A Minimax Algorithm Better Than Alpha-beta?: No and Yes

TL;DR

This paper revisits minimax search algorithms, introduces a practical version of SSS*, and proposes MTD(f), a new best-first search algorithm that outperforms existing methods in game-playing programs.

Contribution

It provides a practical formulation of SSS*, a framework for constructing best-first search algorithms, and introduces MTD(f), a superior fixed-depth minimax algorithm.

Findings

Practical SSS* implementation aligns with alpha-beta plus transposition tables.

MTD(f) outperforms traditional minimax algorithms in experimental tests.

The framework clarifies differences between various best-first search algorithms.

Abstract

This paper has three main contributions to our understanding of fixed-depth minimax search: (A) A new formulation for Stockman's SSS* algorithm, based on Alpha-Beta, is presented. It solves all the perceived drawbacks of SSS*, finally transforming it into a practical algorithm. In effect, we show that SSS* = alpha-beta + ransposition tables. The crucial step is the realization that transposition tables contain so-called solution trees, structures that are used in best-first search algorithms like SSS*. Having created a practical version, we present performance measurements with tournament game-playing programs for three different minimax games, yielding results that contradict a number of publications. (B) Based on the insights gained in our attempts at understanding SSS*, we present a framework that facilitates the construction of several best-first fixed- depth game-tree search algorithms, known and new. The framework is based on depth-first null-window Alpha-Beta search, enhanced with storage to allow for the refining of previous search results. It focuses attention on the essential differences between algorithms. (C) We present a new instance of the framework, MTD(f). It is well-suited for use with iterative deepening, and performs better than algorithms that are currently used in most state-of-the-art game-playing programs. We provide experimental evidence to explain why MTD(f) performs better than the other fixed-depth minimax algorithms.