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bogdan314's picture
Nash Equilibrium for A9o vs. first hand shove in hyper

   Hi everyone,

I've been studying the Nash shove/call tables a little and especially from Will Tipton's explanations of the reasoning behind them, I understood it 100%. I'm also a game theory and statistics passionate, so it all makes perfect sense.

However, I've imagined myself a scenario where I'm starting to doubt the reasoning behind Nash.

Scenario

First hand in a hyper (25BB deep); unknown opponent shoves from the small blind. I have A9o. I know nothing about this guy, so I'm gonna play readless until I get some hints about his strategies and then start to adjust.

Nash solution

I'm looking at the Nash calling table. It's telling me A9o is a call for maximum 38.2 BB deep. I'm 25BB deep so that qualifies.

Understanding Nash solution

1. The Nash table is telling me that if an opponent plays "perfectly", i.e. he is pushing unexploitable (Nash pushing ranges), then I must call with A9o. The reasoning is pretty simple to understand: his 25BB deep pushing range is 22+, A2o+, KTo+, QTo+, J9o+, T9o, A2s+, K4s+, Q6s+, J7s+, T7s+, 96s+, 86s+, 76s, 65s (I got this from Will Tipton's book). My hand is only slightly behind all pocket pair, is dominated by AT-AK,  but it dominates A8-A2, K9, Q9, J9, T9 and is a favourite over all the other holdings. So it's obviously +EV here.

2. According to Nash definition, if my opponent does not push according to Nash push tables, my expectation does not decrease, but rather it increases. I am doing fine by calling with A9o no matter what his range is. On one hand, it is true that for most of his deviations I can find a better strategy (the maximum exploitable strategy), but since right now I don't know what his strategy is, Nash assures me I'm not doing a bad thing calling with A9o. I'm only not doing the best thing, but still better than folding.

Doubting Nash solution

What I said above makes perfect sense, it's what the theory says (plese correct me if I was wrong).

However, I'm starting to have second thought by analyzing the following situation: let's say Villain is in reality a strange player who likes to push 25BB deep only with exactly the hands that have me dominated: 99+, AK-AT. And he is folding everything else.

What does Nash say? It says: don't worry! Indeed if Villain happens to be this type of guy, by calling with A9o you are -EV. But this is balanced by the fact that he is folding too much. So all these blinds you pick up when he folds (which is very often) will make up for the EV you lose when you call A9o against his strong, dominating hands, especially taking into account that even when you call you still have some equity. Indeed, there is a better strategy here (simply fold everything but monsters and you win all his folds - easy game with higher profit), but since you can not know for the moment if his strategy is trully this, you may stick to playing Nash which still gives you a small profit over folding.

This is how I understand the game, please correct me if I'm wrong.

 

Now after all this lengthy introduction (I hope I haven't bored you to death so far), there's my question: what if Villain does not "fold everything else", but rather makes a small raise and play very well postflop with everything except the hands I described (99+, AK-AT) and of course without total junk. In this case, since I'm sticking to Nash because I don't know his strategy, not only am I -EV when calling with my A9o, but also I don't have all his folds anymore to compensate. Am I still fine by calling the shove? Because Nash doesn't take into account the minraise and postflop play, it only assumes Villain is either shoving and folding. Am I still calling unexplotably with A9o?

 

Please I need an answer only from people who understand the Nash perfectly. I don't want answers like "Nash is crap anyway, it's stupid etc".

