Heads up Theory
 

Hey guys. I don't know if this is something that has been talked about or not. I responded to a question/critique in this post and discussed something that I think about while playing. I realized that I'd gotten it from nowhere and that I hadn't heard it talked about, so I wanted to hear some comments.

If it's a new idea, I'd like it to be talked about and named after me, so that I feel important. It's not worded that well so let me know if you need clarification.

Here is a copy of my response:


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***** Hand History for Game 2503905585 *****
500/1000 Tourney Texas Hold'em Game Table (NL) (Tournament 14678948) - Tue Aug 09
02:58:28 EDT 2005
Table Mini Step 5 1014193 (Real Money) -- Seat 8 is the button
Total number of players : 2
Seat 8: skinsftbl (6690)
Seat 10: aks47 (3310)
skinsftbl posts small blind (250)
aks47 posts big blind (500)
** Dealing down cards **
Dealt to skinsftbl [ 2c, 9c ]
skinsftbl raises (6440) to 6690
skinsftbl is all-In.
aks47 folds.
Creating Main Pot with $7190 with skinsftbl
** Summary **
Main Pot: 7190
skinsftbl balance 7190, bet 6690, collected 7190, net +500
aks47 balance 2810, lost 500 (folded)

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I don't think this one pencils out.

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I'm not exactly sure about his range here, and if the push is +EV, but this is an example of a play that I sometimes make with thinking that I haven't heard mentioned anywhere.

Heads-up, I often make very marginal pushes because they will lead to more +EV situations. Here's what I mean...

When you are the bigger stack, generally your pushing range gets wider as the smaller stack-blind ratio goes down. (until he almost is so small that he has to call with any two). This means that the smaller the short stack is, the more advantageous it is for the more aggressive player.

If I fold this hand, he has ~3500 chips next hand. If I push and he folds, he has ~ 2800. Now obviously it's always advantageous to pick up chips, BUT I believe there is an intrinsic advantage to DECREASING the size of the smaller stack.

Therefore, when faced with marginal heads up decisions, I err on the side that decreases the smaller stack. That means folding if I already am the smaller stack, and pushing if I'm the bigger stack.

I'd like to hear thoughts on this one.

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Re: Heads up Theory
 

No $hit. The advantage to heads up play in a game where the stack/blind ratio is so small is making your opponent fold more hands then he is supposed to.

Thats why a pushing any two strategy while heads up isnt too far from optimal.

The secret to heads up poker. Voila!

Re: Heads up Theory
 

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No $hit. The advantage to heads up play in a game where the stack/blind ratio is so small is making your opponent fold more hands then he is supposed to.

Thats why a pushing any two strategy while heads up isnt too far from optimal.

The secret to heads up poker. Voila!

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This response makes you think you didn't read my post.

That, or my post is worded even worse(ly?) than I thought.

Re: Heads up Theory
 

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No $hit. The advantage to heads up play in a game where the stack/blind ratio is so small is making your opponent fold more hands then he is supposed to.

Thats why a pushing any two strategy while heads up isnt too far from optimal.

The secret to heads up poker. Voila!

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This response makes you think you didn't read my post.

That, or my post is worded even worse(ly?) than I thought.

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Its exactly what your post implies, except I would use my statement more as a fundamental theorem than yours.

And I disagree in NOT pushing as the small stack. Why? The same reason you start making desperation pushes at level 4/5.

Make your opponents fold is key to HU, and this applies whether you are a big stack or a small stack. And the way to apply maximum pressure for them to fold is to go all in.

Re: Heads up Theory
 

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No $hit. The advantage to heads up play in a game where the stack/blind ratio is so small is making your opponent fold more hands then he is supposed to.

Thats why a pushing any two strategy while heads up isnt too far from optimal.

The secret to heads up poker. Voila!

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This response makes you think you didn't read my post.

That, or my post is worded even worse(ly?) than I thought.

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Its exactly what your post implies, except I would use my statement more as a fundamental theorem than yours.

And I disagree in NOT pushing as the small stack. Why? The same reason you start making desperation pushes at level 4/5.

Make your opponents fold is key to HU, and this applies whether you are a big stack or a small stack. And the way to apply maximum pressure for them to fold is to go all in.

