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 the first player when blue edges and red edges have respective sizes at most $p$ and $q$. 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 $p,q \geq 3$ even when the 3-edges are the same for both colors. That last result has an interesting consequence on the Maker-Maker convention: for 3-uniform hypergraphs, which is the only case whose complexity is currently open (for starting positions of the game), we show PSPACE-completeness for positions obtained after one round of play.

翻译：我们引入成就性位置博弈——一种涵盖Maker-Maker和Maker-Breaker约定的位置博弈新框架。该博弈定义在共享同一顶点集的两个超图上：红色超图与蓝色超图。左右两位玩家轮流选取未被选中的顶点。率先填满己方颜色超边（左方对应蓝色边，右方对应红色边）的玩家获胜（平局亦可能发生）。我们建立了此类博弈的一般性质，特别证明了Maker-Maker博弈中的许多原理可推广至成就性位置博弈。此外，我们研究了当蓝色边与红色边各自包含至多p个和q个顶点时，判断左方作为先手是否存在必胜策略的算法复杂度：当p,q ≤ 2时问题属于P类；当p ≥ 3且q=2时为NP难问题；当p=2且q ≥ 3时为coNP完全问题；当p,q ≥ 3时，即使两种颜色的3-超边完全相同，问题仍为PSPACE完全问题。最后这一结果对Maker-Maker约定具有重要推论：对于当前复杂度尚待厘清的三一致超图（仅针对游戏起始位置），我们证明了经过一轮对弈后的局面判定问题具有PSPACE完全性。