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Maker-Maker Games of Rank 4 Are PSPACE-Complete

Authors Florian Galliot , Jonas Sénizergues

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LIPIcs.STACS.2026.40.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Game theory
  • Positional games
  • Combinatorial games
  • Complexity
  • Hypergraphs

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Abstract

The Maker-Maker convention of positional games is played on a hypergraph whose edges are interpreted as winning sets. Two players take turns picking a previously unpicked vertex, aiming at being first to pick all the vertices of some edge. Optimal play can only lead to a first player win or a draw, and deciding between the two is known to be PSPACE-complete even for 6-uniform hypergraphs. We establish PSPACE-completeness for hypergraphs of rank 4. As an intermediary, we use the recently introduced achievement positional games, a more general convention in which each player has their own winning sets (blue and red). We show that deciding whether the blue player has a winning strategy as the first player is PSPACE-complete even with blue edges of size 2 or 3 and pairwise disjoint red edges of size 2. The result for hypergraphs of rank 4 in the Maker-Maker convention follows as a simple corollary.

Cite As Get BibTex

Florian Galliot and Jonas Sénizergues. Maker-Maker Games of Rank 4 Are PSPACE-Complete. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.40

Author Details

Florian Galliot
  • Aix-Marseille Université, CNRS, I2M, UMR 7373, 13453 Marseille, France
Jonas Sénizergues
  • Université de Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France

References

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