Heads up Theory

[eastbay]

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If you could show me why what I said is stupid, I'd really appreciate it.

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Come on, Jman.

I said that taking -cEV pushes appears to be always wrong.

You then "countered" (with style, no less) that passing on +cEV pushes can be right.

Don't you see that a strategy can pass on +cEV pushes and yet take no -cEV pushes? That these are two completely different things?


Your "counter" is a non-sequitur. I didn't say passing on +cEV pushes was always wrong. I said taking -cEV pushes is.

eastbay

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Thank you. I guess I misread this:


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there are no circumstances HU where taking -cEV situations is superior to some other strategy which never does.

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to mean that no situation where making a play that is -cEV is best, whether that be pushing or folding.

By 'situation' I assumed you meant 'situation' and not 'push'


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No, that just happened to be the context you had brought up to attempt to display something (which didn't make sense and still doesn't). The principle is the same for calling, for example.

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ICM says a push here would be -EV (only -.125%)


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Why are you invoking ICM? ICM is irrelevant HU.

I suspect at this point there is some confusion over the use of EV of a play vs. differences in EV by comparing different plays. It seems you are using "EV" to actually mean "the difference in EV between two different moves."

This could cause you to confuse things like what I was calling "passing on +cEV" and "taking -cEV." This helps explain your non-sequitur.

The reason you can't discuss EV in terms of differences like this in general is that there's often more than two options (and really there are always a very large number). Now you can't talk about the "EV of the play" in the sense you're using because it isn't clear what you're comparing it to.

So let's get on the same page. When I say EV I am talking about the cEV of the play, not the cEV _difference_ between two different plays. Clear?

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However, Lawanda gets very scared when she gets below 3400 chips. When she has less than 3400 chips, she folds every hand except for QQ+ in the SB, and doesn't call a push without AA.

So, pushing, while slightly -EV according to ICM, would be best since you would immediately get Lawanda to under 3400 chips and then run over her from there.

Please now type some short angry comment, then allow me to ask you to expand on it, and then tell me why this example doesn't satisfy your requirements.

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My requirements for what?

You might want to look at some of my old posts refuting Sklansky's assertion (he calls it a "proof", even) that HU equity must be linear. I think you're getting at what I've said many times in those posts.

I think it is clear that when you add absolute stack dependence to a strategy, linear equity HU is wrong. You can make it a step function in the extreme case.

So I guess I'm agreeing with you that for strategies that exhibit absolute stack dependence, you can generate this kind of pathology.

However, I think what you are saying is that this effect also exists in non-pathological strategies (to be a little more precise 'something that might earn me money to know about'). So I'm back to my original challenge: try to find a reasonable example. I've looked, and never been able to find one. What I found is that strategies blew up when they started choosing -cEV plays. Of course, I didn't use a pathological counter-strategy of folding everything below 5000 chips and pushing everything above. I used counter-strategies based on stack/blind ratios and ranges that approximated both "correct" and "guesses at typical" play.

If you're just saying that you can construct these pathological examples, then there's no disagreement. Try searching for "step function HU" or something to see where I've often said the equivalent in the past.

eastbay

[dfan]

Didn't see your post A_Plus before I wrote essentially the same thing. I did add the general form though, which to me seems like it would be true. Not as sure about the specific blind size hypothesis, but can't see any holes in it either.

[gumpzilla]

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Thank you. I guess I misread this:


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there are no circumstances HU where taking -cEV situations is superior to some other strategy which never does.

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to mean that no situation where making a play that is -cEV is best, whether that be pushing or folding.

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I agree that eastbay's statement was unclear at best, since "situations" are going to include more things than pushing. I also think that the old specter of the double usage of EV - to describe the results of a particular move or a particular situation - is showing up here. One could certainly consider a move -EV in a relative sense - even if it is +EV in an absolute sense in that you expect to gain chips by making the move - if you are passing on better absolute +EV plays to make this one. I don't think this is what eastbay means, but this interpretation would make sense, too - strategies that routinely pick less than optimal moves at each point are going to be inferior to those that always pick optimal moves. This seems kind of tautological, but I guess it depends on how you define optimal.

I've seen people mention both cEV and ICM $EV in this conversation. People do realize that heads up these return exactly the same results, right?

EDIT: I see while I was writing this eastbay wrote and covered pretty much all of the things I just mentioned and then some.

[Jman28]

Thank you for finally contributing positively to this thread.

The reason I used ICM is that your 'Power Tools' is the easiest method I have for quickly calculating the EV of plays. I understand that ICM is unnecesary in heads up calculations, but it doesn't hurt them, so I used what was easiest.

I appreciate you changing your tone, and I will do the same.

I'll search for the thread you are talking about and get back to you on these ideas later tonight (maybe much later) because I'm at 8 tables right now.

[Jman28]

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Dude, you owe me like an hour of my life back. why you introduced the ICM $EV calcs into this I will never know. Is what you are saying that typically, opponents make more mistakes with high blinds and widely varied stacks? So when given a choice that is EV neutral pick the situation they are more likely to make mistakes in?

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I think this is pretty much it, yes. It seems that it can be put into many different forms.

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If so, gotcha. Dont know if I agree, but I see your point.


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Good enough for me.

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Now that $EV stuff was just a bad example. You see why that was complete BS right?

