@article{GroschwitzSzabo2017,
  author    = {Groschwitz, Jonas and Szabo, Tibor},
  title     = {Sharp Thresholds for Half-Random Games II},
  series = {GRAPHS AND COMBINATORICS},
  volume    = {33},
  journal   = {GRAPHS AND COMBINATORICS},
  publisher = {Springer},
  address   = {Tokyo},
  issn      = {0911-0119},
  doi       = {10.1007/s00373-016-1753-4},
  pages     = {387 -- 401},
  year      = {2017},
  abstract  = {We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this work we focus on the case of Breaker playing randomly and Maker being "clever". The reverse scenario is treated in a separate paper. We determine the sharp threshold bias of classical games played on the edge set of the complete graph , such as connectivity, perfect matching, Hamiltonicity, and minimum degree-1 and -2. In all of these games, the threshold is equal to the trivial upper bound implied by the number of edges needed for Maker to occupy a winning set. Moreover, we show that CleverMaker can not only win against asymptotically optimal bias, but can do so very fast, wasting only logarithmically many moves (while the winning set sizes are linear in n).},
  language  = {en}
}
@article{GroschwitzSzabo2016,
  author    = {Groschwitz, Jonas and Szabo, Tibor},
  title     = {Sharp Thresholds for Half-Random Games I},
  series = {European polymer journal},
  volume    = {49},
  journal   = {European polymer journal},
  publisher = {Wiley-Blackwell},
  address   = {Hoboken},
  issn      = {1042-9832},
  doi       = {10.1002/rsa.20681},
  pages     = {766 -- 794},
  year      = {2016},
  abstract  = {We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-k. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias tilts significantly more towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph. (C) 2016Wiley Periodicals, Inc.},
  language  = {en}
}