Towards an Understanding of Minimaxing of Game Trees with Random Leaf Values

Random minimaxing, introduced by Beal and Smith 1], is the process of using a random static evaluation function for scoring the leaf nodes of a full width game tree and then computing the best move using the standard minimax procedure. The experiments carried out by Beal and Smith, using random minimaxing in Chess, showed that the strength of play increases as the depth of the lookahead is increased. We investigate random minimaxing from a combinatorial point of view in order to gain a better understanding of the utility of the minimax procedure and thus obtain a theoretical justiication for the results of Beal and Smith's experiments. The concept of domination is central to our theory. Intuitively, a move by white dominates another move when this move gives less choice for black when it is black's turn to move, and subsequently more choice for white when it is white's turn to move. We view domination as a measure of mobility and show that when one move dominates another then its probability of being chosen is higher. We then investigate when the probability of a \good" move relative to the probability of a \bad" move" increases with the depth of search. We show that there exist situations when increased depth of search is \beneecial", but that this is not always the case. Under the assumption that each move is either \good" or \bad", we are able to state suucient conditions that guarantee that increasing the depth of search increases the strength of play of random minimaxing. If the semantics of the game under consideration match these assumptions then it is fair to say that random minimaxing appears to follow a reasonably \intelligent" strategy. In practice domination does not always occur, so it remains an open problem to nd a more general measure of mobility in the absence of domination.