How to Design a Stable Serial Knockout Competition

We investigate a new tournament format that consists of a series of individual knockout tournaments; we call this new format a serial knockout competition (SKC). This format has recently been adopted by the Professional Darts Corporation. Depending on the seedings of the players used for each of the knockout tournaments, players can meet in the various rounds (e.g., first round, second round ... semifinal, final) of the knockout tournaments. Following a fairness principle of treating all players equal, we identify an attractive property of an SKCl each pair of players should potentially meet equally often in each of the rounds of the SKC. If the seedings are such that this property is indeed present, we call the resulting SKC stable. . In this note, we formalize this notion, and we address the following question. Do there exist seedings for each of the knockout tournaments such that the resulting SKC is stable? We show using a connection to the Fano plane that the answer is yes for eight players, and we prove that the resulting SKC is unique up to permutations of the players. We further prove that stable SKCs exist for any numbers of players that are a power of two, and we provide stable schedules for competitions on 16 and 32 players.