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We settle two long-standing complexity-theoretical questions&#x2014;open since 1981 and 1993&#x2014;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 &#x201C;phase transition to intractability&#x201D;: The Grundy value of the game over any degree-three graph is polynomial-time computable, but over degree-four graphs&#x2014;even when planar &#x0026; bipartite&#x2014;is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value &#x002A;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&#x2014;in polynomial time&#x2014;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 <tex>$\mathrm{P}= \text{PSPACE}$</tex>, there is no polynomial-time reduction from winnability in misere-play setting to the Grundy value, and vice versa (in Undirected Geography).

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