Look-Ahead Strategies in One Person Games with Randomly Generated Game Trees

Abstract

A random method for generated binary trees is presented, ad twp forms of a class of one person games called, "Tree Solitaire" which have such trees as their game trees are defined. After what "look ahead strategy" means in terms of such games is discussed, as theorem on the most efficient use of unlimited look-ahead is proved, and a collection of strategies involving 0, 1, or 2 look-ahead per move is introduced. A method involving diagrams is presented for calculation the probability of winning under the various strategies over a restricted class of games. The superiority of one of the l look-ahead strategies over the other is proved for games of the first form on this restricted class. For games of the second form of this class, all the introduced strategies have their chances of winning calculated, and these results are compared among themselves, with the result for the first form of the game, and with the results of Monte Carlo estimation of the chance of winning in a particular case. An approximate methods for evaluating strategies form any given position is introduced, used to explain some of the previous results, and suggest modifications of strategies already defined, which are then evaluated by Monte Carlo methods. Finally, variants on Tree Solitaire are suggested, their general implications are discussed, and using the methods already developed one of the most suggestive variants is studied and the results show a significant reversal from those of the original game, which is explained by the difference in the games on one particular.