empirical equity study

[eastbay]

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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).

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I am not sure what you are objecting to. Are you objecting to how I computed the icm average for the region in stack-space, or how I am computing the empirical equity? Or both?

You may have a point or you may be confused, depending.

eastbay

[eastbay]

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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?


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It is not either/or, it is an additional variable, and no, it's not one I've separated out in the analysis yet.

But I agree with you that it's an interesting angle to explore. To do so may require more data than I currently have, though.

eastbay

[eastbay]

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Is this in Slansky's book?

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I understand he makes a "purely logical" proof that for "equal skill" the distribution must be linear, yes. I am also convinced this is wrong without further qualification, as my counterexample shows.

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Any strategy that based on stack size instead of cards vialate this relation, like the one you created.


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Yes but that's a fairly huge restriction on the validaty of the 'proof'. Most players do incorporate stack size in their nominal, realistic strategies - as they should. I've simply taken that to an extreme to illustrate the effect in the clearest of terms.

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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.

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I don't follow this sentence. Clearly two identical strategies must be considered to have "equal skill."

eastbay

[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?

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Not virtually - exactly.

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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).


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No. Both players are playing identical strategies. Clearly then both strategies must have equal skill - because they are the same.

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Strategy B takes skill entierly out of the equation.

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Any two identical strategies takes skill out of the game.

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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.

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Right, but there is also a 'purely logical proof.' Can you quote it for us? It seems clearly wrong, at the very least through insufficient qualification.

eastbay

[jcm4ccc]

do all of your tournaments have the same number of data points? for example, does a tournament that lasts one hand on the bubble have the same number of data points (rows on an Excel worksheet) as a tournament that lasts 20 hands on the bubble?

if not, there is a serious flaw in your design. you are overweighting the tournaments that last longer on the bubble. all of your calculations are flawed, if that is true.

[Phil Van Sexton]

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It is not either/or, it is an additional variable

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Why is it either/or? A break down by your position just like your break down by big stack position would be interesting. Whether or not there is a big stack is not important, just the % difference between ICM and actual.

[rachelwxm]

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Is this in Slansky's book?

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I understand he makes a "purely logical" proof that for "equal skill" the distribution must be linear, yes. I am also convinced this is wrong without further qualification, as my counterexample shows.

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Any strategy that based on stack size instead of cards vialate this relation, like the one you created.


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Yes but that's a fairly huge restriction on the validaty of the 'proof'. Most players do incorporate stack size in their nominal, realistic strategies - as they should. I've simply taken that to an extreme to illustrate the effect in the clearest of terms.

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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.

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I don't follow this sentence. Clearly two identical strategies must be considered to have "equal skill."

eastbay

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Point all accepted, all that I want to point out is just stack based strategy does not create linear relationship, like the one you did. Particularly stack size is of opposite for two different people. The fact that one play big stack aggressively and the other short stack play passively lead to the one direction, not random equilibrium (linear relationship). [img]/images/graemlins/cool.gif[/img]

[eastbay]

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do all of your tournaments have the same number of data points? for example, does a tournament that lasts one hand on the bubble have the same number of data points (rows on an Excel worksheet) as a tournament that lasts 20 hands on the bubble?

if not, there is a serious flaw in your design. you are overweighting the tournaments that last longer on the bubble. all of your calculations are flawed, if that is true.

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I see your objection and I think it's legitimate in the coarse-grained results I've shown so far, especially for the fully averaged case. That wasn't my original intention for how to compute it - I wanted to make the discretization fine enough that you wouldn't have more than one instance of any given distribution in a single SnG except as a rare coincidence.

However, it's not clear to me now if there's a problem in using the same tournament results multiple times so long as the chip stack distribution for which you're recording it doesn't occur more than once in the same tournament. Thoughts? My inclination is that it's fine to do that. You just have to not double count in the same range of stack-space.

Which, in the case of lumping all of them together, means no double-counting at all. Oops.

eastbay

[eastbay]

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Point all accepted, all that I want to point out is just stack based strategy does not create linear relationship, like the one you did. Particularly stack size is of opposite for two different people. The fact that one play big stack aggressively and the other short stack play passively lead to the one direction, not random equilibrium (linear relationship). [img]/images/graemlins/cool.gif[/img]

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Agree. The point is mostly academic, but I do think that there is no "purely logical proof" of linearity once reasonable, stack-dependent strategies are considered. The magnitude of the nonlinearity in real-world play is another question.

eastbay

[eastbay]

jcm4cc discovered an error in my initial calculational procedure, so wait on corrected results. Until then...

I wanted to lay out my thoughts about grouping the stack space for a more detailed look at the function equity=f(stacks_k) where stacks_k are the possible distributions of stacks for k=1..N for N-handed play.

Starting with the case 3-handed play because it's easier to visualize:

It's a geometrical property of an equilateral triangle that the normal distances from any point in the triangle to the sides of the triangle sum to a constant, the height of the triangle. So you can visualize the possible combinations of 3 stacks as the points in the interior of an equilaterial triangle, with height equal to the number of chips in the tournament.

Now, you could never in a practical way collect enough data for each of the possible sets of 3 chip stacks. There are too many combinations to ever get a big enough sample for each one to determine its expected equity. So you have to lump certain ranges together to get a reasonable sample. You can visualize this as dividing up the interior of the triangle in some way.

A logical way to do this is to divide the triangle parallel to each side into M equal intervals. For M=2:



I've labeled these regions of the space according to the interval for players (A,B,C). Choosing higher M you can get a finer-grained description of the function over this space.

For N=2 you get a line segment, and for N=4 it's a tetrahedron. In math speak I think in general this is called an n-dimensional simplex.

So when I present results for (1,0,0,0) it's not so much about the "presence of the big stack" but a region of the stack space so described.

My immediate goal is to describe the equity function over this space for me in particular as player A, and compare that to what ICM says the function looks like (which is what I normally use for valuing various stacks). ICM has 3-way symmetry, whereas my function ought to only have 2-way symmetry for players A and B who are chosen at random from a large sample.

There's lots of additional ideas popping up in the thread for other interesting things to try to measure. Keep 'em coming.

eastbay