PreprintCC BY 4.0arXiv
Probabilistic and game thresholds
Abstract
Let X be a finite set and ℱ ⊆ 2X a non-trivial increasing property. We show that the bias-threshold b(ℱ) for the Picker-Chooser ℱ-game satisfies b(ℱ) ≥ α/ pc(ℱ), where pc(ℱ) is the threshold for the property ℱ and α> 0 is an absolute constant. This proves that Picker-Chooser games satisfy one-sided ``probabilistic intuition''. The proof is based on the notion of influence from the theory of boolean functions.
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BibTeX
@misc{nenadov2026probabilistic,
title = {{Probabilistic and game thresholds}},
author = {Rajko Nenadov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30730},
eprint = {2610.30730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30730},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probabilistic and game thresholds
AU - Rajko Nenadov
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30730
UR - https://arxiv.org/abs/2610.30730
AB - Let X be a finite set and ℱ ⊆ 2^X a non-trivial increasing property. We show that the bias-threshold b(ℱ) for the Picker-Chooser ℱ-game satisfies b(ℱ) ≥ α/ p_c(ℱ), where p_c(ℱ) is the threshold for the property ℱ and α> 0 is an absolute constant. This proves that Picker-Chooser games satisfy one-sided ``probabilistic intuition''. The proof is based on the notion of influence from the theory of boolean functions.
ER -