Rollouts

 Confidence intervals

 From: Timothy Chow Address: tchow8@hotmail.com Date: 24 May 2010 Subject: Got Variance? Forum: rec.games.backgammon

```What is the equity of a position? It is the value of the game, averaged
over all possible ways of playing out the game (i.e., all possible
subsequent sequences of dice rolls). We can't compute this directly because
we don't have infinite computing power. So we sample from the space of
possible games by doing a rollout.

Ignoring variance reduction for simplicity, what we get is a long list of
outcomes: some +1, some -1, some +2, some -2, some +3, some -3, some +4,
some -4, etc. We can plot a histogram of these results, with the height of
each bar being proportional to how often we observed that result. The
weighted average of the samples gives us an equity estimate.

Now, where does the confidence interval come from? This comes from the
assumption that, in the long run, if we take enough samples, the shape of
the histogram will approach a bell curve (a Gaussian or normal
distribution). The bell curve has certain properties, such as the fact that
95% of the area lies within 2 standard deviations of the mean.

We can estimate the standard deviation by taking the standard deviation of
our samples. This is what the bots report as the 95% confidence interval:
plus or minus 2 sample standard deviations from the sample mean. (Maybe
they do some slight correction to compensate for the fact that the sample
standard deviation is a biased estimate of the true standard deviation even
in the Gaussian case.)
```

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### Rollouts

Advice  (David Montgomery, Apr 1996)
Cautionary tale  (Kit Woolsey, Sept 1995)
Combining rollouts  (Gregg Cattanach+, Dec 2003)
Confidence intervals  (Bob Koca, Nov 2010)
Confidence intervals  (Timothy Chow, May 2010)
Confidence intervals  (Gerry Tesauro, Feb 1994)
Cubeless vs centered-cube rollouts  (Ron Karr, Dec 1997)
Duplicate dice  (David Montgomery, June 1998)
How reliable are rollouts?  (David Montgomery, Aug 1999)
Level-5 versus level-6 rollouts  (Michael J. Zehr, June 1998)
Level-5 versus level-6 rollouts  (Chuck Bower, Aug 1997)
Positions with inaccurate rollouts  (Douglas Zare, Oct 2002)
Reporting results of rollouts  (David Montgomery, June 1995)
Rollout settings  (Lokicol+, Apr 2010)
Settlement limit  (Michael J. Zehr, Apr 1998)
Settlement limit  (Kit Woolsey, Dec 1997)
Settlement limit in races  (Alexander Nitschke, Dec 1997)
Some guidelines  (Kit Woolsey, Apr 1996)
Standard error and JSD  (rambiz+, Feb 2011)
Standard error and JSD  (Stick+, Oct 2007)
Systematic error  (Chuck Bower, Oct 1996)
Tips for doing rollouts  (Douglas Zare, June 2002)
Truncated rollouts  (Gregg Cattanach, Oct 2002)
Truncated rollouts: pros and cons  (Jason Lee+, Jan 2006)
What is a rollout?  (Gregg Cattanach, Dec 1999)

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