Fair Duels after a False Start

Abstract: In the fair dueling game, two players take turns shooting at each other, with the turn order defined by the following fairness criterion: the next shot is assigned to the player who is less likely to have won the duel so far. Cooper and Dutle showed that when both players have the same probability $p$ of a successful shot, the shooting sequence converges to the Thue--Morse sequence as $p \rightarrow 0$. We extend this model by forcing the sequence to begin with an arbitrary finite prefix before reverting to this fairness criterion. We prove that this initial disruption fundamentally alters the sequence's eventual behavior: in the $p \rightarrow 0$ limit, unless the initial prefix is itself a prefix of the Thue--Morse sequence or its complement, the resulting sequence is eventually periodic. Moreover, the least period is a power of 2 that is at most $4<sup>{n+1}$, where $n$ is the length of the prefix, and the repeating block is itself a prefix of the Thue--Morse sequence.