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Motivated by problems in percolation theory, we study the following 2-playerpositional game. Let $\Lambda_{m \times n}$ be a rectangular grid-graph with$m$ vertices in each row and $n$ vertices in each column. Two players, Makerand Breaker, play in alternating turns. On each of her turns, Maker claims $p$(as-yet unclaimed) edges of the board $\Lambda_{m \times n}$, while on each ofhis turns Breaker claims $q$ (as-yet unclaimed) edges of the board and destroysthem. Maker wins the game if she manages to claim all the edges of a crossingpath joining the left-hand side of the board to its right-hand side, otherwiseBreaker wins. We call this game the $(p,q)$-crossing game on $\Lambda_{m \timesn}$.  Given $m,n\in \mathbb{N}$, for which pairs $(p,q)$ does Maker have a winningstrategy for the $(p,q)$-crossing game on $\Lambda_{m \times n}$? The$(1,1)$-case corresponds exactly to the popular game of Bridg-it, which is wellunderstood 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 toestablish the following transition:  $\bullet$ If $p\geqslant 2q$, then Maker wins the game on arbitrarily longversions of the narrowest board possible, i.e. Maker has a winning strategy forthe $(2q, q)$-crossing game on $\Lambda_{m \times(q+1)}$ for any $m\in\mathbb{N}$;  $\bullet$ if $p\leqslant 2q-1$, then for every width $n$ of the board,Breaker has a winning strategy for the $(p,q)$-crossing game on $\Lambda_{m\times n}$ for all sufficiently large board-lengths $m$.  Our winning strategies in both cases adapt more generally to other grids andcrossing games. In addition we pose many new questions and problems.

Maker-Breaker Percolation Games I: Crossing Grids