Sometimes you need a COTM to demonstrate the above point. I know Vernon remembers back when the fish at the table would talk and weak posters would post about triple range merging like that was the secret to winning low stakes poker.

When in fact the secret is board manipulation.

^ ldo.

Well, who doesn't know that?

I don't see how this is true.

GTO is still a product of its environment, is it not?

Use paper/rock/scissor as an example, if we know that villain will ONLY use paper or rock, then our GTO approach against him vs against someone that will use all three will be different, else it wouldn't be GTO.

So if we know that certain cards and certain players are limited to certain options, then we should still be able to use the same GTO approach while changing some of its underlying variables.

GTO isn't about exploiting your opponent, it is about presenting a strategy that is unexploitable, no matter what range your opponent has. The point is to create a strategy where you don't care if he only throws rock or only throws paper or uses a mix of different options.

Not refuting that, but you're also not following what I said.

Is GTO the same:

-Someone that only plays paper and rock.

vs

-Someone that plays paper, rock, and scissor.

And taking it a bit further.

Is GTO the same in Omaha as in Hold'em?

Environment changes and therefore variables should also change?

Yes, it is the same. GTO is playing a strategy so that anyone playing one of the exploitable strategies you described can't shift to a strategy that exploits your strategy.

Those are different games. You might as well ask if GTO is the same in RPS as in Hold Em.

Disclaimer: I know jack about GTO, hence I am posting and asking questions.

Wouldn't GTO strategy be choosing paper at 100% frequency?

Would a strategy that chooses paper at 100% frequency still be GTO?

That's why I asked if changing environment effects selection of GTO strategy.

Argh, never mind. I am not even having fun nor actually learning anything .

Ignore me. OP, please delete as derail.

In any game, your opponent has strategic options. A GTO strategy is one where your opponent cannot choose options that exploit your strategy. You don't care which options your opponent chooses in determining what GTO strategy is.

Changing environment affects GTO strategy if there are changes in the options available to an opponent or to the payouts of those options. Changing opponent tendencies does not affect GTO strategy if the options and payouts remain the same.

If you are using your opponents range to modify your strategy, you are, by definition, using an exploitative strategy.

Opponent can recognize what you are doing and change his strategy. Therefor your strategy is exploitable.

Thus, not GTO.

What's the difference between opponent never choosing scissor and scissor not being an option?

If the opponent never chooses scissors, he has the ability to change his strategy to include scissors. If scissors is not an option, he does not have the ability to change his strategy to include scissors.

RP- seems you're equating GTO with optimal. Easy mistake. All GTO means is that your opponent will gain nothing more or less with any strategy. Hence, it won't matter if he never chooses scissors or always chooses scissors. Your win % will still be the same.

It matters a great deal if your goal is to maximize your win % vs. him. But GTO is not concerned with this. It simply makes it impossible for either of you to make a profit.

Hey all. I know I'm late to the party, and I apologize for that. I thought of a couple questions to ask in this thread last week but never got around to posting.

First question: I've seen some videos that seem to treat GTO and exploitative play as two ends of a spectrum rather than binary terms. They'll refer to a line as "GTOish," and emphasize the need to exploit while trying to minimize the extent to which a villain can exploit us. Do you think it's better to look at Exploitative vs GTO as two ends of a spectrum or in strictly binary terms?

Second question: In terms of dominated strategies, which lines in NLHE do we think fall into this category? I would guess that "raising to see where we're at" with middle pair falls into this since it's a great way to lose the most when our opponent holds the nuts and win the least when our opponent is bluffing. What else can we include? Loose/passive play in general?

No big deal. Thanks for contributing!

Great question, and the answer I think is pretty complicated. The short answer is neither, but I'm going to try to give a good explanation. It starts with this: it's going to depend on what game you are considering.

For example, let's look at the C/E/G game that I made up for an example earlier in the thread. The GTO strategy for that game is static--100% G. In that case it may make sense to look at GTO vs. non-GTO in strictly binary terms--you are either playing GTO, or you are deviating to exploit an opponent who is playing C, and it basically stops there and is one of those two things.

The times when you might think of your strategies as a spectrum are when the GTO strategy is mixed--say, for a game like RPS. Since you can always have a strategy of x% rock, y% paper, (100-x-y)% scissors, it does make sense to think of RPS strategies as lying on some kind of spectrum.

