Hi,

There's an argument going on in this thread in LLSNL.

pocketzeroes is arguing that given a GTO strat in HUNL, we can vary our frequencies for any of our options and not damage our results vs a GTO opponent. For example, let's say facing a raise preflop with JJ, it turns out to be GTO to threebet 90% of the time and flat 10% (ignore the question of varying threebet sizing, for the sake of argument). pocketzeroes claims that we can instead threebet 10% of the time and flat 90% and that this will not affect our results against an opponent playing GTO. Further, that we can do this with all our hands (i.e. also change our threebet frequency with TT from 90% to 10% but change our threebet freq with 45s from 10% to 90%, say) and that it is guaranteed not to matter against a GTO opponent who will not change their strategy. I say this is nonsense. Who is right?

Start reading at post 114 in the thread, including the post I quote in that. Can post in support of me or pocketzeroes either here or in the LLSNL thread. Thanks for any assistance!

Please hurry. Broken, Arty, Bob, Doctor?

Yes it works like that. EV equal -> frequencies doesn't matter if GTOvillain is not allowed to change his strategy to exploit you.

OK, I'll crosspost my response from LLSNL here. What's wrong with this argument?

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I hate you guys for making me think this hard, but here's a disproof by counterexample of the idea that mixed strategies can always be varied without penalty.

Imagine a game between players A and B, where A has either been dealt a 1 or a 0, with 50% probability each.

In round 1 of the game, if A has a 1, he may choose to turn it face up or keep it hidden. If he chooses to turn it face up, B has to pay him 1 coin.

In round 2 of the game, B may wager that A has a 0. If he is correct, A has to pay him 10 coins. If he is incorrect, he has to pay A 20 coins.

It's clear that A cannot turn up his 1 100% of the time. If he does, B can always wager in round 2 whenever A does not turn his card up, always winning 10 coins off A. What A has to do is ensure that B cannot exploit him in round 2. To break even in that round, he needs B to be wrong once for every two times he is correct. Therefore, he needs to have 1 in 3 odds of having a 1. He can achieve this by mixing his strategy such that he only turns over his 1 50% of the time. This means that heading into round 2, he will have a 0 50% of the time, a hidden 1 25% of the time and an exposed 1 25% of the time. This prevents B from having a profitable wager in round 2. A's expectation with this strategy is 0.25 coins per iteration, gaining 1 coin on the 25% of occasions he turns his 1 over.

So here we have a game where A is playing a mixed strategy. Yet it is not the case that it doesn't matter what his frequencies are in round 1. If he unilaterally changes his strategy such that he never turns over his 1, his expectation in the game is now zero.

I set up this game to mirror my understanding of what is happening in HUNL. Getting dealt JJ is a naturally profitable situation. However, if we go for maximum profit on round 1 of the game (ie preflop), our range may become too weak on later streets. We need to sometimes skip going for the immediate value in order to strengthen our range for later. However, this doesn't mean that it is OK to never go for the immediate value.

pocketzeroes is correct.

Basically it boils down to this:
Against a static strategy, the EV of your non-JJ hands is completely independent from the strategy you use for JJ. So you definitely don't affect the EV of other hands by changing your strategy for JJ, your opponent is still playing exactly the same against those other hands.

And changing the frequencies for JJ between several choices of equal EV also doesn't change your EV for JJ itself. (The choices need to be exactly equal EV in order to be played mixed in a NE.)

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Then the two options were obviously not equal EV, and if they aren't then they can't be played mixed in a NE. You didn't mention B's strategy, but you seem to assume that B doesn't wager at all. Your are not in a NE in that case, A could exploit this by simply turning over all his 1s in round 1.

In your game the NE would look like this:
A: Turn 1s 50% of the time in round 1.
B: Wager 5% of the time in round 2.

A is indifferent to turning, he either receives 1 immediately or 20 in the 5% that B wagers.
B is indifferent to wagering, as he is exactly breaking even.

pocketzeroes is correct in mentioned thread. I have to say that he explained it very good...

That is my understanding too.
If mixed strategies arise with both options having the same EV, and villain never alters his strategy, it doesn't matter if you always pick the same option. It's got the same EV as the other one, after all!

If one option had a higher EV than the other, then you should always take the higher EV line. e.g. it would make no sense to flat with JJ if 3-betting had a higher expectation.

I think it goes back to the definition if NE. Once equilibrium is established, no player can gain an advantage over any other player by changing strategy.

This does not mean that the player who deviates from NE will achieve the same monetary results by doing so. It means the player who stays at NE does not care what the other player does, and will stay at or above the expected monetary results.

If it is the same EV to call or raise with JJ, then both actions are at equilibrium and doing either with 100 percent frequency will not change the GTO player at all.

But I think the confusion arises when we mere humans actually want to play poker and have fun doing so.

For instance, a human will fold the JJ some percentage above zero, and lose money doing so.


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Thanks all for chiming in.

I remember I once thought this concept a little strange too, as it seems to make a GTO opponent weaker, or at least not difficult to break even against... However, once you realize that most actions will still be quite limited vs the number of choices available, it becomes obvious that GTO is still not easy to play against, and will still win a lot of $ from any human. Also, balancing frequencies would happen more across a full range of hands rather than within a single hand. I.e., most situations with a given specific hand probably require a pure (not mixed) strategy - though I'm not 100% certain of this.

Thanks guys. I think I understand.

One of the far more enjoyable and potentially profitable features of poker played between humans is the psychology involved, and the opportunity to consider what a human might think/do in a certain spot. All of that disappears vs. a GTO playing computer, our cunning is rendered useless.