Thanks!

coffeeyay's picture
I think your confusion is

I think your confusion is coming from the fact that you're not describing a Nash equilibrium to 25bb hu poker, but instead to a sub game: one in which villain is either shoving or folding oop and you are only minraising or folding ip. In practice the push fold oop is a bad assumption to make, both because it is definitely better for villain to include other options and because in practice almost no one plays this way. The ip assumption also isn't relevant, but isn't correct (the full game likely includes limping and possibly open shoving) but it's not as big a deal as the oop assumption.
 
it is better to try to model your villains actual 3b shove range (using reads or population data) and then make the correct calls vs this range (ie maximally exploit it). alternatively you could solve for the actual unrestricted Nash equilibrium, but gl with that one ;)
 
hope this his clears it up, let me know if you have further questions.

bogdan314's picture
So what you mean is that A9o

So what you mean is that A9o is a call only if I know he is playing shove/fold? If yes, how could I evaluate whether it's still better calling than folding even if I know he's not pushing all his range?

bogdan314's picture
Oh and please read the post

Oh and please read the post again, I was assuming I was the big blind and Villain is shoving in the small blind.

coffeeyay's picture
Oh my bad. Basically same

Oh my bad. Basically same point though the Nash equilibrium applies to the sub game in which villain plays push/fold ip. At 25bb it is not an aacurate assumption, so it's not useful for decision making.
 
to decide whether to call or fold you need to compare the two options and see which has higher ev. At 25bb you are calling 24bb into a 50bb pot, so you need 224/58 = 48% equity vs his calling range. So put him on a range and stove it :) alternatively you can use software like Icmizer to evaluate  is. I did a ton of this kind of stuff in my Math in HUSNGs if you want to learn more.

bogdan314's picture
Thanks for the tips but I

Thanks for the tips but I know those things already.
My question is what to do readless, i.e. I can't "put him on a range".

cdon3822's picture
Make assumptions based on

Make assumptions based on what the population typically does and design ranges to exploit them. 
Then run a sensitivity analysis to determine the correct adjustments to deviations from typical population ranges. 
Adjust accordingly as you gain information. 

coffeeyay's picture
I think the most accurate way

I think the most accurate way is to use pt4(or hem). You can get population open shove frequencies, and if you build a population alias you can also get a hand range visualizer. This will allow you to build a range (probably mixed). I go into detail in my video pack if you're having trouble. You can also just guesstimate based on frequency (that's easy to get) and glance through the showdown hands to put villain on a range and at least get some idea do what are clear calls and what are clear folds.

3onthego's picture
I find that monsters and

I find that monsters and small pairs alike like to exploit our A9o by open shoving more so than A5o. This is not really good news for A9o and a ton of starting hands like it. 
These big hands are exploiting the fact that they appeared very early on and that you are readless except for the fact that his open shove % is close to 100%.
It is much more plus +ve to wait a bit and see rather than just calling because you have a reasonable hand.
In fact what I like to do in this situation is fold A9o after a long pause. This gives him the impression that you have a hand like A9o and therefore a difficult decision. By folding we are crediting him with a monster & thereby implying that we think open shoving is a strategy employed by monsters.
Then open shove right back at him the next hand with any two cards. I bet he doesn't call with A9o if he has it. And in this way you use his strength against him in a Bruce Lee way. Furthermore if you reply to his open shove with an open shove he will often drop the strategy.

coffeeyay's picture
So I figured I'd give a

So I figured I'd give a couple of results since before I was only talking about method.

Using the method described in my vid pack and database based at the 7s 15s and 30s on Stars Hypers,  25bb deep A8o A6s are the worst Ax with which we should be calling open shoves (we should call with those and all better). I won't go through the whole calling range, but interesting things to note are that KQo and 33 is a fold, while a lot of suited hands (ex T8s) and 44+ are calls. 

cdon3822's picture
It's pretty interesting if

It's pretty interesting if you work through the math facing open shoves @ 20-25BB. 
A lot of people only open shove low pocket pairs at these depths which makes a lot of suited connectors a mistake to fold (as you mentioned above).
Out of curiousity, you calling off here when you hold such hands (eg. Tx9x)?
 
You back your analysis of typical population ranges here and call off.
You implicitly give your opponent credit for playing such holdings this way. 
Are you calling off the next time, knowing that if he adjusted correctly to seeing what you called off with, he now has a whole heap of much stronger hands in his open jamming range? 
 
How much are you preemptively adjusting your ranges to how you expect villain to adjust? 
 
This begs a larger question about readless play => what pecentage of the population can we give credit to being thinking players?