But there's still a problem with the phrasing of your question. It's still not right to look at RPS as a line segment with "GTO" on one end and "exploitive" on the other end. It's more subtle than that.

Instead of doing that, try looking at RPS as a Venn diagram with 3 circles. One circle represents rock, one represents paper, and the third represents scissors. Whatever mix of the three you are choosing, you can visualize as a point in one, two, or all of these circles as follows:

-If your strategy is unbalanced toward rock, but roughly balanced between paper and scissors (for example 60/20/20), you'll be in only the "rock" circle. Same for the other two pure strategies (P and S).
-If your strategy is jointly unbalanced toward rock and paper, at the expense of scissors (for example 40/40/20), you'll be in the intersection of the R and P circles. Same for the other two combinations of 2 pure strategies.
-If your strategy is roughly balanced between all three, you'll be in the intersection of all circles.

In this diagram, the GTO strategy of exactly equal probabilities of R/P/S would be the dead center of the diagram.

The point I'm trying to make is that even in cases where it makes sense to think of a game as a spectrum, the "spectrum" need not be only 1-dimensional, AND it's a spectrum of exploitive plays, with GTO play taking up one point in the middle of the spectrum, not at any end of it.

To use a poker example, let's say you are betting the river with a polarized range (everyone's favorite toy example). It is not correct to say that your value/bluff percentages are a spectrum with "balanced" on one end and "unbalanced" on the other end. Instead, the spectrum is "unbalanced/no bluffs" on one end, "unbalanced/all bluffs" on the other end, and somewhere in the middle (not the exact middle depending on your bet sizing) is the "balanced" percentage of bluffs.

Due to the multitude of different strategies in NL, it's basically impossible to answer this question with any degree of confidence except in a few cases that are so trivial that they're not worth anything.

In order to prove that a strategy is dominated, you'd have to prove that there was a different strategy that did better against all strategies, not just all strategies that you consider to be good.

Let's say you had some specific strategy in mind that you were trying to prove was dominated. You'd have to find a dominating strategy. In practice, this is already pretty much impossible. And then even if you think you could do it, you'd go down the rabbit hole into other virtually-impossible-to-answer questions.

Here's an example. Let's say you are playing 10-handed 1/2 NLHE with no rake, and everyone has $1000 stacks. Let's start by trying to prove that open-folding every hand UTG is a dominated strategy.

In this case, it's easy to prove that it is. The reason is because it's very easy to come up with a strategy that outperforms it across the board. That strategy is to open-jam AA and open-fold everything else UTG.

Even if your opponents were playing perfectly, and knew what you were doing, they couldn't do anything about this strategy. They'd never call unless they also had AA, and then you'd have 50% equity and you'd chop the blinds up. And if no one else has the other two aces, you just win the blinds outright. (Of course, if they ever called you with worse, you'd do even better.)

So the new strategy wins the blinds or more approximately 1/221 of the time, and all other times it equals the strategy of always folding. That means it dominates the "always fold" strategy.

However, probably something is sticking out to you now. The strategy of open-jamming AA and open-folding everything else UTG is probably also dominated, isn't it? It looks like a crappy strategy. It ought to be easy to find a strategy that dominates it.

Well, it is really hard to prove that anything dominates that strategy. Off the top of my head, I can't do it. I would never play this strategy in real life, and neither would you, and probably for good reason, but I can't prove that that strategy is dominated.

Here's an example of a strategy that you might think is easy to prove dominates the "jam AA and fold otherwise" strategy--open-jamming AA and KK and open-folding everything else. The problem with that strategy is that it gives our opponents a viable counter-strategy: call a jam with AA and fold everything else.

If you only open-jam AA and KK, folding everything else, and your opponents call your jam with AA only, you will win the $3 in blinds roughly twice as much as when you were only jamming AA--but in return, you face the hugely negative outcome of jamming with KK and someone calling with AA. It turns out that when stacks are $1,000, the negative associated with running KK into AA outweighs the extra blind wins. (This follows easily from the fact that the Sklansky-Chubukov number for KK is less than 1000.) So because we have found a counter-strategy that "aces only" does better against than "aces and kings only", the latter strategy does not dominate the "aces only" strategy! (However, there is a smaller stack depth at which it actually would.)

