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Variance reduction is unbiased but may be skewed
Posted By: Timothy Chow
Date: Friday, 10 April 2015, at 11:42 p.m.
This is a minor technical observation about variance reduction that occurred to me recently, that I don't recall seeing mentioned anywhere.
Mathematically, the bots' technique of variance reduction amounts to subtracting off a certain quantity called the "luck". It is well known and easy to see that the luck is an unbiased random variable, implying that equity estimates from rollouts with and without variance reduction will have the same expected (a.k.a. average) value.
What occurred to me recently is that luck, while unbiased, could be skewed. The key fact is simply this: Doublets tend to be lucky, often very lucky. In other words, there are 6 rolls that tend to have disproportionately large luck values. This is counterbalanced by 30 other rolls that are clustered nearer the mean value of zero—typically averaging slightly below zero so that the average of all 36 rolls is exactly zero. In probability jargon, such a distribution is said to be "positively skewed."
This skewness suggests that there might be a greater than 50% probability that a rollout with variance reduction will yield an overestimate. This is conjectural, but in principle it ought to be possible to test this hypothesis via computational experiments. Note that if it turns out to be true, then the amount by which a rollout with variance reduction is an overestimate (when it is and overestimate) will generally be smaller than the amount by which it's an underestimate (when it is an underestimate), because the average value still has to be correct.

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