How excluding [some] luck helps for better statistics?

[indianaV8]

You have played 2000 hands. Out of that you can derive EV BB/100h range with some statistical confidence. I.e. I one can calculate with 95% conf level that our BB/100h is e.g. between -0.5BB and 2BB / 100 hands.

Now let's say the only thing you did in these 2000 hands was all-in or fold (to simplify the things). For some of the hands that you were allin you got called => game went to showdown, and you know the cards with which you were called.

Now, you can calculate for all these all in confontations (for which you know opponents cards) how much money you made, and how much you were expected to make.

My question is: How can I transfer this information back to the original problem? I.e. if I still have my 2000 hands history only, but take into account with what hands my allins were called, and what was the expected profit vs the actuall outcome; and I want again (say with 95% statistic confidence) to find what is my BB/100h range (that considers the additional luck involved). I should have now a better (more accurate) estimation, but what is the way to find this out.

I post this particular problem and not any generalization, I simply want by looking at this step by step to understand how you approach it.

Thanks
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indiana

[pzhon]

You must know the total variance because you say you can calculate the confidence interval. The total variance is roughly the sum of the variance from the hands which do not involve an all-in confrontation and the variance from the hands which do. So, estimate the variance from the all-in confrontations, and subtract this from the total variance, to get an estimate of the residual variance, which will give you a new estimated confidence interval.

This isn't trivial. There are a lot of technical difficulties. However, I'm not going to go into detail because you say you aren't interested in any generalization, and your confidence interval is way off.

[indianaV8]

Thanks a lot for your reply.
Yes, you are right. So let's say for this partucular case I know (or have good estimation) of the variance.

The practical thing I'm looking for is to find out a way to judge about a strategy much earlier. So in the jam/fold case, let's assume from many hands I know the variance, and I don't expect it to change a lot with small deviation of the strategy. I'm not 100% sure if this is correct of course.

I don't know - maybe this clarification already answers the particular case. If I have the variance (in the scenarios where we take the expected value of the allin confrontations), then bsed on the smaller sample (2000 hands) I can calculate the EV/100 range, and this will be better estimation of the real EV/100.

Two more questions now ... is there any good statistic available on std and "std of std" in NL games? As I now see that I need to know the std. Is there anything else I can do?

The whole problem stated crysp is: How you can get a better EV/100BB estimation, based not only on win/loss but on more things you have access to (like luck in all in confontations).

Hence come the second question. Are there more things one can look at to improve the esimation (except a situation where all the money are in)?

Again, thanks a lot for looking at that.

[pzhon]

Outcome = net luck + net skill.

Anything you can do to better estimate the luck will give you a better estimate of the skill. A significant theoretical problem is that unless the cards go to showdown, you don't know how lucky you were to get a particular community card. The times you do go to showdown are not an unbiased sample.

For example, suppose you raise preflop with AA, and push on any flop against a set-miner. When you go to showdown, your average flop luck will be terrible. Most of the time when it was good, you don't get to see your opponent's hand.

Nevertheless, you can make some slightly better estimates than saying that the luck was 0, if you are willing to introduce some bias. When you have a draw, you can estimate that it was good luck when it hits, and bad luck when it misses. To estimate how much, you can look at past data. When you have a low flush draw with an effective stack size of 3 times the pot, how much did you make in the past when it missed? How much did you make when it hit? You can use these to produce an estimate for how lucky the next card is.

[indianaV8]

I ran some experiments, and the results are pretty nice looking. For jam/fold in some scenarios the STD (if you take the expectation of the confrontation) was reduced from 45BB/100h to 27BB/100h, and the estimated range for the EVBB/100h by factor 2 (for several confidence intervals I tested).

I can try this out on some normal play (I can do it based on datamined hands, right ...? As the hand should have reached showdown).

Seems that any other case is pretty hard to analyse.

I would like to understand the math behind in more details. Is the draw example the best one? What about this one: everytime you limp with small poker (22-88) you take for your "improved" estimation the average you made (based on past results) instead of what you've made now. What does this mean ...?

This will help you only if you don't change the way you play small poker pairs. If you changed from folding on missed flop to bluffing, this will fail miserably.
And the more you assume that you don't changed your strategy, the more meaningless such EV estimation become.

[pzhon]

[ QUOTE ]
And the more you assume that you don't changed your strategy, the more meaningless such EV estimation become.

[/ QUOTE ]
No, you don't have to assume that you are playing the same way. If your estimate is wrong, you don't cancel out as much of the luck, but you should not ruin your EV estimates. The bias should be small.

Luck averages to 0. If we add something which averages to 0, the result still averages to 0, but we might have reduced the variance.

For example, suppose you have a pure flush draw and are trying to estimate the luck from the turn. You look at past misses with the same pot size, and find that you averaged getting a net of $1 back from the pot. You look at past successes, and find that you took netted $10. You have 9 outs in 47 cards, so you say that your average result is 128/47, and hitting is lucky by 10-128/47, while missing is unlucky by 128/47.

If you were really drawing from the 47 unseen cards, this would average to 0. The problem is that there is a bias, which in many situations is small, that whether you get to that situation depends on the number of your outs your opponent holds. It might be that the probability of hitting is slightly greater than 9/47 or slightly lower. If it were not for this effect, then you could feel free to use a very bad luck estimate, and it would not change the average of your estimated EV. A bad estimate might only reduce your estimate's SD/100 from 45 BB/100 to 25 BB/100, while a good one might reduce it to 20 BB/100. You can bound the size of this bias using the largest changes to the probability possible. In practice, the worst bias will often be much smaller than the amount of luck removed.

The bias is much worse if you try to apply this variance reduction technique to Scrabble, where you don't know your opponent's rack, but your opponent's rack is a much greater fraction of the "deck" and it is very far from random. In backgammon, there is no hidden information, and the variance reduction is unbiased, and very effective. Although the backgammon bots' evaluations are not perfect, unbiased luck estimates using these may reduce the variance by a factor of 10-100.

[rufus]

If you want something sophisticated, you should definitely look into the papers published by the University of Alberta people who are very smart and have been working on this sort of problem for a long time.

[pzhon]

They have developed some variance reduction tools for heads-up limit where you can see both players' hole cards, which works for their context of analyzing poker bots playing each other. It's difficult to extend most of what they have done to multiplayer poker or NL or actual hand histories where you don't see all of the cards.

[indianaV8]

I regularly read Alberta's papers, but did not spend so much energy on the DIVAT work, partly because it was limited and you need to know all the cards. After getting into this problem I'll definetely reread this thou.
Also, knowing all the cards is great as pzhon says for comparing bots against each other, but I'm looking for a methodology to help players in real game situation.

[rufus]

[ QUOTE ]
They have developed some variance reduction tools for heads-up limit where you can see both players' hole cards, which works for their context of analyzing poker bots playing each other. It's difficult to extend most of what they have done to multiplayer poker or NL or actual hand histories where you don't see all of the cards.

[/ QUOTE ]

IIRC one of DIVAT papers mentions some earlier-generation tools that were set-up for dealing with more limited information. I don't have access to a university library, so it's not always easy to track down papers.