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Please reread my OP.

Re: Heads up Theory
 

If your name is "Unexploitable Game Theory" I think you've got a very good chance of having this named after you.

Re: Heads up Theory
 

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If your name is "Unexploitable Game Theory" I think you've got a very good chance of having this named after you.

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[censored], that's my brother's name. He always gets all the credit.

Re: Heads up Theory
 

OMG, I just stated in more fundamental terms WHY you want to push PERIOD, whether you are a big stack or a small stack.

Uh, why is it an advantage for a big stack to take 700 chips from a small stack?

Answer: The small stack has less chips to double up with and is forced to push with a mediocre hand.

Uh, why does a small stack want to steal from a big stack?

Answer: To maintain folding equity and keep afloat in the SNG.

And how do both types of stack obtain these goals?

By going all in.

CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

When the EV difference between pushing and folding are minimal, and you are heads up, err on the side that will decrease the size of the smallest stack (increase stack disparity).

*assuming you are more aggressive than your opponent heads up.

Re: Heads up Theory
 

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OMG, I just stated in more fundamental terms WHY you want to push PERIOD, whether you are a big stack or a small stack.

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OMG OMG

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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When the EV difference between pushing and folding are minimal, and you are heads up, err on the side that will decrease the size of the smallest stack (increase stack disparity).

*assuming you are more aggressive than your opponent heads up.

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DUH. DUH. DUH. I was just explaining why you want to do that.

In fact, Ill put it more succinctly for you:

(PUSH ANY TWO)

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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When the EV difference between pushing and folding are minimal, and you are heads up, err on the side that will decrease the size of the smallest stack (increase stack disparity).

*assuming you are more aggressive than your opponent heads up.

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DUH. DUH. DUH. I was just explaining why you want to do that.

In fact, Ill put it more succinctly for you:

(PUSH ANY TWO)

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You were not explaining what I said at all.

I am saying that there is an intrinsic advantage in increasing stack difference (until the point before you have very little FE).

This means that when a decision to push or fold is EV nuetral, it is better to increase the stack size disparity (THAT MEANS FOLDING AS THE SMALL STACK).

You clearly don't understand this.

I really shouldn't be explaining this to you since you're being a huge floppy weiner, and I will not respond to you further in this thread if you don't say anything constructive.

Re: Heads up Theory
 

Let me make sure I understand...

The bigger the difference between 2 stacks heads up, the more +EV pushing any hand becomes for either the big or small stack. Therefore, as a small stack facing a marginal push/fold situation, it would be correct to fold, sacrificing chips, but increasing the gap between the 2 stacks providing the opportunity for more +EV pushes in later hands. As a big stack it would mean pushing these marginal situations.

Yes?

Re: Heads up Theory
 

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Let me make sure I understand...

The bigger the difference between 2 stacks heads up, the more +EV pushing any hand becomes for either the big or small stack. Therefore, as a small stack facing a marginal push/fold situation, it would be correct to fold, sacrificing chips , but increasing the gap between the 2 stacks providing the opportunity for more +EV pushes in later hands. As a big stack it would mean pushing these marginal situations.

Yes?

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Yes, this is pretty much it. And better worded than my post, I might add.

I'm a little uncomfortable with the 'sacrificing chips' wording, because it may lead some people to think that I'm saying you are better of with 30% of the chips than with 31%. You are not.

I'm pretty confident that you understand this though.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

Hi,
OMGOMGOMG. I understood your post on the first read-through. Not sure what the problem is.. lol.

Anyway, I couldn't think of a logical mathematical argument for why this should be true. I think that a given number of chips in a pot preflop should have the same value regardless of whether you're the big stack or the small stack (assuming exact opposite chip distributions in each case). But I think yourfoxygrandma gave me an idea for a non-mathematical argument.

