2005/02/02 by Robert A. Hearn, Hearn, Robert A.
Computer Science · Decision Sciences · #Artificial Intelligence in Games #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Computer Science and Game Theory (cs.GT) #F.2.2 #FOS: Computer and information sciences #Scientific Computing and Data Management #cs.CC #cs.GT
paper · pdf · doi:10.48550/arxiv.cs/0502013
7 pages, 5 figures. For 2005 Combinatorial Game Theory Workshop at BIRS
arxiv created 2005/02/02 · openalex publication_date 2005/02/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Amazons is a board game which combines elements of Chess and Go. It has become popular in recent years, and has served as a useful platform for both game-theoretic study and AI games research. Buro showed that simple Amazons endgames are NP-equivalent, leaving the complexity of the general case as an open problem. We settle this problem, by showing that deciding the outcome of an n x n Amazons position is PSPACE-hard. We give a reduction from one of the PSPACE-complete two-player formula games described by Schaefer. Since the number of moves in an Amazons game is polynomially bounded (unlike Chess and Go), Amazons is in PSPACE. It is thus on a par with other two-player, bounded-move, perfect-information games such as Hex, Othello, and Kayles. Our construction also provides an alternate proof that simple Amazons endgames are NP-equivalent. Our reduction uses a number of amazons polynomial in the input formula length; a remaining open problem is the complexity of Amazons when only a constant number of amazons is used.