A unified convention for achievement positional games

TL;DR

The paper proposes achievement positional games as a unified framework that subsumes Maker-Maker and Maker-Breaker conventions by modeling two colored edge sets $E_L$ and $E_R$ on a shared vertex set. It establishes fundamental properties (edge updating, outcomes, and disjoint unions) and develops a comprehensive complexity map across small edge sizes, proving $\mathrm{P}$-time solvability for $(2,2)$, $\mathrm{coNP}$-completeness for $(2,3)$, $\mathrm{NP}$-hardness for $(3,2)$, and $\mathrm{PSPACE}$-completeness for $(3,3)$ via reductions from SAT and QBF problems. The results generalize strategy-stealing and monotonicity arguments from Maker-Maker to the broader setting, and reveal how disjoint unions and delay concepts govern composite games. The insights advance theoretical understanding of combinatorial game complexity and provide a foundation for extending the framework to avoidance games, CNF formulations, and vertex-partizan variants, with implications for how hardness arises across game conventions.

Abstract

We introduce achievement positional games, a convention for positional games which encompasses the Maker-Maker and Maker-Breaker conventions. We consider two hypergraphs, one red and one blue, on the same vertex set. Two players, Left and Right, take turns picking a previously unpicked vertex. Whoever first fills an edge of their color, blue for Left or red for Right, wins the game (draws are possible). We establish general properties of such games. In particular, we show that a lot of principles which hold for Maker-Maker games generalize to achievement positional games. We also study the algorithmic complexity of deciding whether Left has a winning strategy as first player when all blue edges have size at mot $p$ and all red edges have size at most $q$. This problem is in P for $p,q \leq 2$, but it is NP-hard for $p \geq 3$ and $q=2$, coNP-complete for $p=2$ and $q \geq 3$, and PSPACE-complete for $p,q \geq 3$. A consequence of this last result is that, in the Maker-Maker convention, deciding whether the first player has a winning strategy on a hypergraph of rank 4 after one round of (non-optimal) play is PSPACE-complete.

Figures (8)

• Figure 1: A blue butterfly.
• Figure 2: The partial order $\leq_L$ on the existing outcomes.
• Figure 3: Example games for each outcome.
• Figure 4: Despite the blue and red hypergraphs being isomorphic, Right has a winning strategy as second player.
• Figure 5: The symmetry of the Hex board allows for strategy stealing.

Theorems & Definitions (35)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• proof