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.

翻译：我们引入达成式位置博弈，这是一种涵盖制造者-制造者与制造者-破坏者两种规范的位置博弈统一框架。我们在同一顶点集上定义两个超图，分别标记为红色与蓝色。两名玩家（左方与右方）轮流选取先前未被选过的顶点。首先完成一条自身颜色边（左方对应蓝色边，右方对应红色边）的玩家获胜（允许平局）。我们建立了此类博弈的通用性质。特别地，我们证明了制造者-制造者博弈中的诸多原理可推广至达成式位置博弈。我们还研究了当所有蓝边规模至多为$p$且所有红边规模至多为$q$时，判定先手玩家左方是否具有必胜策略的算法复杂度：该问题在$p,q \leq 2$时属于P类；当$p \geq 3$且$q=2$时为NP难问题；当$p=2$且$q \geq 3$时为coNP完全问题；当$p,q \geq 3$时为PSPACE完全问题。最后一项结果可推得：在制造者-制造者规范中，判定先手玩家在秩为4的超图上经过一轮（非最优）对弈后是否具有必胜策略属于PSPACE完全问题。