We study two positional games played on hypergraphs, whose edges may be interpreted as winning sets. Two players take turns picking a previously unpicked vertex of the hypergraph. We say a player fills an edge if that player has picked all the vertices of that edge. In the Maker-Maker convention, whoever first fills an edge wins, or we get a draw if no edge is filled. In the Maker-Breaker convention, the first player aims at filling an edge while the second player aims at preventing the first player from filling an edge. Our main result is that, for both games, deciding whether the first player has a winning strategy is a PSPACE-complete problem even when restricted to 4-uniform hypergraphs (of bounded maximum degree). For the Maker-Maker convention, this improves on the known PSPACE-completeness result for hypergraphs of rank 4. For the Maker-Breaker convention, this improves on the known PSPACE-completeness result for 5-uniform hypergraphs, and closes the complexity gap since the problem for hypergraphs of rank 3 is known to be solvable in polynomial time. As a corollary of our construction, we actually get a stronger result: deciding whether the first player has a winning strategy for the vertex-$C_4$-game played on arbitrary graphs, where the winning sets are the vertex sets of 4-cycles, is a PSPACE-complete problem for both conventions.

翻译：我们研究两类在超图上进行的位势博弈，其中超边可解释为获胜集合。两位玩家轮流选取超图中尚未被选取的顶点。当某位玩家选取了某条边的所有顶点时，我们称该玩家填满了这条边。在 Maker-Maker 约定中，率先填满某条边的玩家获胜；若无任何边被填满，则判为平局。在 Maker-Breaker 约定中，首位玩家旨在填满某条边，而第二位玩家则致力于阻止首位玩家填满任何边。我们的主要结论是：对于这两种博弈，即便限制在4一致（且最大度有界）的超图上，判定首位玩家是否存在必胜策略均为PSPACE完全问题。针对Maker-Maker 约定，该结果改进了已知的关于秩为4的超图的PSPACE完全性结论；针对Maker-Breaker 约定，该结果改进了已知的关于5一致超图的PSPACE完全性结论，并填补了复杂度空隙——因为秩为3的超图上的该问题已知可在多项式时间内求解。作为我们构造的推论，我们实际上获得了更强的结论：对于在任意图上进行的顶点-4-圈博弈（其中获胜集合为4-圈的顶点集），在两种约定下，判定首位玩家是否存在必胜策略均为PSPACE完全问题。