[Bug-gnubg] Confidence intervals from rollouts

[Douglas Zare]

There was an interesting question on the Gammonline bulletin board about the 
standard deviation in cubeless rollouts and the standard deviations of live 
cube results. I've included an excerpt below, in which I attempt to estimate a 
confidence interval for the difference between doubling and not doubling. I'm 
not sure what the best way to do this is, but I suggest that an attempt would 
be worth implementing in gnu.

---- http://www.gammonline.com/members/board/config.cgi?read=38103 ----

rew: Snowie says 0,504 cubeless +- 0,034 (95% confidence interval). Naturally 
this means it is 95% certain that the cubeless equity is in the interval 0,470-
0,538. But what is really interesting is the cubeful equities which is 0,669 
for nodouble and 0,663 for double-take. I interpret your above answer that the 
confidence interval applies rougly to these equities as well, e.g it is 95% 
certain that the nodouble equity is between 0,635 and 0,703. Is this correct
(remember I said rougly)? 

There might be some multiplication factor, usually between 0 and 4. The exact 
value of the factor depends on the chances of future cubes being accepted and 
the number of gammons. Since no double and double-take can result in much 
different future cube actions, the multiplication factors are usually 
different, and higher for double-take than for not doubling. 

However, if you are worrying about whether to double, you don't really want the 
equities. Instead, you want to know the difference between the two equities. 
That is usually known with a greater confidence (smaller confidence interval) 
than the absolute equities. 

For example, I rolled out the position after 

5-4 24/20 13/8
3-3 8/5*(2) 6/3(2)
Dance


twice, with a huge 2-ply cubeless rollout, 36 trials. (This is for the purposes 
of demonstarting the differences between rollouts. I don't recommend using so 
few trials. 2-ply is not unreasonable for this type of position, though.) 

seed 2121:
cubeless equity: 0.530 (+- 0.084)
double/take: 0.805
no double: 0.755 

seed 1250:
cubeless equity: 0.592
double/take: 0.953
no double: 0.838 

So, in the second rollout, the cubeless equity increased by 0.062. The equity 
for double/take increased by 0.148, suggesting a multiplier of about 2.3. The 
equity for no double increased by 0.083, suggesting a multiplier of about 1.3. 

The difference between the two, how much of an error it was not to double 
(assuming that it is right to take) increased by only 0.065, so the difference 
in this case seems to have a multiplier of about 1.0. This suggests that the 
difference, initially 0.050 and then 0.115, has a confidence interval of about 
1.0 times the confidence interval of the cubeless equity. In other positions, 
the difference may have a multiplier greater or less than 1. 

So, if you find in a longer rollout that the error from not doubling is greater 
than (1.0 times) the radius of the confidence interval, you can be confident 
that Snowie rollouts indicate that it is right to redouble. 

----

I have a question about the effectiveness of variance reduction. Two common 
techniques are to subtract an unbiased estimate of luck, and to cycle through 
the possible rolls on every 36 rollouts, or every 1296 rollouts. To estimate 
the standard error, one ignores that the second technique is used. This ought 
to overestimate the standard error. Has anyone determined how much the 
overestimate is? 

Douglas Zare