A Statistical Treatment of the Problem of Division

Abstract

The problem of division is one of the most important problems in the emergence of probability. The problem has been long considered ''solved'' from a probabilistic viewpoint. However, we do not find the solution satisfactory. In this study, the problem is recasted as a statistical problem. The outcomes of matches of the game are considered as an infinitely exchangeable random sequence and predictors/estimators are constructed in light of de Finetti representation theorem. Bounds of the estimators are derived over wide classes of priors (mixing distributions). Interestingly, the classical solutions are identified as the most conservative solutions while the plug-in estimators are found to be out of range of the bounds. It implies that, although conservative, the classical solutions are justified by our analysis while the plug-in estimators are too optimistic for the winning player.