On optimal algorithms for solving bi-valued game trees
Harold W. Martin, Terrance Lee Seethoff · 1987
Let each leaf in a bi-valued game tree have the same probability PO of being labeled a WIN. Define a search algorithm D for a bi-valued game tree T to be directed if D consists of searching T by adhering to a predetermined ordering of the leaves of T, an ordering which is independent both of PO and of the assignment of WINs and LOSSes to the leaves of T. The following theorems are established for an arbitrary bi-valued game tree T: (i) An optimal search algorithm for T has a piecewise polynomial cost function V(T) in PO and V(T) is the infimum of the set of all cost functions VA (T) where A ranges over the set of all directed search algorithms. (ii) T has an optimal search algorithm which is directed if and only if it has an optimal search algorithm with a cost function which is a polynomial in PO. (iii) If T has leaves at both even and odd depths from the root, then no optimal algorithm for searching T is directed.