# Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

arXiv: https://arxiv.org/abs/2106.02114 · HTML: https://mlsanity.com/paper/2106.02114/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 47 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 4 of 4)

- [Quantum Combinatorial Games: Structures and Computational Complexity](https://mlsanity.com/paper/2011.03704/) (2020)
- [Strategy-Stealing is Non-Constructive](https://mlsanity.com/paper/1911.06907/) (2019)
- [Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players](https://mlsanity.com/paper/2002.03887/) (2020)
- [Feedback game on Eulerian graphs](https://mlsanity.com/paper/2002.09570/) (2020)

## Built on by (most cited first, 1 of 1)

- [Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings](https://mlsanity.com/paper/2109.05622/) (2021)

## Abstract (arXiv)

We settle two long-standing complexity-theoretical questions-open since 1981 and 1993-in combinatorial game theory (CGT). We prove that the Grundy value (a.k.a. nim-value, or nimber) of Undirected Geography is PSPACE-complete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomial-time solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundy-value computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geography over any degree-three graph is polynomial-time computable, but over degree-four graphs-even when planar and bipartite-is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value ast n and size polynomial in n. We strengthen a result from 1981 showing that sums of tractable partisan games are PSPACE-complete in two fundamental ways. First, since Undirected Geography is an impartial ruleset, we extend the hardness of sums to impartial games, a strict subset of partisan. Second, the 1981 construction is not built from a natural ruleset, instead using a long sum of tailored short-depth game positions. We use the sum of two Undirected Geography positions to create our hard instances. Our result also has computational implications to Sprague-Grundy Theory (1930s) which shows that the Grundy value of the disjunctive sum of any two impartial games can be computed-in polynomial time-from their Grundy values. In contrast, we prove that assuming PSPACE neq P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.