Amazons is PSPACE-complete

TL;DR

This paper proves that the general Amazons game is PSPACE-complete by providing a polynomial reduction from a known PSPACE-complete problem, establishing its computational complexity and placing it among other complex two-player games.

Contribution

It establishes the PSPACE-completeness of Amazons, resolving an open problem and providing a reduction from a known PSPACE-complete game, with implications for game complexity theory.

Findings

Amazons is PSPACE-hard.

Amazons is in PSPACE due to polynomial move bounds.

Simple Amazons endgames are NP-equivalent.

Abstract

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.