Optimal design of random knockout tournaments

Abstract

For many years, researchers have investigated problems related to the design of sports tournaments. Sports competitions involve many logistical and economic issues, which has led many authors to examine them from a more theoretical point of view. Many studies deal with the best tournament type choice, and the optimal way to devise a draw—that is, where to place players in the tournament in order to optimize the winning probability of a given player, deciding the best way to rank players according to several criteria, and other issues. However, there are relatively few studies on the structure (i.e., the skeleton) of these tournaments, although structure has a big impact on the outcome of the competition. The purpose of this thesis is, therefore, to analyze the different tournament structures and to infer which ones maximize or minimize the strongest player’s probability of winning. The research question of this dissertation is: “In a knockout tournament, that is to say, direct elimination tournament, what type of structure optimizes the strongest player’s probability of winning?” During the elaboration of this paper, different sports tournaments and their specific terminology are explained, winning probabilities of random knockout tournament are computed, and an algorithm is developed in order to provide indications of the effectiveness of the tournament structure and to evaluate and draw conclusions on the types of structure to be chosen. As a result, we support the conjectures of Adler et al. (2017), saying that, in a random knockout tournament and in a general case where the players all have different strengths, the balanced structures maximize the chances of victory for the strongest player. In addition, we also achieve that the structures minimizing the winning probability of the strongest player, are the totally unbalanced ones, that is to say, those where only one match per round is played. Concerning the weakest player, the same analyses were carried out and it was concluded, as Adler et al. (2017), that, conversely, balanced tournaments minimize the chances of victory of the weakest player and totally unbalanced structures maximize them.