Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

Abstract

AbstractWe consider random‐turn positional games, introduced by Peres, Schramm, Sheffield, and Wilson in 2007. A p‐random‐turn positional game is a two‐player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random‐turn version of several classical Maker–Breaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the k‐vertex‐connectivity game (both played on the edge set of ). For each of these games we provide each of the players with a (randomized) efficient strategy that typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.