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-vertexconnectivity 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 ( lnn n ) , G ∼ G(n, p) is typically such that the critical bias for all the aforementioned Maker-Breaker games is asymptotically b∗ = np lnn . We also prove that in the case p = Θ ( lnn n ) , the critical bias is b∗ = Θ ( np lnn ) . These results settle a conjecture of Stojaković and Szabó. For Avoider-Enforcer games, we prove that for p = Ω ( lnn n ) , the critical bias for all the aforementioned games is b∗ = Θ ( np lnn ) .