Tournament form

A tournament form , a tournament format or a tournament mode describes the structure of a tournament or a league with which the best player, athlete or the best team is to be determined. Usually it is about sports that do not allow direct measurement, but in which two participants have to compete against each other.

Note: The term game is to be understood here in an abstract sense: In a chess tournament one can play as consisting of two parts (each player plays once with the white pieces and once with the black pieces) as well, in a football tournament , a game of two games exist (a first and a second leg), in tennis a game is decided as soon as a participant has won two or three “sets”. The following does not deal with the exact design of the gamein the sense of the respective sport, but only on the outcome that is significant for the further course of the tournament. In this case, reference is made to the article Best-of-Modus as an example .

Requirements for a tournament form

The comparison of sporting performances is - depending on the sport - differently difficult. In the case of a marathon, for example, the performance comparison is comparatively easy, since the fact that all tournament participants compete against each other at the same time allows a direct performance comparison. An objective performance measurement, such as timing, is in principle not even necessary.

In many cases, however, it is not possible to have all participants compete against each other. In addition, performance can often not be measured objectively, but only relative to an opponent ( football , tennis , chess , bridge ). This means that the “best” participant in these sports can only be determined through a tournament consisting of several comparative matches.

Desirable properties of a tournament format are therefore:

The final ranking should reflect the true player strength as accurately as possible:
- If the best player - the favorite - wins every single game, he should also win the tournament.
- The second best player should finish second.
- The third-best player should reach third place and the same applies to the other places.
For organizational reasons, the number of games or game rounds should be as small as possible.
The decision about the tournament victory should - to increase the tension - only be made at the end of the tournament in a final of the two best players.
Exciting games are desirable for the spectators. That is why every game should have about equally strong participants.
The form of the tournament is intended to encourage every player to play as well as possible in every game:
- A tactical defeat or a tactical draw in a game must not be rewarded with a better final ranking.
- Agreements between two participants to improve their own situation at the expense of a third party (so-called collusion , see Bundesliga scandal or Gijón non-aggression pact ) should be prevented.
The duration of the tournament participation should be known in advance for each participant.

No tournament form can meet all of these desirable properties, which is why there is no “ideal” tournament format. The following different tournament formats therefore only partially meet these criteria.

Overview of the tournament forms

Tournament form	advantages	disadvantage	Number of games required with n players
Knockout system	Exciting, spectator-friendly games, after all, every encounter leads to the elimination of one player Suitable for many participants	An early encounter between two top players inevitably leads to a poor final placement for one player Many players left early and thus had little tournament experience For a large number of participants, no rank is determined at all	${\ displaystyle n-1}$
Round tournament	The final result shows the relative skill level of the players very precisely Predictability: It is already known when planning the tournament which player meets which opponent and when	Unsuitable for tournaments with many participants, especially for home and return matches Danger of tactical considerations, kingmaking opportunities, fraudulent interaction The decision about a tournament victory is usually not made by a direct encounter between the top two, but in a remote duel	${\ displaystyle {\ frac {n} {2}} \ cdot (n-1)}$ or with simple execution, with the second half of the season ${\ displaystyle {\ binom {n} {2}}}$ ${\ displaystyle n \ cdot (n-1)}$
Swiss system (Danish system)	Exciting games throughout the tournament, as players who are almost equally good meet each other Suitable for many participating players who should be present for the entire duration of the tournament	In the midfield, the ranking does not reflect the player's strength very well	${\ displaystyle {\ frac {n} {2}} \ cdot \ log _ {2} n}$
Ladder tournament	Depending on the design, the final result reflects the relative strengths of the game very well If the field of participants is well placed, very few games are necessary Flexibility for the participants as they can choose number of games and opponents	If the ranking is drawn at the beginning, this form of tournament is very inefficient Risk of arbitrariness (players subjectively choose defeatable opponents) unequal number of games per participant	${\ displaystyle {\ frac {n} {2}} \ cdot (n-1)}$ in the worst case, in the best case ${\ displaystyle {\ frac {n} {2}}}$

Examples: Number of games required for different types of tournaments
Number of players	Knockout system	Swiss system	Round tournament	with the second half of the season
4th	3	4th	6th	12
8th	7th	12	28	56
16	15th	32	120	240
30th	29	75	435	870
50	49	150	1225	2450
100	99	350	4950	9900
${\ displaystyle n}$	${\ displaystyle (n-1)}$	${\ displaystyle {\ tfrac {n} {2}} \ cdot \ log _ {2} n}$	${\ displaystyle {\ binom {n} {2}}}$	${\ displaystyle n \ cdot (n-1)}$

Knockout system

Single knockout

The knockout system (from English knockout ; about: "put out of action") in the narrower sense is a form of tournament in which two participants meet in each game round and the loser is eliminated until the winner of the tournament is finally in the final is determined.

Double knockout

When double knockout (also double-elimination ) refers to a tournament format, which, in essence, on the Knockout K.-system is based, however, tries to circumvent its weaknesses. The main differences between the double knockout format and the simple knockout system ( single knockout or single elimination ) are:

A participant is only eliminated from the tournament after the second defeat.
A participant can win the tournament despite a one-time loss. This is also ruled out in the simple knockout system if additional games are held in order to obtain a continuous sequence: the loser of a semifinal game can at best take 3rd place, the loser of a quarterfinal match at best 5th place, etc. .
Even without seeding, it is ensured that the second best player takes second place (provided that the respective favorite wins every single match).
With the same number of participants, about twice as many games have to be played as in the knockout system. (If n players take part, 2n-2 games are required.)

