A braid is described by both the series of permutations performed on the strands and the choice for each crossing of which strand lies on top. For many classes of braids, ranging from the traditional three-strand braid to more complicated weaves of several strands, a single rule of precedence is used for every crossing of two strands, and thus the rules describing which strand lies on top in each crossing are completely characterized by a tournament whose vertices represent the strands. Braids are often constructed with attention to structural cohesion and ease of design, and this work will associate these desiderata with a mathematical property of the underlying tournaments.