Maker–Breaker percolation games I: crossing grids

Abstract

Abstract 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.