The structure, efficacy, and manipulation of double-elimination tournaments

A double-elimination tournament is a competition where no participant is eliminated until they have lost two matches. It is structured as two single-elimination tournaments: the winner bracket and the loser bracket. Players who lose once in the winner bracket are mapped to positions in the loser bracket, according to a mapping called the link function. Surprisingly, although the same structure of the winner and loser brackets is used universally, there is no standard definition of the link function. By investigating several design goals, we show that the functions used in practice are not optimal. We propose a similar function that is optimal with respect to our design goals. In order to demonstrate some of the possible research questions about double-elimination tournaments, we address the manipulability of the outcome of a double-elimination tournament. We show that they are vulnerable to manipulation by a coalition of players who can improve their chance of winning by throwing matches, a phenomenon recently observed in Olympic Badminton. We also discuss the computational complexity of manipulation by a tournament organizer (agenda control) in two settings: by changing the player seeding in the winner bracket, or by picking the mapping of losers to the loser bracket. We provide algorithms, hardness proofs, and we formulate open problems for future research.