{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:40:47Z","timestamp":1776782447563,"version":"3.51.2"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T00:00:00Z","timestamp":1600128000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Motivated by problems in percolation theory, we study the following two-player positional game. Let \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic><\/jats:sub> be a rectangular grid-graph with <jats:italic>m<\/jats:italic> vertices in each row and <jats:italic>n<\/jats:italic> vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims <jats:italic>p<\/jats:italic> (as yet unclaimed) edges of the board \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic><\/jats:sub>, while on each of his turns Breaker claims <jats:italic>q<\/jats:italic> (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 (<jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>)-crossing game on \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>Given <jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic> \u2208 \u2115, for which pairs (<jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>) does Maker have a winning strategy for the (<jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>)-crossing game on \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic><\/jats:sub>? 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 (<jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>)-case. Our main result is to establish the following transition.\n<jats:list list-type=\"bullet\"><jats:list-item><jats:p>If <jats:italic>p<\/jats:italic> \u2265 2<jats:italic>q<\/jats:italic>, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2<jats:italic>q<\/jats:italic>, <jats:italic>q<\/jats:italic>)-crossing game on \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7(<jats:italic>q<\/jats:italic>+1<jats:italic>)<\/jats:italic><\/jats:sub> for any <jats:italic>m<\/jats:italic> \u2208 \u2115.<\/jats:p><\/jats:list-item><jats:list-item><jats:p>If <jats:italic>p<\/jats:italic> \u2264 2<jats:italic>q<\/jats:italic> \u2212 1, then for every width <jats:italic>n<\/jats:italic> of the board, Breaker has a winning strategy for the (<jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>)-crossing game on \u039b<jats:sub><jats:italic>m<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic><\/jats:sub> for all sufficiently large board-lengths <jats:italic>m<\/jats:italic>.<\/jats:p><\/jats:list-item><\/jats:list><\/jats:p><jats:p>Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.<\/jats:p>","DOI":"10.1017\/s0963548320000097","type":"journal-article","created":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T07:24:03Z","timestamp":1600154643000},"page":"200-227","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Maker\u2013Breaker percolation games I: crossing grids"],"prefix":"10.1017","volume":"30","author":[{"given":"A. Nicholas","family":"Day","sequence":"first","affiliation":[]},{"given":"Victor","family":"Falgas-Ravry","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,9,15]]},"reference":[{"key":"S0963548320000097_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2006.01.029"},{"key":"S0963548320000097_ref17","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2010-00678-9"},{"key":"S0963548320000097_ref11","article-title":"Maker\u2013Breaker percolation games II: Escaping to infinity","author":"Day","year":"2020","journal-title":"Journal of Combinatorial Theory, Series B."},{"key":"S0963548320000097_ref7","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200103)18:2<141::AID-RSA1002>3.0.CO;2-W"},{"key":"S0963548320000097_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139167383"},{"key":"S0963548320000097_ref2","first-page":"7","volume-title":"Random Graphs \u201983","author":"Beck","year":"1985"},{"key":"S0963548320000097_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01197577"},{"key":"S0963548320000097_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000936"},{"key":"S0963548320000097_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01897304"},{"key":"S0963548320000097_ref19","doi-asserted-by":"publisher","DOI":"10.1137\/0112059"},{"key":"S0963548320000097_ref13","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20279"},{"key":"S0963548320000097_ref9","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/42\/001"},{"key":"S0963548320000097_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(73)90005-8"},{"key":"S0963548320000097_ref18","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20829"},{"key":"S0963548320000097_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735202"},{"key":"S0963548320000097_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5060(08)70335-2"},{"key":"S0963548320000097_ref20","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1996.12004732"},{"key":"S0963548320000097_ref3","first-page":"51","volume-title":"Combinatorics: Paul Erd\u00f6s is Eighty","volume":"1","author":"Beck","year":"1993"},{"key":"S0963548320000097_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070002"},{"key":"S0963548320000097_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100034241"},{"key":"S0963548320000097_ref21","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20059"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548320000097","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,4]],"date-time":"2021-03-04T12:51:48Z","timestamp":1614862308000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548320000097\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,15]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["S0963548320000097"],"URL":"https:\/\/doi.org\/10.1017\/s0963548320000097","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,15]]},"assertion":[{"value":"\u00a9 The Author(s) 2020. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}