by Douglas Zare
3 December 2014
Let's continue our study of roll variance in backgammon bearoffs. In Part 1, we saw that in close take/pass decisions, the effective pip count can be off by 2 pips, and that the roll variance of the trailer explains a lot of what the effective pip count misses. In Part 2, we saw what happens when you vary the length of the race, and the roll variance of the player doubling. Each unit of roll variance for the trailer was worth about 2 pips, more in shorter races and when the leader had a low roll variance. In this column, we will look at estimating the roll variance of actual positions.
When we want to estimate the effective pip count, there are two easy cases. Walter Trice found that the effective pip count of a pure n-roll position is close to 7n+1. It is remarkable that this is accurate to within a fraction of a pip. Pure rolls positions are very inefficient. Very efficient positions have effective pip counts about 6-7 pips more than the nominal pip count, or a little less for efficient races with very low pip counts or few checkers. Most positions fall between these two extremes, and you can interpolate. Are there analogues of these for roll variance?
Pure roll positions do not have the lowest roll variances. Positions with no misses but fewer working doubles have lower roll variance. However, the difference is small.
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