If you are a better heads up player than your opponent (which often means more aggressive in these games), you should forego marginal pushes if getting called and losing will bust you. This is because you can find more +EV spots later as long as you're still in the game. However, if you're the chip leader, then you should take these small +chip EV pushes because even if you lose the hand you can still continue to play, and continue to exploit your +EV opportunities. Basically you're saying there's an intrinsic value for simply surviving if you're a better HU player than your opponent, whereas most models suggest (I think) that it's all about your chip stack. I like the idea, but I'd like a more rigorous proof to fully convince me that this is true, and the chips you're sacrificing by folding as the short stack really are less valuable in terms of $EV than if you're the big stack.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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If you are a better heads up player than your opponent (which often means more aggressive in these games), you should forego marginal pushes if getting called and losing will bust you. This is because you can find more +EV spots later as long as you're still in the game. However, if you're the chip leader, then you should take these small +chip EV pushes because even if you lose the hand you can still continue to play, and continue to exploit your +EV opportunities. Basically you're saying there's an intrinsic value for simply surviving if you're a better HU player than your opponent, whereas most models suggest (I think) that it's all about your chip stack. I like the idea, but I'd like a more rigorous proof to fully convince me that this is true, and the chips you're sacrificing by folding as the short stack really are less valuable in terms of $EV than if you're the big stack.

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Yeah. This is an interesting thought. Makes sense to me.

However, it isn't the same thing I said so you and Foxy can go get your own theory and name it after yourselves.

My idea is more about the value of chip disparity than the value of surviving. However, both theories would often yeild the same results.

Here's where they differ/don't differ:

Hero (3500)
Villain (6500)

Push and fold are EV nuetral.

Jman's Theory says: FOLD
Matt's Theory says: FOLD

Hero (6500)
Villain (3500)

Push and fold are EV nuetral.

Jman's Theory says: PUSH
Matt's Theory says: Doesn't matter

If you want one where we fully disagree:

Hero (6500)
Villain (3500)

Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD

Am I making sense?

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

I'm going to add two other theories and their veiws on these hands. Probably a combination of these thoughts should work together in close situations and you should weigh the consequsnces. I'll recap the main ideas here.


Matt's Theory: In EV Nuetral spots (or close to it) where hero may be eliminated, fold in order to take advantage or your skill edge.
Jman's Theory: In EV Nuetral spots (or close to it) do whatever increases stack disparity(to a point).
Theory ICM: Do whatever ICM says
Theory 2+2*: Adhere to ICM. When decisions are very close, fold in order to help your image.


*(as I understand it, what we usually recommend)

Here's where they differ/don't differ:

Hero (3500)
Villain (6500)

Push and fold are EV nuetral.

Jman's Theory says: FOLD
Matt's Theory says: FOLD
Theory ICM: Doesn't Matter
Theory 2+2: FOLD

Hero (6500)
Villain (3500)

Push and fold are EV nuetral.

Jman's Theory says: PUSH
Matt's Theory says: Doesn't matter
Theory ICM: Doesn't Matter
Theory 2+2: FOLD

If you want one where we fully disagree:

Hero (6500)
Villain (3500)

Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD
Theory ICM: Doesn't Matter
Theory 2+2: FOLD

I use a lot of these guys ((())) (afraid to spell them wrong so I made a visual aid)

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

It took me like fifty tries but I finally get it. And I like what you're saying. I hadn't thought of anything quite like that before. Probably not even intuitively.

I think your post would have gotten a lot more credit if you added about 40 pages of math that nobody bothered to read but assumed was well thought out.

Disclaimer
 

Please note that the actual consequences of these thoughts are pretty minimal.

Even if there is merit to my theory, ignoring it will not hurt your ROI much at all.

I just like to think things through sometimes.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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It took me like fifty tries but I finally get it. And I like what you're saying. I hadn't thought of anything quite like that before. Probably not even intuitively.

I think your post would have gotten a lot more credit if you added about 40 pages of math that nobody bothered to read but assumed was well thought out.

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Haha. Yeah, I should've done that. I prefer word problems myself though.

Anyways, glad you got it, and that you liked it.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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DUH. DUH. DUH. I was just explaining why you want to do that.

In fact, Ill put it more succinctly for you:

(PUSH ANY TWO)

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At what ratio of blinds to small stack does pushing any two become correct, then?

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD

Am I making sense?

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Here's a novel idea: test your theory.

I think you will find it is false.

If you can't test it on full scale poker (although I don't see why it's not possible), come up with a simplified model that retains the properties that you think are essential to making your theory correct.

eastbay

Re: Heads up Theory
 

My boring theory is that the better player is able to more precisely put his opponent on a range and then do the math to calculate the best EV decision. Seems to me that every time you make a -EV decision you are losing chips and that doesn't seem good.

