### A Note on the First Occurrence of Strings

#### Abstract

We consider the context of a three-person game in which each player selects strings over $\{0,1\}$ and observe a series of fair coin tosses. The winner of the game is the player whose selected string appears first. Recently, Chen et al. showed that if the string length is greater and equal to three, two players can collude to attain an advantage by choosing the pair of strings $11\ldots 10$ and $00\ldots 01$. We call these two strings "complement strings", since each bit of one string is the complement bit of the corresponding bit of the other string. In this note, we further study the property of complement strings for three-person games. We prove that if the string length is greater than five and two players choose any pair of complement strings (except for the pair $11\ldots 10$ and $00\ldots 01$), then the third player can always attain an advantage by choosing a particular string.