The impossibility of a perfect tournament

Abstract

Is there a way to determine which tournament format is better than another? Are round-robin tournaments superior to a single-elimination bracket? Using elements from game design, we generalize the existing literature on tournament design and establish metrics to evaluate the effectiveness of different tournament formats. Our approach does not make any assumptions about the sport or game being played, allowing us to analyze tournaments in their basic form. We establish two basic metrics for determining the effectiveness of a tournament schedule: fairness and balance. These metrics follow logically from the purpose of a tournament, which is to determine the best player. We prove an impossibility theorem that most tournament schedules cannot minimize both fairness and balance, including all those used in practice.