I think that 92 push cost you about 600 Sklansky chips.

Re: Disclaimer
 

I understand what your saying here jman, I've been thinking along the same lines at times when heads....
At times I will be ahead say 6000-2000 with blinds of 150-300

I will push Every hand at this point. My reasoning being that I'f I can get him down to the point where he has to call he has to win like 3 hands in a row to get even again. If I lose now he thinks I'm pushing 9 2 all the time and will call me with any 2 when We are even and I have a2 or k3 or something like this... which again is +ev cause he will be callling with worse hands now..

Not sure if I made any sense or if this is the same thing your talking about Jman, But just some thoughts I had on the same situation.

Re: Heads up Theory
 

yes but Can It gain you more on later hands.. By loosing 600 sklansky chips can you gain say 1000 ?

Re: Heads up Theory
 

I don't think so.

Say you are in this situation 1,000,000 times and on average you lose 600 chips. How are you going to turn that to your advantage?

A big percentage of the time, maybe 60%, you will just pick up the blinds. The fact that you won those chips is already accounted for in the average amount won/lost. You have pushed one more time (already pushed quite a bit recently) and have thus opened your opponents calling range even further. This means you will have fewer pushable hands. This isn't a huge factor, but I don't think it is positive.

A decent percentage of the time you will lose - about 25 - in those cases you will lose a bunch of chips (which are already accounted for) and you will show down 92, which will widen your opponents calling range and leave you the short stack and much more likely to be called.

The rest of the time you win. All good there.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

hi Jman,

what kind of blind sizes are you assuming here?

thanks,

- Kenny

Re: Heads up Theory
 

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My boring theory is that the better player is able to more precisely put his opponent on a range and then do the math to calculate the best EV decision. Seems to me that every time you make a -EV decision you are losing chips and that doesn't seem good.

I think that 92 push cost you about 600 Sklansky chips.

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Slightly off topic, but what calling range do you put my opponent on in the specific hand we were discussing?

I'm giving him a range of 22+,A2+,K5+,Q8+,J9+,T9 and getting a +EV push, right? (This of course, before applying my theory)

Edit: I think his actual calling range is even less.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD

Am I making sense?

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Here's a novel idea: test your theory.

I think you will find it is false.

If you can't test it on full scale poker (although I don't see why it's not possible), come up with a simplified model that retains the properties that you think are essential to making your theory correct.

eastbay

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I would like to do this.

As for full scale poker, I think it would take many years to come up with a sample significant enough.

A full scale model, sounds good. Unfortunately I have no idea whatsoever how to do this. Wanna help?

Re: Heads up Theory
 

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Say you are in this situation 1,000,000 times and on average you lose 600 chips. How are you going to turn that to your advantage?


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I wouldn't turn this to my advantage.

I'm talking about situations where I average losing, say, 20 chips.

Re: Heads up Theory
 

You had just pushed a bunch. I think that calling range is fair. I might have used a bit broader range though.

I don't have any tools with me now - what's the result with that range?

BTW, what part of the world are you in? I'm in Southern California.

Re: Heads up Theory
 

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You had just pushed a bunch. I think that calling range is fair. I might have used a bit broader range though.

I don't have any tools with me now - what's the result with that range?

BTW, what part of the world are you in? I'm in Southern California.

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That range made a push +.3%.

I'm in the midwest, so central time zone. Usually not awake at this hour (I like to wake up at 3:30 pm and go to bed at 4 am) but I got two phone calls while sleeping and couldn't go back to bed. I assume that's what this question was about. right?

Re: Heads up Theory
 

Well, I get it, so thats a start.

I agree when you have more chips. Basically, it takes a situation where against a perfect opponent it would be EV neutral and puts him ina spot to makea decision which can only increase your EV over time.

Pretty much standard aggressive poker theory.

Now onto the real 'theory'

From a game theory perspective, I can see where this comes from.

As the chip disparity grows, the leading player can afford to make riskier and riskier plays b/c losing a hand still leaves him with enough chips to win a reasonable amount of the time.

So, your premise is that when given a situation to which we are EV indifferent, we should choose the one which will make our opponent make riskier moves going forward. If I am wrong with my thoughts so far, skip the rest.

