Maker-Breaker Percolation Games I: Crossing Grids

Abstract

Motivated by problems in percolation theory, we study the following 2-player positional game. Let Λ m×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 Λ m×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 Λ m×n .Given m, n ∈ N, for which pairs (p, q) does Maker have a winning strategy for the (p, q)crossing game on Λ m×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, i.e. Maker has a winning strategy for the (2q, q)-crossing game on Λ m×(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 Λ m×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.

Search for an immobile hider on a stochastic network

“…Bollobás et al (2013) considered a cop and robbers games played on a random graph. Some intriguing interactions between percolation and game theory have been recently studied by Day andFalgas-Ravry (2018), Holroyd et al (2019), who considered two-person zero-sum games on a graph with alternating moves.…”