empirical equity study

[eastbay]

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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.

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Of course. Whose equity do you think I am interested in?

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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).

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All players on all hands? I have doubts about the usefulness of that model. How does position then matter?

eastbay

[eastbay]

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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.

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I noticed this as well. It may be that the typical player misplays the big stack, sitting patiently waiting for someone to bust, and leaving my blinds alone.

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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?

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If I follow you - yes, that that's the point of knowing the equity as a function of stack distribution. For any decision, say, pushing, if you estimate what hands you'll be called by, you can calc. the probability of the call, and get the probability of the resulting stack distributions, and finally determine whether your push or a fold results in more equity. Until now, ICM has been the tool of choice for determining equity=f(stacks). I'm seeing if I can improve on that. Obviously, every person has a different such function, so in the first instance, I'm interested in my own function. I might also try to compute an average function for general interest, however.

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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.


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Yes.

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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?


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Yes, I think that's right.

eastbay

[hansarnic]

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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).

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All players on all hands? I have doubts about the usefulness of that model. How does position then matter?

eastbay

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Position doesn't matter. Nor do blinds or skill levels. But it would validate (or disprove) ICM, which would be useful in itself.

[eastbay]

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Position doesn't matter. Nor do blinds or skill levels. But it would validate (or disprove) ICM, which would be useful in itself.

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I have some doubts about that. It raises an interesting question though: can a game of players who all play the same generate different equities for the same stack distribution, by the manner in which they all play? It seems there should be an easy logical proof or an easy counterexample, but neither came to me in the 10 seconds it took me to write this post.

eastbay

[eastbay]

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Position doesn't matter. Nor do blinds or skill levels. But it would validate (or disprove) ICM, which would be useful in itself.

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I have some doubts about that. It raises an interesting question though: can a game of players who all play the same generate different equities for the same stack distribution, by the manner in which they all play? It seems there should be an easy logical proof or an easy counterexample, but neither came to me in the 10 seconds it took me to write this post.

eastbay

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Ok, I thought of one. Consider HU match. Strategy A is to push when you have half or more of the chips, and fold if you have less than half. Clearly equity distribution is a step function for two players both playing this strategy.

Strategy B is push everything, call everything. We know that equity distribution is basically linear for this strategy.

So the answer is yes: the equity function of a game of players all playing the same strategy does depend on the strategy employed. And this is why I am not too fond of your "validation" of ICM for all pushes. It may generate a reasonable distribution, but it also may not, and I don't think it will close the case either way.

eastbay

[eastbay]

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Position doesn't matter. Nor do blinds or skill levels. But it would validate (or disprove) ICM, which would be useful in itself.

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I have some doubts about that. It raises an interesting question though: can a game of players who all play the same generate different equities for the same stack distribution, by the manner in which they all play? It seems there should be an easy logical proof or an easy counterexample, but neither came to me in the 10 seconds it took me to write this post.

eastbay

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Ok, I thought of one. Consider HU match. Strategy A is to push when you have half or more of the chips, and fold if you have less than half. Clearly equity distribution is a step function for two players both playing this strategy.

Strategy B is push everything, call everything. We know that equity distribution is basically linear for this strategy.

So the answer is yes: the equity function of a game of players all playing the same strategy does depend on the strategy employed. And this is why I am not too fond of your "validation" of ICM for all pushes. It may generate a reasonable distribution, but it also may not, and I don't think it will close the case either way.

eastbay

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It is interesting to note that strategy A employs "equal skill" and yet the distribution is not linear. I understand Mr. Sklansky provides a "flawless logical proof" in TPFAP that any such distribution must be linear.

I don't own the book. Anyone care to comment?

eastbay

[rachelwxm]

Is this in Slansky's book?

Any strategy that based on stack size instead of cards vialate this relation, like the one you created. For example push more if big stack, etc because while it's vague when you talk about same strategy but chip stack is certainly not symmetric, you either have more or less chips.

[hansarnic]

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Position doesn't matter. Nor do blinds or skill levels. But it would validate (or disprove) ICM, which would be useful in itself.

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I have some doubts about that. It raises an interesting question though: can a game of players who all play the same generate different equities for the same stack distribution, by the manner in which they all play? It seems there should be an easy logical proof or an easy counterexample, but neither came to me in the 10 seconds it took me to write this post.

eastbay

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Ok, I thought of one. Consider HU match. Strategy A is to push when you have half or more of the chips, and fold if you have less than half. Clearly equity distribution is a step function for two players both playing this strategy.

Strategy B is push everything, call everything. We know that equity distribution is basically linear for this strategy.

So the answer is yes: the equity function of a game of players all playing the same strategy does depend on the strategy employed. And this is why I am not too fond of your "validation" of ICM for all pushes. It may generate a reasonable distribution, but it also may not, and I don't think it will close the case either way.

eastbay

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Don't think I understand Eastbay.

Strategy A leads to the player with the most chips at the start of the HU match winning virtually 100% of the time, no?

So that player has more skill (or rather the lower stacked player has no skill as he just folds every hand giving the game to his opponent).

Strategy B takes skill entierly out of the equation.

And yes, there are a couple of proofs of $EV = Chip count in HU in TPFAP, one of which is calculated on the basis of both players going all-in every hand.

[jcm4ccc]

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

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

eastbay

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If I understand you correctly, you are plugging in data points for every 4-handed hand. So, for example, some tournaments might have 12 data points, and some might have only one. The data might look like this:

tourney stack icm actual
1111111 1500 $10 $15
1111112 1500 $10 $25
1111112 1500 $10 $25
1111112 1450 $9.5 $25

if that's true, I would say that this is a flaw in your research design. Your data points are not independent. They would be correlated with data points from the same tournament. Tournaments that lasted longer on the bubble would have more influence on your data than other tournaments.

This is a signficant problem, because there is probably a correlation between the size of the big stack and the length of the bubble. In other words, tournaments in which one stack has more than half the chips would have fewer hands on the bubble, and therefore have less influence on your data (fewer data points from those tournaments).

I would do as irieguy suggested. Use only the first hand in which you are four-handed. Alternatively, you could use the first four hands of a tourney (so that everybody gets the SB and BB once), average the results, and use that as your data point. Tournaments that lasted less than 4 hands would not be included in the data (though that adds a confounding factor--you would want to track how many tournaments are not included).

[Phil Van Sexton]

Apologies if this has been done 1 millions times already...

Have you run this analysis based on the position of the button rather than the position of the big stack? Clearly, if 4 players have equal stacks, the ICM will be .25 for all, but the button's equity is really higher and the BB's lower.

How much, I'm not sure, but likely significant if the blinds were 300/600 since the BB will need to post 30%+ of his stack over the next 2 hands. Meanwhile, the button posts nothing and can steal if he chooses.