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Maker-Maker games of rank 4 are PSPACE-complete

Florian Galliot, Jonas Sénizergues

TL;DR

This work resolves the complexity of rank-4 Maker-Maker games by establishing PSPACE-completeness through a polynomial-time reduction from 3-QBF to an achievement positional game with blue edges of size at most 3 and pairwise disjoint red edges of size 2. The core technique encodes a quantified Boolean formula as a network of variable gadgets, clause gadgets, and edge types (butterfly, destruction, link, trap) that force a valuation during Phase 1 and leverage clause satisfaction in Phase 2. The reduction implies PSPACE-completeness for starting positions of Maker-Maker games on rank-4 hypergraphs, significantly advancing the known complexity landscape beyond the prior rank-6 results. The results also suggest new directions for understanding the remaining cases in rank-4 and higher, and illustrate the potential of the achievement-position framework as a tool for hardness constructions in positional games.

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.

Maker-Maker games of rank 4 are PSPACE-complete

TL;DR

This work resolves the complexity of rank-4 Maker-Maker games by establishing PSPACE-completeness through a polynomial-time reduction from 3-QBF to an achievement positional game with blue edges of size at most 3 and pairwise disjoint red edges of size 2. The core technique encodes a quantified Boolean formula as a network of variable gadgets, clause gadgets, and edge types (butterfly, destruction, link, trap) that force a valuation during Phase 1 and leverage clause satisfaction in Phase 2. The reduction implies PSPACE-completeness for starting positions of Maker-Maker games on rank-4 hypergraphs, significantly advancing the known complexity landscape beyond the prior rank-6 results. The results also suggest new directions for understanding the remaining cases in rank-4 and higher, and illustrate the potential of the achievement-position framework as a tool for hardness constructions in positional games.

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.

Paper Structure

This paper contains 11 sections, 4 theorems, 2 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and statements of the main results
  3. Definitions
  4. Elementary strategic principles
  5. Statement of the main results
  6. Proof of Theorem \ref{['theo:32-new']}
  7. Idea of the construction
  8. Definition of the game
  9. Regular play
  10. Optimality of regular play
  11. Conclusion

Key Result

Lemma 2.1

JonasFlorian Let $\mathcal{G}=(V,E_L,E_R)$ be an achievement positional game. If the hypergraph $(V,E_R)$ (resp. $(V,E_L)$) admits a complete pairing, then Left (resp. Right) has a non-losing strategy on $\mathcal{G}$ both as the first player and as the second player.

Figures (2)

  • Figure 1: A blue butterfly plus a red edge.
  • Figure 3: The updated variable gadget associated with $x_i$ and $y_i$ at the start of round $i$ of regular play. For clarity, we use a shaded blue edge $\{u,u'\}$ to represent $\{u,u'\}^{(i)}$ (so that the left part and the right part are actually connected to each other).

Theorems & Definitions (13)

  • Lemma 2.1: Pairing Strategy
  • Lemma 2.2: Greedy Move
  • Theorem 2.3
  • Corollary 2.4
  • proof : Proof of Corollary \ref{['coro:makermaker4']} assuming Theorem \ref{['theo:32-new']}
  • Claim 1
  • proof : Proof of Claim \ref{['cla:regular0']}
  • Claim 2
  • proof : Proof of Claim \ref{['cla:regular1']}
  • Claim 3
  • ...and 3 more