Backgammon is Hard

TL;DR

This paper proves that determining the winner in backgammon is computationally very hard (NP-Hard, PSPACE-Hard, and EXPTIME-Hard) depending on game conditions, answering an open question from 2001.

Contribution

It establishes the computational complexity of backgammon under various game settings, including real-life scenarios with unknown strategies and dice rolls.

Findings

Deciding a win is NP-Hard, PSPACE-Hard, and EXPTIME-Hard under different conditions.

Proves EXPTIME-Hardness in real-life game settings with unknown strategies.

Highlights the complexity due to potential infinite gameplay from capture rules.

Abstract

We study the computational complexity of the popular board game backgammon. We show that deciding whether a player can win from a given board configuration is NP-Hard, PSPACE-Hard, and EXPTIME-Hard under different settings of known and unknown opponents' strategies and dice rolls. Our work answers an open question posed by Erik Demaine in 2001. In particular, for the real life setting where the opponent's strategy and dice rolls are unknown, we prove that determining whether a player can win is EXPTIME-Hard. Interestingly, it is not clear what complexity class strictly contains each problem we consider because backgammon games can theoretically continue indefinitely as a result of the capture rule.