empirical equity study
 

I finally coded up the icm comparison/validation I talked about doing awhile ago. The idea was to empirically determine my equity as a function of chip stack distribution and compare that to what ICM says.

A brief look at some bubble numbers:

Overall average equity on the bubble:

ICM: .232
Actual: .277

Looks pretty good. I've got about 4% more of the prize pool than ICM says I "deserved".

Breaking it down into situations where someone has more than half the chips:

stacks: actual equity, icm predicted equity, difference, standard error of sample, number of data points in this range
0,0,0,0: 0.282925; icm = 0.239223; delta = 0.0437015; std err=0.00508585; N=1265
0,0,0,1: 0.266667; icm = 0.195756; delta = 0.0709111; std err=0.0189215; N=90
0,0,1,0: 0.244944; icm = 0.203231; delta = 0.0417128; std err=0.0190025; N=89
0,1,0,0: 0.215385; icm = 0.18758; delta = 0.0278049; std err=0.0148834; N=117
1,0,0,0: 0.426316; icm = 0.393979; delta = 0.0323365; std err=0.0221329; N=19

0,0,0,0 means no one has more than half the chips.
0,1,0,0 means the guy to my immediate left does. etc.

This is kind of interesting, if it really holds up. You can see the increasing advantage of having the big stack further away to the left. This is an effect that is completely missing from the ICM, which takes no heed of position.

Another interesting fact is that the other players are playing the big stack at this phase of the game much more often than I am (N=19). And yet I have significant equity advantage over them on average.

Discuss. Probably more to come.

eastbay

Re: empirical equity study
 

Quick question, for the ICM numbers did you just plug in whatever the stacks were on the 1st 4-handed hand?

Re: empirical equity study
 

[ QUOTE ]
Quick question, for the ICM numbers did you just plug in whatever the stacks were on the 1st 4-handed hand?

[/ QUOTE ]

For the ICM numbers I plugged in for every 4-handed hand.

eastbay

Re: empirical equity study
 

Interesting.

If ICM is to be trusted, I suppose your extra equity can be accounted for by difference in playing ability.

What I'm really wondering though, is if your results seem to indicate a different model would be more appropriate.

[ QUOTE ]
This is an effect that is completely missing from the ICM, which takes no heed of position.


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I'm thinking about this, and I am not sure that this is missing from ICM. After all, ICM assumes equal playing ability and I'm thinking that this position related difference in equity might have everything to do with you, and the fact that you have an equity edge.

I think these results might mean more if you only ran them on bubble histories that you were actually not involved in. In this way, it would always be different players and might more accurately reflect the equal skill assumption. Then again, it could be even more important to look at the advantage biased results.

...Already I'm thinking I'm wrong about the position differences being a factor of your advantage. It would make sense that these positional distances from the big stack would affect equity.

Can you run this and calculate equity actual/predicted for the other three positions as well?

I have a feeling this is going to become a very important thread. I can't wait to see where this goes. Very ambitious putting this all together. Has the potential for a major contribution to tourney theory as a whole.

Regards
Brad S

Re: empirical equity study
 

[ QUOTE ]
Interesting.

If ICM is to be trusted, I suppose your extra equity can be accounted for by difference in playing ability.

[/ QUOTE ]

That's certainly in there. I originally had thought that was the purpose of the investigation, since I was approaching this from the perspective of determining my equity to the best approximation. Right now I do that with ICM, for lack of something better. That something better might be a curve fit on empirical data, that includes things like my playing advantages.

But, now you're getting me thinking about something a little more general and theoretical: an empirical distribution based on the player pool as a whole. That has some merit too, I think.

[ QUOTE ]


What I'm really wondering though, is if your results seem to indicate a different model would be more appropriate.

[ QUOTE ]
This is an effect that is completely missing from the ICM, which takes no heed of position.


[/ QUOTE ]

I'm thinking about this, and I am not sure that this is missing from ICM. After all, ICM assumes equal playing ability and I'm thinking that this position related difference in equity might have everything to do with you, and the fact that you have an equity edge.

I think these results might mean more if you only ran them on bubble histories that you were actually not involved in.


[/ QUOTE ]

Yes, that's an interesting idea.

