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

We settle two long-standing complexity-theoretical questions—open since 1981 and 1993—in combinatorial game theory (CGT). We prove that the Grundy value 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 sharp “phase transition to intractability”: The Grundy value of the game over any degree-three graph is polynomial-time computable, but over degree-four graphs—even when planar & bipartite—is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value *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, we extend the result 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. Our result also has computational ramification 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 is not equal to P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum. Our proof enables us to answer another long-term structural question in the field. We establish the following complexity independence: Unless
$\mathrm{P}= \text{PSPACE}$
, there is no polynomial-time reduction from winnability in misere-play setting to the Grundy value, and vice versa (in Undirected Geography).