Wednesday, April 26, 2006

How rollouts work I

The million dollar question is simple enough: out of all the games thatcould result from playing this position, how many do we win (and how manyof our wins and losses are gammons, and how many are backgammons)? Themodel is exactly the same as if we had an urn with a googol balls in it(it's a big urn), and many of the balls have "win" written on them, andsome say "gammon loss", and if we look hard enough there are a few thatread "backgammon win", and so on. (Balls and urns are to probabilitytheorists what teapots and chequerboards are to computer graphicsresearchers, or "squeamish ossifrage" is to cryptographers -- they seem tocome with the territory.) Instead of having the patience to count thegoogol balls, we just give the urn a really good shake and then pull 100balls out without looking, and say for instance "Well, I got 53 wins, 31losses, 9 gammon wins, 6 gammon losses, and a backgammon win -- looks likemy equity's roughly +0.26." and go home. If we were a bit more thorough(but there's still a long way between my "thorough" and yours!), we couldgo a bit further and figure out that by cheating and measuring the sampleproportions instead of the population proportions, we introduced a standarderror of 0.06 into our result. (Of course, the trick is to select a samplesize that's big enough that you reduce the standard error to a tolerablelevel, but small enough that the answer arrives before you get bored.)


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