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

@article{Burke2022WinningTW,
  title={Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography},
  author={Kyle G. Burke and Matthew Ferland and Shang-Hua Teng},
  journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2022},
  pages={1217-1228}
}
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… 
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