[ QUOTE ]

In this way, it would always be different players and might more accurately reflect the equal skill assumption. Then again, it could be even more important to look at the advantage biased results.

[/ QUOTE ]

For me, yeah. In general, probably not.

[ QUOTE ]

...Already I'm thinking I'm wrong about the position differences being a factor of your advantage. It would make sense that these positional distances from the big stack would affect equity.


[/ QUOTE ]

I think it's both effects mixed together.

[ QUOTE ]


Can you run this and calculate equity actual/predicted for the other three positions as well?


[/ QUOTE ]

Yes, I could do that. Time permitting.

eastbay

Re: empirical equity study
 

Pretty interesting. So if I read it correctly, your ROI is 18% (.277-rake) and you make extra 4% from bubble on.

I have looked at ROI boost for each stages and found for low limit, you can make 50% of you equity from bubble on. I think that high limit player are generally much better in short handed games.

What most interesting is the fact you look at the position vs big stack. Although those numbers are subjet to sample size bias since the difference of each cases is similar to the mean standard deviation. So 1000 mean you are the big stack, I was a little bit surprised that this case compared unfavorably to 0000 which is a big sample. [img]/images/graemlins/smile.gif[/img]

I have not look at anything regarding second part in my analysis though. [img]/images/graemlins/tongue.gif[/img]

question
 

Why wouldn't the overall average equity on the bubble according to ICM be 25%?

If you are looking at the "overall average equity," it seems like everybody would have equal equity in the ICM.

Irieguy

Re: question
 

[ QUOTE ]
Why wouldn't the overall average equity on the bubble according to ICM be 25%?

If you are looking at the "overall average equity," it seems like everybody would have equal equity in the ICM.

Irieguy

[/ QUOTE ]

They would if we played the bubble with even stacks on average. We don't. Prior play and bubble play influences the average stack distribution when it's 4-handed.

eastbay

Re: empirical equity study
 

[ QUOTE ]

0,0,0,1: 0.266667; icm = 0.195756; delta = 0.0709111; std err=0.0189215; N=90


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Is this the situation when the big stack is to your right so you get to act just after him? (I want to make sure I've got yout 0,0,0,1 down). It is interesting that in this you far outstrip the ICM average (more than any other 4-handed situation). It seems either there is a larger advantage to being behind the big stack than I thought, you excel vs. other smallish stacks when the big stack isn't playing the hand, or that the large difference in delta could be an anomoly.

Believe it or not I have been eagerly awaiting this thread for some time and along with Aleo, would be *very* interested to see what one comes up with when analyzing 4-handed play on Party Poker with random participants.

I'm interested in what else you think your numbers may mean other than ICM cannot take into account position to the big stack. Would running this to determine equity of pushers 4-handed or folders 4-handed be meaningful in any way (i.e. on average is it better to push rather than fold, and if so how close is it)? Would this be some sort of factor of the buyin you're at (55s I assume?) where general tightness/looseness would determine the equity of each action?

I'm also interested to know if anything meaningful can be gleaned from the equity your opponents had vs. the big stack -- I'm wondering if the average player may generally outperform or significantly underperform in different situations vs. the big stack and/or with the big stack. Hmmm, perhaps this question is better answered by running ICM vs. actual equity #s for 4-handed hands with random players.

Also, do your numbers mean you're making an extra 4% ROI from bubble on and making 14% by just getting to the bubble with a playable stack?

Very intersting stuff Eastbay, major props to you.

Yugoslav

Re: empirical equity study
 

This is very interesting, but as others have said, it really only demonstrated YOUR equity at certain stages and that comes from your realtive skill level.

What would be interesting (and not that hard to do) would be to sim certain player / stack situations & compare the results to ICM.

One can do this by just getting each player to move in on every single hand (thus negating the effects of blinds & skill).

This is something I've been meaning to set-up as & when I have time, so will have a go over the next week or so.

Of course this only validates ICM which as we know doesn't take account of certain key variables in tournament structure, (position, skill and blinds) being the key ones. And as Aleo said the best way to do this would be to crunch huge amounts of actual data.

Strikes me this should be a job for Party or Stars!!