My problems:
-Your opponent gets off easy. He increases his equity without having to make a decision. When we put out opponent on a range and get an EV neutral spot, any hand he plays outside of that range is +EV. So you are putting a lot of faith in your reads and your opponent here.

-So for this to make any sense, we are assuming our opponents are playing near optimal poker. So his range will slide wider by a small amount given the new disparity in stack sizes. For us to take advantage of this. He needs to be dealt a hand in exactly that new portion of his range that was adde, coupled with us being dealt a hand that makes a call +EV.

-The problem with this is, his increased range is directly affected by the stack/blind ratio of both players. So, the higher the blinds the more likely we are to be able to take advantage of it. BUT, the higher the blinds the less equity we will have if we do win the favorable hand after folding the neutral hand.

**Basically, I think this strategy would work in a game where the cost associated with waiting (paying blinds) was lower, and/or the edge you expected to gain was larger.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD

Am I making sense?

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Here's a novel idea: test your theory.

I think you will find it is false.

If you can't test it on full scale poker (although I don't see why it's not possible), come up with a simplified model that retains the properties that you think are essential to making your theory correct.

eastbay

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I would like to do this.

As for full scale poker, I think it would take many years to come up with a sample significant enough.

A full scale model, sounds good. Unfortunately I have no idea whatsoever how to do this. Wanna help?

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Not really, no.

I will just say that I have done investigations along these lines before, however, both with simpler model games as well as full preflop push/fold NL Hold'Em and have consistently found that there are no circumstances HU where taking -cEV situations is superior to some other strategy which never does. In fact, very strong strategies will collapse very quickly once you add -cEV moves to them.

eastbay

Re: Heads up Theory
 

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So, your premise is that when given a situation to which we are EV indifferent, we should choose the one which will make our opponent make riskier moves going forward. If I am wrong with my thoughts so far, skip the rest.

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You are close, but this is a misunderstanding.

The reason we want to increase chips disparity is that it widens OUR pushing range, creating more +EV opportunities, whether we are the small stack or the big stack.

To reword, 'widening our own pushing range' is the same thing as 'finding ourselves in more +EV situations' since we should only widen our range if the extra hands in there are +EV.

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So for this to make any sense, we are assuming our opponents are playing near optimal poker.

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I know you said to ignore the rest, but I think this is where your confusion might be though.

For my idea to work, we are assuming the opposite of this. We assume that WE play near optimally, and our opponent does not.

When chip disparity increases:

It becomes +EV to push more.
It becomes +EV to call more pushes.

These are both adjustments that we count on our opponent to NOT make.

For example:

Hero 4600
BB 5400

Blinds 300/600

Hero has 86s.
BB will call with 22+,A2+,KT+,QJ+
Push is +.9%

Now...take 1k and move it.

Hero 3600
BB 6400

Blinds 300/600

Hero has 86s.
BB will call with 22+,A2+,KT+,QJ+
Push is now +1.1%

This is the effect of our villain not adjusting his calling range. Pushes become better for us as the stack disparity increases.

I realize (now more than before) that the difference is minimal. I don't think this idea is a very big deal, as I said before, in that it will have much impact on your game. It will not.

Re: Heads up Theory
 

In addition to this last post, I want to add that if your opponent has different leaks (calls/pushes too frequently) you would want to employ a different strategy.

This time, leaning toward DECREASING chip disparity, since then he will often be making pushes and calls that are even more -EV.

The reason my basic idea is to increase the difference is because generally, our opponents leaks are not pushing/calling enough.

Re: Heads up Theory
 

Ok, It looks like a marginal spot. Maybe a few hands could be added to the range, maybe not. Villian himself would be unlikeky to be able to answer the question.

As far as asking where you are, I was just wondering if you might be in SoCal (20 million people are).

I'm thinking about looking for a live tourney tonight. Anyone interested? Hopefully Yugo can come. He wouldn't have a date, would he?

Re: Heads up Theory
 

I think you are making the mistakes I mentioned, but I likely did a bad job explaining what I mean

I think you are confusing the EV difference (pushing vs folding) and overall equity. You mention how the same situation turns from +.9% to 1.1%. This is a direct result of the size of our stacks. We have less equity to begin with in the 2nd example, so increasing it by a larger % still makes us worse off.

