Mathematics of tournaments

SUMMARY: Mathematical methods can be used to seed the bracket for a tournament.

A tournament is any of a variety of competitions in which a relatively large number of players or teams compete at a sport, game, or other competitive activity. While formats differ widely, tournaments generally involve teams or individuals playing a large number of games in a relatively brief period of time. Typically, the ostensible purpose is to determine a single overall winner when the total number of players is (much) larger than the number of players who can participate in a single match. Tournaments of various kinds are held for most competitive activities.

Considerable mathematics goes into the design of tournaments and the choice of format for a particular tournament, often drawing from disciplines such as combinatorics and graph theory. Different choices about the rules of the tournament affect the appeal of the tournament for participants and spectators and, more importantly, can affect which players will be more likely to win. The situation is somewhat analogous to voting systems in which the outcome of a decision can change based on the form of ballot, even when the voters’ preferences are unchanged.

Common Types of Tournaments

In a single-elimination knockout tournament, the players compete in pairs. The loser of each game is eliminated from the tournament; the winners go on to the next round. This process continues until only one player is left, who is declared the winner. If the number of competitors is not a power of two, then some competitors sit out one or more initial rounds, automatically advancing to the next round. Which players sit out can be determined randomly or based on some prior rankings. The schedule for which players meet in the first round, the winners of which of these games will meet in the second round, and so on, is called the bracket for the tournament. In situations where the competitors are ranked in advance (for example, seeds in a tennis tournament), care must be taken in designing the bracket. It would be undesirable for a player to gain an advantage in a tournament by deliberately underperforming in order to obtain an artificially low prior ranking. The most commonly used brackets involve the highest ranked player meeting the lowest-ranked player in the first round and are used because they are optimized to prevent such manipulation. Double-elimination and triple-elimination tournaments, in which participants are not eliminated until suffering a second or third loss, also exist, though the latter are rather rare. These formats are tolerant of one (or two) lost matches by the player or team that will go on to be champion, but the problem of arranging the brackets and scheduling the matches can be more complicated.

In a round-robin tournament, each participant competes against every other participant. Typically, each pairing competes in a single match, but variants exist in which more games are played. Such a format gives more information about the relative strength of the players at the expense of requiring more games. Another drawback is that it is generally difficult to identify a canonical choice for first-place champion after a round-robin tournament.

Of course, much more complicated systems exist. Consider, for example, the FIFA World Cup tournament. In the 2010 format, the thirty-two competing teams are first randomly divided into eight groups. The teams within each group all play against one another. Based on the results of these round-robin matches, a winner and a runner-up emerge from each group. These sixteen teams then compete in a single-elimination knockout; the first round of knockout matches involve the group winners each competing against the runner-up from another group. However, a new format was adopted in 2017 and modified in 2023, in which the forty-eight teams will be split into twelve groups of four teams with the top two of each group and the eight best third-placed teams progressing to a new round of thirty-two. From there, the tournament resumes as usual. This was the first change in tournament style since 1998.

Tournaments can also be used as teaching tools in classrooms as a cooperative learning method. Competition can be used as a motivating tool for students of all ages. Students can be individually seeded or can work in pairs or groups in nearly any content area in order to solve problems, present findings or arguments, or share information. One study demonstrated that when tournament-style teaching was used in a junior high mathematics classroom, students developed greater self-efficacy in mathematics and confidence and learning achievement in the academic setting.

Graph-Theoretic Tournaments

The term “tournament” is also used with a specialized meaning in the subject of graph theory. A tournament in this sense is a collection of any number of vertices and arrows, where each pair of vertices is connected by a single arrow. Such a picture can represent a round-robin tournament in which each participant competes against every other participant exactly once, and there are no ties. The vertices are the players, and the direction of the arrow indicates who won each game (the arrow points from the winner of the game to the loser). Such configurations were originally studied by H. G. Landau to study the dominance relationships among populations of chickens. Tournaments have gone on to find important applications to social voting theory and public choice.

Bibliography

Annurwanda, Pradipta. “The Effect of Teams Games Tournament on Mathematics Self-Efficacy in Junior High Schools.” SHS Web of Conferences, vol. 42, 2018. EDP Sciences, www.shs-conferences.org/articles/shsconf/pdf/2018/03/shsconf‗gctale2018‗00079.pdf. Accessed 11 Oct. 2024.

“How the FIFA World Cup 26™ Will Work with 48 Teams.” FIFA, 1 Mar. 2023, www.fifa.com/en/articles/article-fifa-world-cup-2026-mexico-canada-usa-new-format-tournament-football-soccer. Accessed 11 Oct. 2024.

Kolesnik, Brett, and Mario Sanchez. "The Geometry of Random Tournaments." Discrete & Computational Geometry, vol. 71, 2024, pp. 1343–1351. Springer Link, doi.org/10.1007/s00454-023-00571-4. Accessed 11 Oct. 2024.

Schwenk, A. J. “What Is the Correct Way to Seed a Knockout Tournament?” Journal of American Mathmatical Monthly 107, no. 2 (2000).

“Tournament.” Encyclopedia of Mathematics, 15 Mar. 2023, encyclopediaofmath.org/wiki/Tournament. Accessed 11 Oct. 2024.