The speed of convergence in greedy Galois games

2026-04-30

AI Summary

This work investigates the rate at which the shooting sequence in the greedy Galois game converges to the Thue–Morse sequence as the hit probability \( p \to 0 \). In this two-player game, Alice and Bob alternately take shots, with each turn assigned to the player whose cumulative success probability is currently lower. By integrating probabilistic analysis with combinatorial sequence theory, we provide a refined estimation of the recursive structure of the sequence and its deviation from the Thue–Morse limit. Our analysis yields the first rigorous quantification of the convergence rate, resolving an open problem posed by Cooper and Dutle. Specifically, we precisely characterize the asymptotic behavior of the convergence error, establishing its explicit order as \( \Theta(p \log(1/p)) \) in terms of \( p \).

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 $p \rightarrow 0$, 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.

Problem

• greedy Galois games
• Thue-Morse sequence
• convergence speed
• shooting sequence
• probability

Innovation

• greedy Galois games
• Thue-Morse sequence
• convergence rate
• probabilistic dueling
• combinatorial game theory