need to know the GTO strategy for that spot in order to be sure.

We know certain properties of GTO strategies. For example, we know that a GTO strategy will be one where you can't unilaterally improve your expectation by changing your strategy. We also know that certain plays dominate others in certain situations.

Folding Aces pre-flop is always dominated by going all-in with Aces pre-flop. So although we can't prove that always going all-in is an optimal strategy (calling or raising less with some frequency may be involved), we can prove that folding Aces is not part of an optimal strategy. Any strategy that involves folding Aces pre-flop could be improved by doing something other than folding.

Edit: Matt Janda's Applications of NLHE does an excellent job of this in a more practical way, using what we know about game theory to determine approximately how strong a hand needs to be to be three-bet pre-flop, raised for value on the flop, etc. If you want something other than a trivial example like I gave here, check out his book.

If you want to sound like an expert on Game Theory, always mention "Mathematics of Poker", the AKQ-game and Janda's book. It may help to add random insight, for instance that folding Aces preflop in cash games is dominated by not folding them. Very smart indeed! Well, by the same logic, amongst the many things you can do to waste your time, reading those books is clearly dominating shooting yourself in the foot.

I really love what Magriel says about Chen's book, starting at min 7:47, because it is exactly the same that I have been advocating on this very board for years.

http://hwcdn.libsyn.com/p/5/b/7/5b79...f4fd032de85cc4

What I like about Janda's attempt to reverse-engineer the Poker Betting-Pyramid is, that if you do his math starting from the river, working yourself back to earlier streets, you end up with pre-flop ranges > 100%.

I guess the mistake is that in order to keep up the 2:1 air:nuts-ratio, he is treating strong hands like they were nut-hands even though you may still end up losing with them. Maybe he just read too much of this AKQ-bull****, who knows?

OP is completely clueless about GTO and it's implementations. He is also the reason poker will always remain profitable, exploitative minds struggle to grasp its truths.

(I have spent the last 10 years of my life mastering GTO LHE, fwiw)

Congratulations that you have mastered GTO LHE! Unfortunately nobody plays it anymore. The good news is, that you didn't waste your time analyzing 5-card-draw also.

Now here is my point: I am not against GTO as a concept. I am against treating it as a religion. Right now we got a bunch of fake priests running around spreading the "holy word" and trying to sell us the secret of Poker for $39,99. Yes, in contrast to god or gods, we can prove the existence of GTO, but not more than that! Even if we knew the exact GTO strategy for LHE, it would be at least 8 TB (terra!) in size. This by the way proves that you are either talking out of your behind or you are Cepheus and learned how write postings on 2+2boards.

Also there is a common misunderstanding that GTO wins money. No, the strategy itsself doesn't win anything more than the exact value of the game. The GTO-player can win additional money if a bad player shows up, but only because he happens to be there and not because of his strategy. If a bad player throws away equity, it ends up with his opponent regardless of his strategy. Every strategy gains this additional EV. The only difference between GTO and any other strategy is, that GTO doesn't give anything back. It is a defensive strategy, not an offensive tool. You cannot become rich by force!

Simple example: If a player busts out of a tournament, every other participant gains EV. Why? Simply because they were still in the tournament. Their strategy didn't have anything to do with that. Nits, callingstations, lagtards, fish, whales, you name them, they all gain EV, because they simply didn't bust out yet for one reason or the other.

Same thing with GTO: The GTO-player is doing nothing other but sitting at the table counting peas all day long, but he gains money when someone else at the table opens up his wallet and gives him $100. That doesn't happen because the GTO-player counted his peas exceptionally well, it happend simply because the other guy was giving away money and the GTO-player was there to take it.

Now you can decide what is better: Sitting at the table, waiting for someone to gift you $100 or trying to hustle the guys for more, because you are good enough to do it. Ask Phil Ivey about it. He may answer that cheating the guys with all sorts of hidden edges is even better, but I am sure that he prefers hustling over just sitting there pushing chips back and forth.

After sinking a ton of bucks in books on GTO and after realizing what it is all about, I noticed that there is too much effort required for too little result, even if I could remember all of it. The GTO-strategy for Poker is not as simple as the GTO-strategy for Blackjack. This is the simple truth. We don't like to hear this, but there is nothing we can do about it. We would love to sit at the Poker table, raking money from fools like Blackjack-Dealer, but we can't. The real world is more complex than a world build from LEGO. Kids learn that eventually, Poker players have to learn it also.

What we can learn is an approximation to perfect play, some cookbook type of recipe. A nice example would be the HU-section in "Easy Game". Stuff like that works well enough to dry out the games up to the point where the rake turns playing into negative EV. Unless the fishtank gets refilled the game is dead and will stay dead and I am not sure if streaming on Twitch or getting people interested in GTO can change it. The "targets" simply noticed that playing onlinepoker hurts their bankroll even more than playing the stockmarket. Where are the customers yachts? There are none!

Your passion to really decipher and debate GTO is certainly great, but you managed to get answers from one person who has really studied the subject and could provide very thorough perspectives, and you then sent that person away.

You are probably picking on something about GTO but still can't quite work it out.

The above scenario with AA starts preflop, and Player A who never bluffs will bet big preflop but not necessarily on later streets. The bluff catcher matters on the river, by which there is a board to take into account and more bets and decisions which have happened by then. There are no preflop bluff catchers. Something in this problem is wrong.

I think that what you are trying to say is that although GTO wins in the long term, one can still lose in the short term playing GTO because poker is still a game of chance.

This is what I was thinking in a way.

So someone who has a $1,000 bankroll and plays $1/2 GTO against someone who has $1,000,000 and makes some mistakes may still go broke before he sees profit from GTO and never recover. But the guy with the $1,000,000 who can take some swings playing GTO should win in the long term.

Because I think bankroll management is not covered by GTO. So perhaps GTO won't work for the highest stakes. If you play GTO on nosebleed stakes against a billionaire who plays good but not GTO, will you survive?

That's one of the things I was thinking about GTO. That perhaps it has an upper cap on stakes one can play on depending on their bankroll where it doesn't have less risk than other styles. But I don't know if this thought is correct.

Or basically, you can't just get a GTO piece of software and start playing at the highest stakes with a small bankroll hoping to become an insta-millionaire.