TY  - JOUR
A1  - Groschwitz, Jonas
A1  - Szabo, Tibor
T1  - Sharp Thresholds for Half-Random Games II
JF  - GRAPHS AND COMBINATORICS
N2  - 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).
KW  - Positional games
KW  - Randomized strategy
KW  - Sharp threshold
KW  - Fast win
KW  - Hamiltonicity
KW  - Connectivity
Y1  - 2017
U6  - https://doi.org/10.1007/s00373-016-1753-4
SN  - 0911-0119
SN  - 1435-5914
VL  - 33
SP  - 387
EP  - 401
PB  - Springer
CY  - Tokyo
ER  -