Sharp Thresholds for Half-Random Games II

  • 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).

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Author details:Jonas GroschwitzORCiD, Tibor Szabo
Title of parent work (English):GRAPHS AND COMBINATORICS
Place of publishing:Tokyo
Publication type:Article
Date of first publication:2017/01/19
Publication year:2017
Release date:2022/06/23
Tag:Connectivity; Fast win; Hamiltonicity; Positional games; Randomized strategy; Sharp threshold
Number of pages:15
First page:387
Last Page:401
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
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