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Rollouts

by Douglas Zare
22 October 2002

Douglas Zare

Statistics are the triumph
of the quantitative method,
and the quantitative method is
the victory of sterility and death.
-- Hillaire Belloc in The Silence of the Sea

Rollouts are one of the most powerful tools for assessing the value of a backgammon position. They have the unfortunate tendency to end rather than to aid a discussion, since it requires some familiarity with both backgammon and basic statistics to argue with them. They ought to be steps on the path toward understanding. Further, they can be used to assess the playing strength of backgammon programs

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Subject: RE: Rollouts

From: MSquonk

Date: 25 Oct 2002 05:35 EST

"When stratification is used, this tends to increase the estimated statistical error when it actually should decrease the statistical error." Could you please elaborate a bit on what is and what should be? As I understand it, when you do a rollout without variance reduction and stratified first two rolls, the estimated error will contain the volatility on the first two rolls, which it shouldn't. Would applying variance reduction on those first two rolls be a feasible solution? Is your above statement also true for stratified rollouts with VR, and why? Thanks.

Subject: Re: Rollouts

From: zare

Date: 25 Oct 2002 11:42 EST

By far the largest effect of stratification is that the luck on the first roll or two completely cancels. This is true whether or not you use variance reduction (by an unbiased estimate of the luck). Stratification makes the rollout results converge to the rollout limit faster.

A relatively minor effect is to increase the difference between the results of individual trials. Independent random points tend to form clumps, so the average variance of an independently chosen sample of size n is (n-1)/n times the variance of the distribution. However, the average variance of the outcomes of stratified rollouts, truncated just after the stratification, will equal the variance of the distribution.

A bit more concretely, let A be the volatility of the rollout limit on the first roll, and let B be the volatility in the rest of the rollout. The trials of an unstratified rollout of 36 games would have an average variance of 35/36 A + 35/36 B. The trials of a stratified rollout of 36 games would have an average variance of 1 A + 35/36 B. The average of the trials of the stratified rollout would have a variance of 1/36 B, whereas the average of the trials of the unstratified rollout would have a variance of 1/36 A + 1/36 B.

Multiplying the observed variance of A + 35/36 B by 1/35 will not produce the correct variance for the average of 1/36 B, though that is what worked (on average) for the unstratified rollouts.

What do you do to A + 35/36 B to obtain 1/36 B? I don't know how to do this, since the rollout does not produce separate values for A and B. There are some clumsy tricks to use in some cases, e.g., you can break up a rollout of length 360 into 10 rollouts of length 36.

Douglas Zare

Subject: Re: Rollouts

From: Larrikin

Date: 18 Nov 2002 06:56 EST

Hi Douglas,

At this point I just wanted to say how much I have enjoyed your articles. There definately seems to be a place for more mathematically orientated discourses on Backgammon. Thank you for your contributions in this regard.

I am new to GammonVillage and I havn't yet read all of your articles, let alone digested them. Thus far, however, I've been particularly impressed by your "Bot Confusion" and "Rollouts" installments.

I would like to raise a few things once I feel I have a firmer grasp of both these articles but I'm wondering if "here" or the Strategy Forum is the best place to do so?

Thanks,

Ian Dunstan.

Subject: Re: Rollouts

From: zare

Date: 18 Nov 2002 19:52 EST

I'm glad you've liked my articles so far.

I think messages attached to the articles work well only if they are posted within the week or perhaps month that the articles are published. If it is more than a month, I think too few people would see the comment, and I would recommend the strategy forum.

Another nice thing about the strategy forum is that it is easier to insert board diagrams there.

Douglas Zare

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