Biased games on random boards1. Issue 4 (24th February 2014) - Ferber, Asaf
ABSTRACT 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⁡nn), G∼G(n, p) is typically such that the critical bias for all the aforementioned Maker‐Breaker games is asymptotically b*=npln⁡n. We also prove that in the case p=Θ(ln⁡nn), the critical bias is b*=Θ(npln⁡n). These results settle a conjecture of Stojaković and Szabó. For Avoider‐Enforcer games, we prove that for p=Ω(ln⁡nn), the critical bias for all the aforementioned games is b*=Θ(npln⁡n). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46, 651–676, 2015