We have 392$ in equity to start #1
We increase this by .9% with a push = $395.5

We have 372$ in equity to start #2
We increase this by 1.1% with a push = $376.1

So yeah, we have more higher % pushes in case #2, but I'd rather have an EV neutral push when I start with 392$ in equity than a 1% positive spot when I star t with 372$.

That is the basics as to why I think it is wrong. For this strategy to work, you would need the starting equity of #1 and #2 to be closer (392$ ~ 390), and/or the difference in EV % to be much higher.

Re: Heads up Theory
 

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I think you are making the mistakes I mentioned, but I likely did a bad job explaining what I mean

I think you are confusing the EV difference (pushing vs folding) and overall equity. You mention how the same situation turns from +.9% to 1.1%. This is a direct result of the size of our stacks. We have less equity to begin with in the 2nd example, so increasing it by a larger % still makes us worse off.

We have 392$ in equity to start #1
We increase this by .9% with a push = $395.5

We have 372$ in equity to start #2
We increase this by 1.1% with a push = $376.1

So yeah, we have more higher % pushes in case #2, but I'd rather have an EV neutral push when I start with 392$ in equity than a 1% positive spot when I star t with 372$.



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I think you're still misunderstanding. I realized this would be a problem for some:

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I'm a little uncomfortable with the 'sacrificing chips' wording, because it may lead some people to think that I'm saying you are better of with 30% of the chips than with 31%. You are not.

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I know that you are ALWAYS worse off with less chips. My point is, in a sense, that you aren't AS bad off as ICM leads you to believe. (because of the more +EV pushes you can make)

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That is the basics as to why I think it is wrong. For this strategy to work, you would need the starting equity of #1 and #2 to be closer (392$ ~ 390), and/or the difference in EV % to be much higher.

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I'm quite sure that you will always be better off with more chips than with less, (all other factors the same) no matter how small the difference.

Re: CONCLUSION OF MY THEORY IN PLAIN ENGLISH
 

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Pushing has EV difference of -.000001% of prize pool (edited this for clarity)

Jman's Theory says: PUSH
Matt's Theory says: FOLD

Am I making sense?

[/ QUOTE ]

Here's a novel idea: test your theory.

I think you will find it is false.

If you can't test it on full scale poker (although I don't see why it's not possible), come up with a simplified model that retains the properties that you think are essential to making your theory correct.

eastbay

[/ QUOTE ]

I would like to do this.

As for full scale poker, I think it would take many years to come up with a sample significant enough.

A full scale model, sounds good. Unfortunately I have no idea whatsoever how to do this. Wanna help?

[/ QUOTE ]

Not really, no.

I will just say that I have done investigations along these lines before, however, both with simpler model games as well as full preflop push/fold NL Hold'Em and have consistently found that there are no circumstances HU where taking -cEV situations is superior to some other strategy which never does. In fact, very strong strategies will collapse very quickly once you add -cEV moves to them.

eastbay

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Here's one for you.

You're playing against opponent X. You've been playing with him for 6 hands heads up.

Hero: 6000
X: 4000
Blinds 250/500

Opponent X is a tall slender man with a rugged handsome face. He has been folding every hand except for AA.

He's waiting for those aces baby! And he's gonna bust you so good when he gets em.

Now you are dealt 94s in the sb. He will only call with AA. However, this opponent X has decided that if you push 4 times in a row into him (you've already pushed 3) that he will adjust his range for the rest of the tournament to calling and even pushing himself with 22+, Ax, Kx, Qx.

Clearly, the optimal strategy is to fold this 94s, even though pushing the hand is +cEV. Then push the next three chances you get, then fold again.

Now, in real life, examples aren't this clear cut. They are more like the one's which I am trying to describe in this thread.

Would you mind opening your mind and thinking about them now that I have shown you how +cEV plays are not the optimal strategy 100% of the time? I would like to hear your thoughts on the idea based on it's own merit rather than based on the fact that you have tested different situations and come up with the conclusion that what was +cEV in those situations was always best.

I would be glad to help set up a simulation if anyone with the knowledge to do something like that would assist me.