The speed of convergence in greedy Galois games

TL;DR

This paper investigates how quickly the sequence of shots in a probabilistic dueling game converges to the Thue-Morse sequence as the success probability approaches zero.

Contribution

It precisely determines the rate at which the greedy shooting sequence converges to the Thue-Morse sequence as success probability tends to zero.

Findings

The sequence converges to the Thue-Morse sequence at a specific rate.

The convergence speed is characterized mathematically.

The result clarifies the dynamics of the greedy algorithm in the game.

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.