We study multi-player turn-based games played on (potentially infinite) directed graphs. An outcome is assigned to every play of the game. Each player has a preference relation on the set of outcomes which allows him to compare plays. We focus on the recently introduced notion of weak subgame perfect equilibrium (weak SPE). This is a variant of the classical notion of SPE, where players who deviate can only use strategies deviating from their initial strategy in a finite number of histories. Having an SPE in a game implies having a weak SPE but the contrary is generally false.

We propose general conditions on the structure of the game graph and on the preference relations of the players that guarantee the existence of a weak SPE, that additionally is finite-memory. From this general result, we derive two large classes of games for which there always exists a weak SPE: (i) the games with a finite-range outcome function, and $(ii)$ the games with a finite underlying graph and a prefix-independent outcome function. For the second class, we identify conditions on the preference relations that guarantee memoryless strategies for the weak SPE.

我们研究在（可能是无限的）有向图上玩的多人回合制游戏。将结果分配给游戏的每个游戏。每个玩家在一组结果上都有偏好关系，这使他可以比较比赛。我们关注于最近引入的弱子博弈完美均衡（弱SPE）的概念。这是SPE经典概念的一种变体，在SPE中，偏离的玩家只能在有限的历史记录中使用偏离其初始策略的策略。在游戏中拥有SPE意味着拥有较弱的SPE，但相反通常是错误的。

我们提出了关于游戏图结构和玩家偏好关系的一般条件，这些条件保证了弱SPE的存在，这又是有限内存。从这个一般结果中，我们得出两类大游戏，它们的SPE总是很弱：（i）具有有限范围结果函数的游戏，以及$（\mathrm{一世}\mathrm{一世}）$带有有限基础图和不依赖前缀的结果函数的游戏。对于第二类，我们确定了偏好关系上的条件，这些条件可以保证弱SPE的无记忆策略。