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Other than the fact that ICM is of no more value than cEV, which I just addressed in my last response to Eastbay, no, I don't see why.

It was another way to explain myself since I wasn't coming across as clearly as I wanted to. Why is it a bad example?

[microbet]

Ok, I didn't notice, but this was my Carpal Tunnel post. I was hoping it was the one in the Science, Math, Philosophy room that David Sklansky called a stupid comment.

Carpal Tunnel in under a year; that's a lot of goofing off.

[Jman28]

Hey... I'm having trouble finding your (eastbay's) posts that you (eastbay) mentioned.

If anyone could point me in their general direction, I'd appreciate it. Thanks.

Are they in the archives?

[gumpzilla]

I think your best plan is to look through the shadow's favorite threads post. I'm pretty sure some of the stuff is linked in there. There's also this lengthy thread from May that is very ICM focused that I think will lead you one way or another to where you want to go.

[Jman28]

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I suspect at this point there is some confusion over the use of EV of a play vs. differences in EV by comparing different plays. It seems you are using "EV" to actually mean "the difference in EV between two different moves."


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You are right. This is how I was using it.
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This could cause you to confuse things like what I was calling "passing on +cEV" and "taking -cEV." This helps explain your non-sequitur.

The reason you can't discuss EV in terms of differences like this in general is that there's often more than two options (and really there are always a very large number). Now you can't talk about the "EV of the play" in the sense you're using because it isn't clear what you're comparing it to.


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I'll take your word that I'm using the vocabulary incorrectly because you know more about this than me. However, is it a big deal? at all? I'm honestly asking.

It's clear that we are either pushing or folding in all of the situations I described (I think) so they are the only two options we might compare. Anyways, I don't wanna get hung up on this technicality.

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So let's get on the same page. When I say EV I am talking about the cEV of the play, not the cEV _difference_ between two different plays. Clear?


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Cool. I'll do my best to use it the same. No promises though.



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You might want to look at some of my old posts refuting Sklansky's assertion (he calls it a "proof", even) that HU equity must be linear. I think you're getting at what I've said many times in those posts.


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I'm still searching for these posts, but I'm thinking I may not be arguing the same thing that you were.

If David specified that the two players are of equal skill (with no weird-ass absolute stack dependent strategies like I made up), I think he is correct that heads up equity is a linear function. (This is just what I believe intuitively, and from what I've read in his books. I've done nowhere near the research you have into it)

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However, I think what you are saying is that this effect also exists in non-pathological strategies (to be a little more precise 'something that might earn me money to know about').

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Yes. This is what I'm saying, however with the disclaimer that the difference it makes is probably very minimal.

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So I'm back to my original challenge: try to find a reasonable example.

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I think I've already explained what you are looking for. Let me know if I misunderstood:

When HU, with chip stacks both around 5k, correct strategy suggests pushing less and calling less than if the stacks were, say 3k and 7k. Since our average opponent doesn't push and call pushes much, he doesn't play THAT far from optimally

When the stacks are far apart (up to a point), say 2500 and 7500, our average opponent is making more mistakes than he does.

This is somewhat similar to the example of the beautiful Lawanda, although much less obvious. Her leak was that when below 3400 chips, she didn't push or call enough. Our average opponents leak is that they never push or call enough heads up. However, that leak is more pronounced when one of the stacks is below 3k chips. (more pronounced at 4k /6k than at 5k/5k, most pronounced probably around 2k/8k)

Therefore, just like we brought out Lawanda's leak by getting her below 3400 chips, we can bring out (or really just magnify) the leaks of our average opponent by getting them into situations where the chip stacks are nearest to (complete estimate) 2k/8k.

Just it was in our best interest to sometimes make a -cEV play (am I still saying this wrong?) against Lawanda in order to take advantage of her leak, it is in our advantage to make similar plays (although they have to be very close, I assume) against our average opponents because of their leaks.

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I used counter-strategies based on stack/blind ratios and ranges that approximated both "correct" and "guesses at typical" play.

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My assumptions are that players don't adjust properly to stack/blind ratios, and that the smaller the stack/blind ratio gets, the further away our average opponent gets from optimal strategy.

I believe that I can define the parameters for a simulation of this for you, but I have no idea how to do it on my own. Nor do I have any idea how hard it is. If it's a huge deal, obviously, don't worry about it.

I think we are on the same page now. I hope.

[Jman28]

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I think your best plan is to look through the shadow's favorite threads post. I'm pretty sure some of the stuff is linked in there. There's also this lengthy thread from May that is very ICM focused that I think will lead you one way or another to where you want to go.

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Thanks. I skimmed this thread and a few of the ones it links to. Still haven't found Sklansky v Easbay debate. Anyone who can find it = my hero.

I looked at JNash's S-Curve Hypothesis and though I've only been thinking about it for the past 5 minutes, I think there MAY be some merit in it. However, it doesn't or shouldn't apply to heads up poker, so I'm not gonna worry about it for now.

I'm confused as to why it was brought up so often in the post you linked me to.

My current stance:

I agree with Sklansky that two evenly matched players' equity is linear when heads up.

I believe that since our average opponent is not evenly matched with us, there are some exceptions to the linearity based on the way our opponents play at different stack sizes.