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 $p \rightarrow 0$, 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.

翻译：2013年，Cooper与Dutle设计了一个对决场景：Alice和Bob相互射击直至一方被命中。每次射击以固定概率$p$（$0<p<1$）成功。射击顺序由贪婪算法决定——每一步将射击权分配给当前成功概率较小的玩家。Cooper与Dutle观察到，当$p\rightarrow0$时，产生的射击序列（由Alice或Bob执行）收敛至无限瑟-莫尔斯序列t，但收敛速度问题仍悬而未决。本文确定了该收敛速度。