Fair Duels after a False Start
Abstract
In the fair dueling game, two players take turns shooting at each other, with the turn order defined by the following fairness criterion: the next shot is assigned to the player who is less likely to have won the duel so far. Cooper and Dutle showed that when both players have the same probability p of a successful shot, the shooting sequence converges to the Thue--Morse sequence as p → 0. We extend this model by forcing the sequence to begin with an arbitrary finite prefix before reverting to this fairness criterion. We prove that this initial disruption fundamentally alters the sequence's eventual behavior: in the p → 0 limit, unless the initial prefix is itself a prefix of the Thue--Morse sequence or its complement, the resulting sequence is eventually periodic. Moreover, the least period is a power of 2 that is at most 4n+1, where n is the length of the prefix, and the repeating block is itself a prefix of the Thue--Morse sequence.
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BibTeX
@misc{khovanova2026fair,
title = {{Fair Duels after a False Start}},
author = {Tanya Khovanova and Tejo Madhavarapu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00880},
eprint = {2610.00880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00880},
}
RIS
TY - UNPB
TI - Fair Duels after a False Start
AU - Tanya Khovanova
AU - Tejo Madhavarapu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00880
UR - https://arxiv.org/abs/2610.00880
AB - In the fair dueling game, two players take turns shooting at each other, with the turn order defined by the following fairness criterion: the next shot is assigned to the player who is less likely to have won the duel so far. Cooper and Dutle showed that when both players have the same probability p of a successful shot, the shooting sequence converges to the Thue--Morse sequence as p → 0. We extend this model by forcing the sequence to begin with an arbitrary finite prefix before reverting to this fairness criterion. We prove that this initial disruption fundamentally alters the sequence's eventual behavior: in the p → 0 limit, unless the initial prefix is itself a prefix of the Thue--Morse sequence or its complement, the resulting sequence is eventually periodic. Moreover, the least period is a power of 2 that is at most 4^n+1, where n is the length of the prefix, and the repeating block is itself a prefix of the Thue--Morse sequence.
ER -