Error Propagation in Game Trees

Abstract

Game tree search is the core of most attempts to make computers play games. We present a fairly general theoretical analysis on how leaf evaluation errors influence the value estimation of a game position at the root. By an approach using prime factorization arguments in the ring of polynomials, we show that in this setting the maximum number of leaf-disjoint strategies proving a particular property is a key notion. This number precisely describes the quality of the heuristic game value in terms of the quality of the leaf evaluation heuristics. We extend this model to include random nodes (rolls of a die). Surprisingly, this changes the situation: utill the number of leaf-disjoint strategies ensures robustness against leaf evaluation errors, but the converse is not true. An average node may produce additional robustness similar to additional leaf-disjoint strategies. This work extends earlier ones which only deal with 0, 1 valued nodes, or without randomness.