The optimal design of elimination tournaments with a superstar

Tournament is a widespread tool for organizing competition and identifying the most effective agent. We study single- and double-elimination tournaments with heterogeneous players of two types: several regular players and a unique superstar. Players simultaneously choose efforts in each match, winning with a probability calculated with the Tullock success function, the costs are linear with respect to player's efforts. We consider well-known designer's objective functions, i. e. the total efforts associated with the performance of the tournament and the probability of winning the strongest player, and introduce the novel weighted composed function accounting for various related organizer costs. Analysis shows that a double-elimination tournament is less profitable in most cases, except when the tournament organizer is concerned about the probability that the superstar wins the tournament. Special attention is paid to the case of an extremely strong superstar. It is shown that attracting such a player is suboptimal for the organizer, both due to the large payments to this player, which increase with his strength, and due to the strategic demotivation of other tournament participants from exerting high effort, which reduces the overall entertainment value of the tournament. The study is valuable for organizers of various tournaments, as it allows them to choose the optimal tournament format and predict its relationship with future entertainment value.