PreprintCC BY 4.0arXiv
The speed of convergence in the Cooper-Dutle dueling game
Abstract
In 2013 Cooper and Dutle invented a dueling scenario where Alice and Bob shoot at each other until one is hit. Each shot is successful with some fixed probability p, 0 < p < 1. The shooting order is given by a greedy algorithm, where at each step a shot is assigned to the player whose current probability of success is smaller. Cooper and Dutle observed that as p → 0, the resulting sequence of shots (by Alice or Bob) converges to the infinite Thue-Morse sequence t, but left the speed of convergence as an open problem. In this note we determine the speed of this convergence.
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BibTeX
@misc{shallit2026speed,
title = {{The speed of convergence in the Cooper-Dutle dueling game}},
author = {Jeffrey Shallit},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.00194},
eprint = {2605.00194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00194},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The speed of convergence in the Cooper-Dutle dueling game
AU - Jeffrey Shallit
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2605.00194
UR - https://arxiv.org/abs/2605.00194
AB - In 2013 Cooper and Dutle invented a dueling scenario where Alice and Bob shoot at each other until one is hit. Each shot is successful with some fixed probability p, 0 < p < 1. The shooting order is given by a greedy algorithm, where at each step a shot is assigned to the player whose current probability of success is smaller. Cooper and Dutle observed that as p → 0, the resulting sequence of shots (by Alice or Bob) converges to the infinite Thue-Morse sequence t, but left the speed of convergence as an open problem. In this note we determine the speed of this convergence.
ER -