Efficient winning strategies in random-turn Maker-Breaker games

TL;DR

This paper analyzes random-turn Maker-Breaker positional games, providing efficient strategies for players and determining the threshold probabilities for winning in classical games like Box, Hamilton cycle, and k-vertex-connectivity.

Contribution

It introduces randomized strategies for both players in random-turn Maker-Breaker games and identifies asymptotic winning thresholds for classical positional games.

Findings

Players have efficient randomized strategies ensuring wins at certain probabilities.

The paper determines asymptotic thresholds for winning in classical games.

Strategies are effective under optimal play assumptions.

Abstract

We 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 Chv\'atal and Erd\H os in 1987), the Hamilton cycle game and the $k$-vertex-connectivity game (both played on the edge set of $K_n$). For each of these games we provide each of the players with a (randomized) efficient strategy which 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.