The *-minimax search procedure for trees containing chance nodes

摘要

An extention of the alpha-beta tree pruning strategy to game trees with ‘probability’ nodes, whose values are defined as the (possibly weighted) average of their successors' values, is developed. These ‘*-minimax’ trees pertain to games involving chance but no concealed information. Based upon our search strategy, we formulate and then analyze several algorithms for *-minimax trees. An initial left-to-right depth-first algorithm is developed and shown to reduce the complexity of an exhaustive search strategy by 25–30 percent. An improved algorithm is then formulated to ‘probe’ beneath the chance nodes of ‘regular’ *-minimax 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 deeper trees, two additional algorithms are presented and analyzed.