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Variance reduction is unbiased but may be skewed
Posted By: Timothy Chow
Date: Saturday, 11 April 2015, at 3:38 p.m.
In Response To: Variance reduction is unbiased but may be skewed (Phil simborg)
Phil Simborg wrote:
There is an assumption that in every rollout that XG plays every roll and cube decision perfectly
Well, it's true that some people make this assumption. However, the point I'm making doesn't make that assumption.
Suppose that what I'm interested in is the equity when XG plays XG. That is, suppose that for whatever reason, I don't care what the "right" play is, or what GNU vs. GNU would say, or what SuperBotoftheFuture vs. SuperBotoftheFuture would say. I am interested in XG for its own sake and don't care what's right or wrong.
Under that assumption, an infinitely long rollout with XG vs. XG would give me exactly what I'm interested in. Exactly. After all, what I want is the result of XG playing XG, warts and all.
The point of variance reduction is that an infinitely long rollout with variance reduction will also give me exactly what I want. This is what it means when we say that "luck is unbiased."
The observation I'm making here is that despite the above, a finitely long rollout with variance reduction (which is all we ever do in the real world) may not have the property that it is an overestimate 50% of the time and an underestimate 50% of the time, even if a rollout without variance reduction has that property.

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