Round tournament

A "round- robin tournament" (also "everyone against everyone", English round robin or league system ) is a form of tournament in which each tournament participant competes the same number of times against all other tournament participants.

This system is used in most league systems used in football, where on the basis of a solid game plan , each team against every other team once at home and once away plays (exact name: "Double Round Robin" because each team each other two times applies). This is also used in bowling.

A tournament “everyone against everyone” simply requires games (without return games) with n participants . Points are awarded for every win and every draw. At the end there is a complete ranking list according to points, in the event of a tie, further criteria are used, the goal difference or the direct encounters. ${\ displaystyle {\ begin {matrix} {\ frac {n} {2}} \ end {matrix}} (n-1)}$

Combinations of knockout and round-robin tournaments

In many cases a combination of a knockout system and a round robin tournament is used.

The first section of the tournament is usually held as a round-robin tournament with several groups or leagues, and another as a knockout tournament. Examples of this are the soccer World Cup and EM (after the qualification phase) and most of the European soccer cup competitions. Here the teams are mostly divided into groups of four teams each. Everyone plays against everyone between the teams in a group. This is called the group stage, sometimes called the preliminary round. The group first and second in the group then play against each other in the knockout system ("elimination games, knockout phase"), at the European Football Championship 2016 the four best of a total of six thirds in the group qualify for the K. -o.-phase, as this European Championship is the first time there is a round of 16. In some cases there is another intermediate round between the preliminary round and the knockout phase. E.g. in the UEFA Champions League until 2002, when this intermediate stage was known as the second group stage . As a final round at is confusingly, only the knockout phase referred to the UEFA Champions League, World Cups and EMs the entire tournament incl. Group stage.

This mixed system has the advantage that good teams can largely be avoided too early if the group winners meet as late as possible. In the first elimination round, the first group always play against second group.

With appropriate staking, you can also ensure that two teams that played against each other in the preliminary round in the same group can only meet again in the final. Before the actual tournament begins, there are often qualification games in which either the knockout system or the round-robin tournament is used as a mode.

There are also many other ways to combine these two types of tournaments. In North American club sport in particular, different variants are used ( NBA , NHL , MLB , NFL or MLS ). Play-off is the name of a knockout system that is played by the top players in a single league after the main season, which is carried out in the everyone-against-everyone system.

Swiss system

The “Swiss system” can best be described as a special form of round-robin tournament. The first round is set or drawn, after which the intermediate result is determined after each round. In the following rounds, the leader always plays against the runner-up, the third against the fourth and continues in this row.

However, it is excluded that two players meet twice, the pairings are therefore determined before each round as follows:

The leader plays against the top ranked player he has not yet played against.
The leader of the remaining players will play against the top ranked player he has not yet played against.

So if the same players are in the lead after a round, the first will play against the third (if he has not yet played against him, therefore: against the best-placed player, against whom he has not yet played), the second against the fourth. Abort criteria can be set - as is common in pétanque : new rounds are played until only one participant has remained without defeat. Without an abort criterion, a tournament according to the Swiss system lasts as long as a round-robin tournament, which, however, negates a significant advantage of the Swiss system.

If indicators such as goal difference or number of goals scored are missing, a detailed evaluation may be necessary in order to establish a clear ranking in the event of a tie. If no detailed evaluation is made, the respective new opponent can also be drawn if there is a tie.

Swiss system (variants)

The Danish system differs from the Swiss system in that the same pairings can occur several times: In each round, the first placed against the second placed, and the next placed in the same way. In Switzerland this game system is known as the "Schoch system".

A special form of the Swiss system is the Mac-Mahon system , which is often used in Shogi and Go tournaments. The players do not start with zero points, but instead start the tournament with different numbers of points depending on their classification in Kyu or Dan grades for Go or their Elo number for Shogi.

ladder

In a ladder tournament, all players are first placed in a provisional ranking or drawn. The exact design of the tournament format can be different, but what all ladder tournaments have in common is:

Any player can challenge a better placed player to a game.
If a player does not meet a challenge within a certain period of time, he loses forfait.
If a player does not challenge another player within a certain period of time, he will either be removed from the tournament or he will be placed last.
If the worse placed player wins, the places in the ranking are swapped.
In the event of a tie or a defeat of the challenger, the ranking remains unchanged.

Leader tournaments are often played within clubs, while this type of tournament rarely occurs between players or teams from different clubs. If each player challenges only the next best player, this tournament procedure corresponds to the so-called bubble sort method. In the worst case - the best player is in last place at the beginning of the tournament, the second best in second from last place and so on - the number of games required corresponds to that of the simple round-robin tournament, since everyone has to play against each other.

Ladder (variants)

The rules can dictate what challenges are allowed. For example, each player can only challenge the next best or the next best player in order to protect the top players from time-wasting, hopeless challenges from beginners.

A similar procedure is common in some martial arts (e.g. boxing and wrestling ): Whoever challenges and beats the master, receives the current master's title from him. The reigning world chess champion was also determined at times: the winner of the candidate tournament challenges the world champion and becomes the world champion himself if he beats him.

Sources and explanations

↑ Logarithms rounded to full rounds.
↑ Plauschturniere on badminton-plauschturniere.ch, accessed on September 4, 2014.

[1] Logarithms rounded to full rounds.

[2] Plauschturniere on badminton-plauschturniere.ch, accessed on September 4, 2014.