We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart showed that in this setting one of the players always has a winning strategy, though it is not known in advance which player. We present simple necessary conditions for the first player to have a winning strategy, and establish an equivalence between winning sets that guarantee a win for the first player and maximal prefix codes. Using this equivalence, we derive a necessary algebraic condition for winning, and exhibit a family of games for which this algebraic condition is in fact equivalent to winning. We introduce the concept of coverings, and show that by covering the tree of the game with an infinite labeled tree corresponding to the free group, we can use "game-theoretic tools" to derive a simple trait of maximal prefix codes.

翻译：我们研究在无限树上进行的交替移动双人博弈。主要关注树为完全（正则）且获胜集合为开集（相对于树的乘积拓扑）的情形。Gale和Stewart证明在此设定下总有一方拥有必胜策略，尽管事先无法确定是哪一方。我们提出了首玩家拥有必胜策略的简单必要条件，并建立了保证首玩家获胜的获胜集合与极大前缀码之间的等价关系。利用该等价关系，我们推导出获胜的必要代数条件，并构造了一族博弈使得该代数条件实际上等价于获胜。我们引入覆盖的概念，证明通过用对应于自由群的无限标记树覆盖博弈树，可以运用"博弈论工具"推导出极大前缀码的简单特征。