Abstract: In competition law, market power requirements are often articulated in terms of market shares. The use of market share thresholds, however, conflates two distinct questions: First, how much market power exists in a given situation? Second, how much market power should the law require? As a consequence, neither question is answered, or even directly illuminated. Furthermore, because market shares are not themselves measures of market power but instead merely a factor that bears on its magnitude in a given setting, they are inapt answers to both inquiries. Their use involves a category mistake. The identified problems are illustrated by unpacking Learned Hand's famous pronouncement in Alcoa of the market shares required for the offense of monopolization, but the core defects characterize all market share declarations.