Greedy Galois Games

TL;DR

This paper analyzes the strategic firing sequence of two duelers with poor shooting skills, showing they follow the Thue-Morse sequence when acting greedily, contrasting with more patient duelers' behavior from approximation theory.

Contribution

It reveals that greedy decision-making in a Galois dueling scenario results in the Thue-Morse sequence, linking game theory with complex function approximation and fractional base expansions.

Findings

Greedy duelers follow the Thue-Morse sequence in firing order.

Contrasts with more patient duelers' fairer strategies from approximation theory.

Connects the sequence to expansions in fractional bases near 1.

Abstract

We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other's a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions that says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue-Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.