Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n.Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ.If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m.Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

中文翻译：

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Maker-Breaker 渗透博弈 I：穿越网格

受渗流理论问题的启发，我们研究了以下两人位置游戏。让Λ米×n是一个矩形网格图米每行的顶点和n每列的顶点。Maker 和 Breaker 两个玩家轮流进行游戏。在她的每一个回合中，梅克声称p（尚未认领的）棋盘边缘 Λ米×n，而在他的每个回合中，Breaker 声称q（尚未认领）板的边缘并摧毁它们。如果 Maker 设法占据连接棋盘左侧和右侧的交叉路径的所有边缘，则 Maker 赢得游戏，否则 Breaker 获胜。我们称这个游戏为（p,q)-Λ 上的穿越游戏米×n.鉴于米,n∈ ℕ，其中对 (p,q) Maker 是否有针对 (p,q)-Λ 上的穿越游戏米×n? (1, 1)-case 完全对应于流行的 Bridg-it 游戏，由于它是旧版 Shannon 切换游戏的一个特例，所以很好理解。在本文中，我们研究一般（p,q）-案子。我们的主要结果是建立以下过渡。如果p≥ 2q, 那么 Maker 在可能的最窄棋盘的任意长版本上赢得游戏，也就是说，Maker 的获胜策略是 (2q,q)-Λ 上的穿越游戏米×(q+1)对于任何米∈ℕ。如果p≤ 2q− 1，然后对于每个宽度n在董事会中，Breaker 有一个获胜策略（p,q)-Λ 上的穿越游戏米×n对于所有足够大的板长米.我们在这两种情况下的获胜策略更普遍地适应其他网格和交叉游戏。此外，我们提出了许多新的问题和问题。