We address two-player combinatorial games whose graph of positions is a directed Galton–Watson tree. We consider normal and misère rules (where a player who cannot move loses or wins, respectively), as well as an “escape game” in which one designated player loses if either player cannot move. We study phase transitions for the probability of a draw or escape under optimal play, as the offspring distribution varies. Across a range of natural cases, we find that the transitions are continuous for the normal and misère games but discontinuous for the escape game; we also exhibit examples where these properties fail to hold. We connect the nature of the phase transitions to the length of the game under optimal play. We establish inequalities between the different games. For instance, the draw probability is no smaller in the misère game than in the normal game.

中文翻译：

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高尔顿-沃森博弈

我们解决了两人组合博弈，其位置图是一个有向高尔顿-沃森树。我们考虑正常规则和悲惨规则（不能移动的玩家分别输或赢），以及“逃脱游戏”，其中指定的玩家如果任一玩家不能移动就输。随着后代分布的变化，我们研究了最佳游戏下平局或逃跑概率的相变。在一系列自然情况下，我们发现正常游戏和悲惨游戏的过渡是连续的，但逃逸游戏的过渡是不连续的；我们还展示了这些属性不成立的例子。我们将相变的性质与最佳游戏下的游戏长度联系起来。我们在不同的游戏之间建立不平等。例如，