Sharp thresholds for half-random games I

Jonas Groschwitz, Tibor Szabó*

*Corresponding author for this work

Research output: Contribution to journalConference paperpeer-review


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.

Original languageEnglish
Pages (from-to)766-794
Number of pages29
JournalRandom Structures and Algorithms
Issue number4
Publication statusPublished - 1 Dec 2016
Externally publishedYes
EventInternational Conference on Random Structures and Algorithms (17th : 2015) - Pittsburgh, United States
Duration: 27 Jul 201531 Jul 2015


  • graph games
  • Hamiltonicity
  • positional games
  • randomized strategy
  • sharp threshold


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