The speed of convergence in greedy Galois games

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 $p\to 0$, 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.