Maker–Breaker percolation games I: crossing grids

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.

The Maker-Breaker percolation game on the square lattice

“…Our second tool is considering variations on Lemma 2.1 (Lemma 2.3 in [7]). This simple result tells us that the edge-boundary of any connected finite subgraph of Z 2 is at most 'a bit' larger than twice the number of edges of this subgraph.…”

“…Useful lemmas. Throughout the paper, we use several times the following reverse isoperimetric inequality observed by Day and Falgas-Ravry [7]. We present two more versions of this lemma, for which we need some definitions.…”

“…In fact, we do not know any such integers even in the boosted version of the game, where Maker is allowed to claim arbitrary but finite number of edges before her first turn. Maker's strategies in the papers of Day and Falgas-Ravry [7,8] suggest that it may be beneficial for Maker to play as a dual Breaker in some auxiliary game. Hence, perhaps some of the techniques developed in the present paper could help in answering those questions.…”

“…In this paper, we address the following Maker-Breaker game played on an infinite board, introduced by Day and Falgas-Ravry [7,8]. Let Λ be an infinite connected graph and let v 0 ∈ V (Λ) be a vertex.…”