There's a ton more I could say, but I'm going to stop here. Hopefully this little example is enough to show how ridiculously hard it is to be able to claim and prove that a strategy is dominated (and this is only for a single street of all-in or fold play). It's a much, much higher standard than just claiming a strategy is bad.

The push/fold situation you described reminded me a bit of the Nash Equilibrium chart.

http://www.holdemresources.net/h/pok...une/about.html

The "About the Solution" page mentions the opinion that due to numerous "arbitrary choices" at work, it would be quite unlikely for two independently produced charts to perfectly match. It seems really interesting to me to consider the point at which a HU tournament fold becomes exploitable at a certain stack depth. Also makes me wonder how many other decisions might be analyzed in the coming years.

People seem to have this misconception that GTO play is different for max exploitable. It actually means the same thing, GTO never makes a negative ev play for the sake of balance. GTO is always trying to maximize the ev of it's play.

No.

Try actually reading the thread.

I had actually never seen one of these things before.

My immediate reaction is that if the strategy were truly GTO, all independently produced charts should match (unless the Nash equilibrium is not unique, which is unlikely but possible, but somehow I doubt that's what's causing the discrepancy). Probably what happened is that the authors made some simplifying assumptions to do their calculations, and those assumptions are not entirely true.

I'm now kind of curious as to exactly what they did to produce this, because I expect they would have had to make a ton of simplifying assumptions and there'd be a lot of room for a wrong assumption. Let me try to explain why I think this is the case.

Let's say you were playing push/fold and you fixed a stack depth. Now let's say that you were trying to create a game matrix for the push/fold game, like I did for RPS in the OP. The rows would be pushing strategies, the columns would be calling strategies, and the entries in the matrix would be the EV of pushing a certain range when your opponent is using a certain calling range.

How big would this matrix be?

Well, there are 169 different equivalence classes of hands, right? (13 pocket pairs, 78 suited hands, and 78 offsuit hands.) So for each hand, there are two possibilities: either you push it or fold it. Each individual "push" strategy has to make a push/fold decision 169 times. Similarly, each individual "call" strategy has to make a call/fold decision 169 times.

So the game matrix will be a square matrix where the number of different strategies is 2^169--which is on the same order of magnitude as a 2 followed by 56 zeroes.

This number is so huge that it cannot be the case that they computed the entire game matrix. They must have made many simplifying assumptions, and I don't know all the assumptions they must have made. I will say, though, that the matrix is way easier to simplify correctly for short stacks, because a LOT of strategies can be taken out as being dominated when stacks are short, and that might make the reduced game manageable. I would tend to believe that they got it right or almost right for short stacks, and that is probably why the caveat about not using the chart once you get past a certain stack depth is there.

And by the way, this level of computational complexity is just for push/fold at a fixed stack depth. Each stack depth has a matrix this huge, and also, once you allow for varied bet sizing, it gets exponentially more complex, because now the number of decisions increases and the number of things you can do at each decision point also increases.

In MOP, Chen and Ankenman describe using fictitious play iteratively to determine their tables for the jam-or-fold game. (For those who haven't read it, I will tease you with their determination that at around 833BB, they go from jamming with only AA to jamming AA plus another hand and the second hand they add to the mix is not KK or AK). Since they say that the strategy will converge on optimal strategy given enough iterations, one arbitrary choice is how many iterations is enough. Using their method, one can decide to stop after n iterations or to stop when the difference between n and (n+1) is sufficiently small. The mixing weight in fictitious play is possibly also something that has some arbitrariness to it. It's easy to believe that this could account for small differences, but keep in mind that I know nothing about FP.

You are the one that needs to actually read the thread if you don't understand that simple concept about it.

You are incorrect. There have been several posts showing why you are incorrect.
GTO is about playing a strategy that your opponent has no incentive to modify their play to improve their expected value. It ignores the actions of your opponents - your opponent could fold to 5bb bets and call 100bb bets 100% of the time, and you would still make the same plays.

An EV maximization strategy is entirely dependent on how your opponent plays.

Look up how an equilibrium is reached.