假开局后的公平决斗
Fair Duels after a False Start
AI总结：
本研究扩展公平决斗模型，证明强制任意有限前缀会改变序列极限行为：除非前缀符合Thue–Morse模式，否则序列最终周期化，周期为2的幂且不超过$4^{n+1}$。
AI中文摘要：
在公平决斗游戏中，两名玩家轮流向对方射击，回合顺序由以下公平性准则定义：下一枪分配给迄今为止赢得决斗可能性较小的玩家。Cooper 和 Dutle 表明，当两名玩家成功射击的概率相同，均为 $p$ 时，随着 $p \ ightarrow 0$，射击序列收敛于 Thue–Morse 序列。我们通过强制序列在恢复此公平性准则之前以任意有限前缀开始来扩展此模型。我们证明，这种初始干扰从根本上改变了序列的最终行为：在 $p \ ightarrow 0$ 极限下，除非初始前缀本身是 Thue–Morse 序列或其补序列的前缀，否则所得序列最终是周期的。此外，最小周期是 2 的幂，且至多为 $4^{n+1}$，其中 $n$ 是前缀的长度，并且重复块本身是 Thue–Morse 序列的前缀。
英文摘要：
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 $4^{n+1}$, where $n$ is the length of the prefix, and the repeating block is itself a prefix of the Thue--Morse sequence.
