empirical equity study - Page 6

eastbay · #51 05-09-2005, 05:33 PM

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I'm sorry. I walked in late and I need a clarification. Why do you argue in this thread that your data points within the same tournament can be treated as independent trials,

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I don't think I did. From thew few posts I re-read, I asked a question, and then the thread went sideways about something else and we never resolved it.

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in The Shadow's recent thread, you seem to make the opposite argument?

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Intuitively I agree that only one data point is permissible on independence of observations grounds. I'm not sure how to prove it, but I'm not sure that it needs proving.

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This case is four-handed, so there is a chip distribution, and the other is two-handed, so only your chip count is variable. Is that it? If so can you explain why?

Slim

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Truth is I'm not certain which methodology is better: use all the data points or use one.

Certainly using one is "safer" as it removes the independence question, but it severely compounds the sample size problem.

eastbay

eastbay · #52 05-09-2005, 05:41 PM

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Interesting post.

I'm thinking however that this kind of empirical $EV model, might create some paradoxical implications.

Any thoughts about this problematic side of what you call empirical equity?

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I've thought about this, and concluded that I don't think it's paradoxical. It will simply take awhile (possibly a very long time) for the system to feed back on itself to reach the best strategy which is self-consistent, where the results which influence your strategy are consistent with the strategy you are currently using, and not "lagged" to the strategy you were using before.

In your example, say you tighten up too much based on your results which say you can pass some edges. Well, if this is indeed wrong, your results will begin to converge back to the mean, which will take you back towards your original strategy. Presumably there is a happy medium where the strategy and the results are consistent.

There is actually an algorithm in game theory, I've forgotten the name of it, which is guaranteed to find optimal strategies that works pretty much exactly like this.

eastbay

the shadow · #53 05-09-2005, 10:46 PM

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There is actually an algorithm in game theory, I've forgotten the name of it, which is guaranteed to find optimal strategies that works pretty much exactly like this.

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Trial and error? [img]/images/graemlins/smile.gif[/img]

Seriously now, were you thinking of backwards induction a/k/a rollback a/k/a Zermelo's algorithm? If so, my recollection is that it applies to games of perfect information. Cite. If not, I'd like to hear more about what you were thinking of, since I'm trying to learn some more game theory.

The Shadow

eastbay · #54 05-10-2005, 01:33 AM

[ QUOTE ]
[ QUOTE ]
There is actually an algorithm in game theory, I've forgotten the name of it, which is guaranteed to find optimal strategies that works pretty much exactly like this.

[/ QUOTE ]

Trial and error? [img]/images/graemlins/smile.gif[/img]

Seriously now, were you thinking of backwards induction a/k/a rollback a/k/a Zermelo's algorithm? If so, my recollection is that it applies to games of perfect information. Cite. If not, I'd like to hear more about what you were thinking of, since I'm trying to learn some more game theory.

The Shadow

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Ha. No, that's not it. It's in every game theory 101 book I've ever read, though. The analogy may not be perfect, but the basic idea of iterating strategies and adjusting based on past results is there.

eastbay

Slim Pickens · #55 05-10-2005, 03:14 PM

[ QUOTE ]
[ QUOTE ]
I'm sorry. I walked in late and I need a clarification. Why do you argue in this thread that your data points within the same tournament can be treated as independent trials,

[/ QUOTE ]

I don't think I did. From thew few posts I re-read, I asked a question, and then the thread went sideways about something else and we never resolved it.

[ QUOTE ]

in The Shadow's recent thread, you seem to make the opposite argument?

[ QUOTE ]
Intuitively I agree that only one data point is permissible on independence of observations grounds. I'm not sure how to prove it, but I'm not sure that it needs proving.

[/ QUOTE ]

This case is four-handed, so there is a chip distribution, and the other is two-handed, so only your chip count is variable. Is that it? If so can you explain why?

Slim

[/ QUOTE ]

Truth is I'm not certain which methodology is better: use all the data points or use one.

Certainly using one is "safer" as it removes the independence question, but it severely compounds the sample size problem.

eastbay

[/ QUOTE ]

It seems like jcm gave a compelling argument that multiple points from the same tournament violates the necessary condition that observations be independent of each other. I have only basic training in statistics, but it seems to me that since having X chip distribution necessarily leads to a distribution of the number of observations of finish A that is different than Y chip distribution, then without removing this effect from ther results, they are not independent. I know that's nothing new but I guess I cast my vote on the side of one point per tournament.

Slim

eastbay · #56 05-10-2005, 09:05 PM

[ QUOTE ]

It seems like jcm gave a compelling argument that multiple points from the same tournament violates the necessary condition that observations be independent of each other. I have only basic training in statistics, but it seems to me that since having X chip distribution necessarily leads to a distribution of the number of observations of finish A that is different than Y chip distribution, then without removing this effect from ther results, they are not independent. I know that's nothing new but I guess I cast my vote on the side of one point per tournament.

Slim

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First problem is that independence is not a binary thing. It's shades of gray. If you want to be a purist, you could say that no two games where two players faced each other twice were permissible, since they may have learned about each other, and the resulting data is not independent in that case. The second result depends on what was learned in the first. But, at some point you have to draw the line, and do some computing rather than just disposing.

Second issue is that even two most certainly dependent data points may get used in such a way that dependence doesn't matter.

In any case, we vote the same way about the first thing to try.

eastbay

the shadow · #57 05-10-2005, 10:55 PM

I'm glad that a consensus is forming about the data dependence/independence issue.

However, the more I think about it, the bigger the selection bias issue appears to be. Let's analogize a SNG to a basketball game or a football game. If we take a look at the score at halftime or the start of the fourth quarter, I'll bet that the team with the greater points is more likely to win the game. But that lead is due, at least in part, to a difference in skill between the two teams and their coaching staffs.

Now, if we take a look at the chip count in a SNG anytime after the starting position, we have to recognize that at least some of the difference is due to relative differences in skills. After all, it's a fact that some players have higher ITM%s/ROIs than others. That's one of the "shortcomings" of ICM and why eastbay has considered modifying ICM with a skill factor.

If we use chip counts from the middle of actual HU tourneys, I cannot see a way to get around selection bias. The same applies with even greater force if we use chip counts once a SNG has become heads up. After all, the two players accumulated their chips at least in part through their relative skill over their opponents and those skill differences had more time and chips with which to express themselves.

As a result, the only way that I can see to use data from live tourneys to negate the null hypothesis (i.e., that the equity function is linear in a HU freezeout) is to use random starting chip stacks. That way the differences in the initial condition don't reflect skill differences. Of course, that pretty much rules out using data from traditional online tournaments.

If data from actual SNGs were used, the results may not be sufficient to negate the null hypothesis, but if that data, notwithstanding the selection bias, was still consistent with the null hypothesis, it might tend to make the hypothesis more likely. Given the difficulty of data collection and selection bias issue, it seems to me that the most fruitful approach at this point is to double-check gumpzilla's argument.

The Shadow (who's more impressed by gumpzilla's argument than David Sklansky's proof)

holdem2000 · #58 05-13-2005, 09:10 PM

I haven't read every reply so maybe this was already discussed...

ICM assumes players have the same strategies, which whether or not it is true, increasing blinds ruins. If every players plays a shorstack tight and a tallstack loose on the bubble at 25/50, but is loose/agressive on any stack at 200/400, then for a given tournament with rapidly rising blinds, the player on a large stack will have a "different" strategy I think.

If the blinds stay constant ICM claims the symmetry of strategies ensures linear equity functions; however, I don't think this applies to constantly rising blinds.