Better than Average

by Douglas Zare
5 May 2014

Douglas Zare

We would like to estimate many quantities in backgammon. What is the value of this position if I take? What is my chance to win against player X? What is my error rate? A standard method for all of these is to collect some data and use the average value as the estimate of the quantity studied. What could be better than this?

The average value is just one possible estimate, and it is not always the most reasonable. The average number of children in a family might be 2.3, but 2.3 would be a bad guess for a particular family because we know that the number of children should be a whole number. Similarly, suppose we roll out a take/pass decision and get results of +2, +2, and then -64. The total is -60, and the average is -20. At this point, is our best guess that the position is worth -20? No, if the cube is on 2 then the position is worth between -6 and +6. A value of -20 is impossible. This suggests that there might be better estimates than the average even if the average is within the logically possible range.

Very short rollouts are close to useless if you only average the results. The noise dominates, and you get less information than an evaluation. When there is a small amount of data, or there is the potential of large outliers, other approaches may be much better. An alternative parametric approach is to build a model incorporating more of our understanding of backgammon, and then estimate parameters of that model. If we have a parametric model, then we can combine several sources of information, and even a short rollout can be useful.

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