The *-Minimax Search Procedure for Trees Containing Chance Nodes

An extention of the alpha-beta tree priming strategy to game trees with "probability" nodes, whose values are defined as the (possibly weighted) average of their successors' values, is developed. These '*-minhnax' trees pertain to games involving chance but no concealed infornzation. Based upon our search strategy, )re formulate and then analyze several algorithnzs for *-minimax trees. An initial left-to-right depth-first algorithm is developed and shown to reduce the conzplexity of an exhaustive search strategy by 25-30 percent. An improved algorithnz is then formtdated to 'probe" beneath the chance nodes of 'regular" *-nzininzax trees, where players alternate in making moves with chance events interspersed. With random ordering of successor nodes, this modified algorithm is shown to reduce search by more than 50 percent. With optimal ordering, it is shown to reduce search complexity by an order of magnitude. After examining the savings of the first two algorithms on deep(r trees, two additional algorithms are presented and analyzed.