Abstract
AbstractMotivated 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.
Highlights
1.1 Results and organization of the paper Biased Maker–Breaker games are a central area of research on positional games, in particular due to their intriguing and deep connections to resilience phenomena in discrete random structures
Much of the research on Maker–Breaker games has focused on the case where the ‘board’ is a complete hypergraph, or an arithmetically defined hypergraph corresponding to all the solutions to a system of equations in some finite integer interval
In this paper we focus on boards and winning sets with rather different properties: we consider rectangular grid-graphs, and our winning sets consist of crossing paths, whose sizes can vary wildly
Summary
1.1 Results and organization of the paper Biased Maker–Breaker games are a central area of research on positional games, in particular due to their intriguing and deep connections to resilience phenomena in discrete random structures. A natural question to ask is, given positive integers m, n, p, q, which player has a winning strategy for the (p, q)-crossing game on m×n?. If p 2q and n q + 1, Maker has a winning strategy for the (p, q)-crossing game on m×n. There exists a natural number m0 = m0(n, q) such that if p 2q − 1 and m m0, Breaker has a winning strategy for the (p, q)-crossing game on m×n. If Maker has strictly less than twice Breaker’s power, Breaker has a winning strategy on all boards that are sufficiently long (with respect to the board width n and Breaker’s power q).
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