Best-First Minimax Search: Othello Results

We present a very simple selective search algorithm for two-player games. It always expands next the frontier node that determines the minimax value of the root. The algorithm requires no information other than a static evaluation function, and its time overhead per node is similar to that of alpha-beta minimax. We also present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, at the cost of increased time complexity. In the game of Othello, using the evaluation function from BiIl (Lee & Mahajan 1990), bestfirst minimax outplays alpha-beta at moderate depths. A hybrid best-first extension algorithm, which combines alpha-beta and best-first minimax, performs significantly better than either pure algorithm even at greater depths. Similar results were also obtained for a class of random game trees. Introduction and Overview The best chess machines are competitive with the best humans, but generate millions of positions per move. Their human opponents, however, only examine tens of positions, but search much deeper along some lines of play. Obviously, people are more selective in their choice of positions to examine. The importance of selective search was first recognized by (Shannon 1950). Most work on game-tree search has focussed on algorithms that make the same decisions as fullwidth, fixed-depth minimax. This includes alpha-beta pruning (Knuth & Moore 1975), fixed and dynamic node ordering (Slagle & Dixon 1969), SSS* (Stockman 1979), Scout (Pearl 1984), aspiration-windows (Kaindl, Shams, & Horacek 1991), etc. We define a selective search algorithm as one that makes different decisions than full-width, fixed-depth minimax. These include B* (Berliner 1979), conspiracy search (McAllester 1988), min/max approximation (Rivest 1987), meta-greedy search (Russell & Wefald 1989), and singular extensions (Anantharaman, Campbell, & Hsu 1990). All of these algorithms, except singular extensions, require exponential memory, and most have large time overheads per node expansion. In addition, B* and meta-greedy search require more information than a single static evaluation function. Singular extensions is the only algorithm to be successfully incorporated into a high-performance program. We describe a very simple selective search algorithm, called best-first minimax. It requires only a single static evaluator, and its time overhead per node is roughly the same as alpha-beta minimax. We describe an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth. We also explore best-first extensions, a hybrid combination of alpha-beta and best-first minimax. Experimentally, best-first extensions outperform alpha-beta in the game of Othello, and on a class of random game trees. Earlier reports on this work include (Korf 1992) and (Korf & Chickering 1993). Best-First Minimax Search The basic idea of best-first minimax is to always explore further the current best line of play. Given a partially expanded game tree, with static evaluations of the leaf nodes, the value of an interior MAX node is the maximum of its children’s values, and the value of an interior MIN node is the minimum of its children’s values. There exists a path, called the principal variation, from the root to a leaf node, in which every node has the same value. This leaf node, whose evaluation determines the mini max value of the root, is called the principal leaf. Best-first minimax always expands next the current principal leaf node, since it has the greatest affect on the minimax value of the root. Consider the example in figure 1, where squares represent MAX nodes and circles represent MIN nodes. Figure 1A shows the situation after the root has been expanded. The values of the children are their static values, and the value of the root is 6, the maximum of its children’s values. Thus, the right child is the principal leaf, and is expanded next, resulting in the situation in figure 1B. The new frontier nodes are statically evaluated at 5 and 2, and the value of their MIN parent changes to 2, the minimum of its children’s values. This changes the value of the root to 4, the maximum of its children’s values. Thus, the left child of the root is the new principal leaf, and is expanded next, resultTwo-Player Games 1365 From: AAAI-94 Proceedings. Copyright © 1994, AAAI (www.aaai.org). All rights reserved.