Maker-Breaker Percolation Games I: Crossing Grids

Abstract

Motivated by problems in percolation theory, we study the following 2-player positional 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, Maker and 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 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 $\Lambda_{m \times n}$. Given $m,n\in \mathbb{N}$, for which pairs $(p,q)$ does Maker have a winning strategy 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 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: $\bullet$ If $p\geqslant 2q$, then Maker wins the game on arbitrarily long versions of the narrowest board possible, i.e. Maker has a winning strategy for the $(2q, q)$-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 and crossing games. In addition we pose many new questions and problems.