On optimal algorithms for solving bi-valued game trees

Abstract

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 by adhering to a predetermined ordering of the T leaves of T, an ordering which is independent both of P 0 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 P d . (iii) If T has leaves at both even and od depths from the root, then no optimal algorithm for searching T is directed.