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BGonline.org Forums
a bias in rollouts
Posted By: Matt Cohn-Geier In Response To: a bias in rollouts (Stick)
Date: Friday, 2 May 2008, at 7:14 a.m.
You're assuming the rollout results are wrong and overestimating from the go - which I also don't understand.
I have centuries of mathematics to back me up ;)
Look, backgammon is a game of chance. In statistics, you aren't going to get the right answer. You're going to get near the right answer with a certain margin of error (confidence interval, or Bayesian probability, or whatever).
So far so good, I think. Now take any one play. Let's say that if we roll out this one play, we are EQUALLY likely to get a result that is lower than the true equity as we are to get a result that is higher. This may be somewhat of a bastardization of Bob's original intent but I think it expresses the general sentiment.
Now take three plays which give exactly the same true equity, 0. Play A will roll out lower than the true equity with probability .5 or higher than the true equity with probability .5. Likewise for play B and C.
What's the probability that all three plays roll out to lower than 0? (1/2)^3 = 1/8. The probability that any one play is higher than 0 is 7/8. So with a probability of 87.5% the rollout result will be higher than the true equity, whereas it's only 12.5% that it's lower than the true equity. Meaning that the method we use is biased for equities higher than the true one.
You can't create a position to demonstate what you're talking about? Leads me to believe such a bias is either non existant or not worth worrying about.
The best known method we have of determining the equity of a position is rollouts. Short of that, you can calculate the equity directly, but that's extremely difficult even for simple bearoffs. You can't use rollout results to demonstrate a bias in rollouts without getting into some vicious circles. The bias should be there as a result of certain statistical principles and methods, not because I unjustifiably assumed it.
I also roll out my permanent rollouts so many trials that, as you mentioned, who gives a shit if after 15k trials it's +.5506 or +.5505.
Like I said, I'm certain that you personally don't care, but I'm not certain that it doesn't make a difference in other ways, like the odd-ply vs. even-ply mentioned on this thread. I also don't believe that there is any better method than rollouts, if that's what your issue is.
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