Error analysis in minimax trees

Abstract

Game tree search deals with the problems that arise, when computers play two-person-zero-sum-games such as chess, checkers, othello, etc. The greatest success of game tree search so far, was the victory of the chess machine ‘Deep Blue’ vs. G. Kasparov (ICCA J. 20 (1997) 95), the best human chess player in the world at that time. In spite of the enormous popularity of computer chess and in spite of the successes of game tree search in game playing programs, we do not know much about a useful theoretical background that could explain the usefulness of (selective) search in adversary games. We introduce a combinatorial model, which allows us to model errors of a heuristic evaluation function, with the help of coin tosses. As a result, we can show that searching in a game tree will be ‘useful’ if, and only if, there are at least two leaf-disjoint strategies which prove the root value. In addition, we show that the number of leaf-disjoint strategies, contained in a game tree, determines the order of the quality of a heuristic minimax value. The model is integrated into the context of average-case analyses.