Biased games on random boards

In this paper we analyze biased Maker‐Breaker games and Avoider‐Enforcer games, both played on the edge set of a random board G ∼ G ( n , p ) . In Maker‐Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of G and Breaker responds by claiming b previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the k‐vertex‐connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider‐Enforcer games are the reverse analogue of Maker‐Breaker games with a slight modification, where the two players claim at least 1 and at least b previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker‐Breaker games are known to be “bias‐monotone”, that is, if Maker wins the (1, b ) game, he also wins the ( 1 , b − 1 ) game. Therefore, it makes sense to define the critical bias of a game, b * , to be the “breaking point” of the game. That is, Maker wins the (1, b ) game whenever b < b *and loses otherwise. An analogous definition of the critical bias exists for Avoider‐Enforcer games: here, the critical bias of a game b * is such that Avoider wins the (1, b ) game for every b ≥ b * , and loses otherwise. We prove that, for every p = ω (ln ⁡ n n ) , G ∼ G ( n , p ) is typically such that the critical bias for all the aforementioned Maker‐Breaker games is asymptoticallyb * = n p ln ⁡ n. We also prove that in the case p = Θ (ln ⁡ n n ) , the critical bias isb * = Θ (n p ln ⁡ n) . These results settle a conjecture of Stojaković and Szabó. For Avoider‐Enforcer games, we prove that for p = Ω (ln ⁡ n n ) , the critical bias for all the aforementioned games isb * = Θ (n p ln ⁡ n) . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,651–676, 2015