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Originally Posted by Axel Foley
Hey all. I know I'm late to the party, and I apologize for that.
No big deal. Thanks for contributing!
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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?
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.
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Second question: In terms of dominated strategies, which lines in NLHE do we think fall into this category?
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.