[Paper Review] Analysis of misere Sprouts game with reduced canonical trees

Julien Lemoine, Simon Viennot|ArXiv.org|Aug 30, 2009

Artificial Intelligence in Games4 references3 citations

TL;DR

This paper introduces the reduced canonical tree (RCT) as a novel theoretical and computational tool to analyze misère Sprouts, overcoming the limitations of traditional nimber-based methods that fail in misère play. By efficiently representing and comparing game positions through RCTs, the authors compute the outcome of misère Sprouts up to 17 spots, confirming a winning position for the first player when p ≡ 0, 4, or 5 mod 6 (excluding p=1,4), and demonstrating that the misère game's complexity grows more regularly than in the normal version.

ABSTRACT

Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots, lasts at most 3p-1 moves, and the player who makes the last move wins. In the misere version of Sprouts, on the contrary, the player who makes the last move loses. Sprouts is a very intricate game, and the first computer analysis in 1991 reached only p=11. New results were made possible in 2007 up to p=32 by using combinatorial game theory: when a position is a sum of independant games, it is possible to replace some of these games by a natural number, called the nimber, without changing the winning or losing outcome of the complete position. However, this reduction does not apply to the misere version, making the analysis of Sprouts (and more generally of any game) more difficult in the misere version. In 1991, only p=9 was reached in misere Sprouts, and we describe in this paper how we obtained up to p=17. First, we describe a theoretical tool, the reduced canonical tree, which plays a role similar to the nimber in the normal version. Then, we describe the way we have implemented it in our program, and detail the results it allowed us to obtain on misere Sprouts.

Motivation & Objective

• To overcome the lack of effective combinatorial game theory tools in misère impartial games, where standard nimber reductions fail.
• To develop and implement a practical method for analyzing misère Sprouts beyond the limits of prior computational approaches.
• To extend the known outcome of misère Sprouts from p=9 to p=17, resolving open conjectures and revealing a periodic pattern in game outcomes.
• To provide a scalable computational framework using RCTs that can be adapted to other impartial misère games with decomposable positions.

Proposed method

• The paper introduces the reduced canonical tree (RCT), a canonical representation of Sprouts positions that preserves outcome equivalence under misère play, analogous to nimbers in normal play.
• RCTs are constructed by recursively simplifying game trees using indistinguishability principles, where positions with identical outcomes in all contexts are merged.
• A depth-first search algorithm with manual pruning of less promising branches is used to explore game trees, guided by the RCT structure to reduce redundant computation.
• A post-processing 'check algorithm' is applied to minimize the database of necessary positions, retaining only those essential to prove the outcome of each starting position.
• The method leverages the fact that many complex positions are indistinguishable from simple nim-heap positions, enabling efficient compression and reuse of results.
• The implementation uses a precomputed game tree for S6 as a base, storing only additional positions needed for higher p, significantly reducing memory overhead.

Experimental results

Research questions

• RQ1 Can a canonical tree representation be developed for misère impartial games that functions analogously to the nimber in normal play?
• RQ2 To what extent can the structure of Sprouts positions be simplified using indistinguishability under misère rules?
• RQ3 What is the outcome of misère Sprouts for p=17, and does it confirm or contradict the observed periodic pattern in earlier results?
• RQ4 How does the computational complexity of misère Sprouts grow with p, and can it be managed effectively using RCT-based compression?

Key findings

• The authors successfully computed the outcome of misère Sprouts up to 17 spots, confirming that S17⁻ is a win for the first player.
• The number of essential positions required to prove S17⁻ is 17,583, significantly reduced from the 170,000 nodes explored during computation via the check algorithm.
• The outcome pattern for misère Sprouts shows a period-6 cycle (W for p ≡ 0,4,5 mod 6), with exceptions at p=1 and p=4, contradicting earlier false conjectures.
• The complexity of misère Sprouts increases more regularly with p than in the normal version, suggesting that future computation beyond p=17 will require substantial resources.
• The RCT-based method reduced memory usage to under 100 MB and completed in just 20 hours on a 1.8 GHz processor, demonstrating high efficiency.
• The implementation, including source code and databases, is publicly available under the GNU license, enabling reproducibility and extension to other games.