The speed of convergence in greedy Galois games
This resolves a specific open problem in combinatorial game theory, but the result is incremental and of interest primarily to researchers in that niche area.
The paper determines the speed at which the sequence of shots in a greedy Galois game converges to the Thue-Morse sequence as the success probability approaches zero, solving an open problem posed by Cooper and Dutle.
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.