In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e. which player has a winning strategy) is PSPACE-complete even when restricted to 5-uniform hypergraphs (Koepke, 2025). On hypergraphs of rank 3, a structural characterization of the outcome and a polynomial-time algorithm have been obtained for two subcases: one by Kutz (2005), the other by Rahman and Watson (2020) who conjectured that their result should generalize to all hypergraphs of rank 3. We prove this conjecture through a structural characterization of the outcome and a description of both players' optimal strategies, all based on intersections of some key subhypergraph collections, from which we derive a polynomial-time algorithm. Another corollary of our structural result is that, if Maker has a winning strategy on a hypergraph of rank 3, then she can ensure to win the game in a number of rounds that is logarithmic in the number of vertices. Note: This paper provides a counterexample to a similar result which was incorrectly claimed (arXiv:2209.11202, Theorem 22).

翻译：在 Maker-Breaker 位置博弈中，Maker 和 Breaker 轮流选取超图 $H$ 的顶点，当且仅当 Maker 占据 $H$ 中某条边的所有顶点时获胜。判定博弈结果（即哪位玩家拥有必胜策略）是 PSPACE 完全的，即使限制在 5-一致超图上也是如此（Koepke, 2025）。对于秩为 3 的超图，此前已在两个子情形中获得了博弈结果的结构性刻画及多项式时间算法：其一是 Kutz（2005）的工作，其二是 Rahman 与 Watson（2020）的工作，他们猜想其结论应能推广至所有秩为 3 的超图。我们通过基于关键子超图族交集的结构性刻画以及对双方最优策略的描述，证明了该猜想，并由此导出了一个多项式时间算法。我们结构性结果的另一个推论是：若 Maker 在秩为 3 的超图上拥有必胜策略，则她能确保在轮次数目为顶点数对数的范围内赢得博弈。注：本文为一项被错误宣称的类似结果（arXiv:2209.11202，定理 22）提供了反例。