PROJECT
IN
P.E.

Submitted by:
Eia Celine Dg. Arellano

Submitted to:
MR. Alfred sta. maria

SINGLE ELIMINATION
A single-elimination tournament, also called an Olympic system tournament, a knockout, cup or sudden death tournament, is a type of elimination tournament where the loser of each match or bracket is immediately eliminated from winning the championship or first prize in the event.
This does not always mean that the defeated competitor will not participate further in the tournament; in some such tournaments, consolation or "classification" contests are subsequently held among those already defeated to determine the awarding of lesser places, for example, a third place playoff.
Where more than two competitors can play in each match, such as in a shootout poker tournament, players are removed when they can no longer play until one player remains from the group. This player moves on to the next round.
A simple way to describe a single-elimination tournament is that the winner of each match moves on and the loser fails to progress in the tournament.
Some advantages of single elimination tournaments are: * the participants understand them easily * they are the simplest tournaments to conduct * they are useful in determining a champion for preliminary tournaments, such as a round robin * they determine the champion in the shortest time compared with other tournaments * they can be conducted with limited facilities * they can accommodate a large number of entries * they are interesting for spectators * they are the most appropriate for a one-day event * they are economical to conduct
Some disadvantages of single elimination tournaments are: * they involve minimum participation * they place maximum emphasis on winning * the champion may not represent the best team or player...this also applies to the second place finisher because entries in the other half of the bracket may be better (if you have no true method for seeding) * they do not allow for an off-day * competition may become too intense because the entry must win every contest or face elimination * outdoor sport programs, with their potential for weather-related postponements, cause scheduling problems because contests must be played sequentially * they provide the least flexibility for the participant

Formulas for Single Elimination Tournaments: with N= total number of entries, you can use the following formulas:
Formula
Number of games = N-1 13-1 = 12 games

Number of 2 - number of times 2 has to be multiplied to equal or exceed the number of entries
2x2 = 4
2x2x2 = 8
2x2x2x2 = 16
2x2x2x2x2 = 32
Number of Byes = Next Highest Power of 2 - N 16-13 = 3 byes
Number of Rounds = the power to which 2 must be raised 2x2x2x2 = 4 rounds to equal or exceed "N"
Number of first round games = N- (next lowest power of 2) 13-8 = 5 first round games
G = 9
SChEDULE:
G1 2 VS. 3
G2 4 VS. 5
G3 6 VS. 7
G4 8 VS. 9 G5 Bye VS. W1 G6 W5 VS. W2 G7 W3 VS. W4 G8 W6 VS. W7

N-1 = 9-1 = 8 games, 1 bye
Diagram
1 Bye
2 G1 G5 W5
3 W1 G6
4 G2 W6
5 W2
6 G3 W3 G8 - Championship
7 G7
8 G4 W7
9 W4

SCHEDULE:
G1 1 VS. 2
G2 3 VS. 4
G3 5 VS. 6
G4 7 VS. 8
G5 W1 VS. W2
G6 W3 VS. W4
G7 W5 VS. W6
H= 8
8-1 = 7 games
DIAGRAM
1 G1 W1
2 G5 W5
3 G2
4 W2 G7 - Championship
5 G3 W3
6 G6
7 G4 W6
8 W4

DOUBLE ELIMINATION
A double-elimination tournament is a type of elimination tournament competition in which a participant ceases to be eligible to win the tournament's championship upon having lost two games or matches. It stands in contrast to a single-elimination tournament, in which only one defeat results in elimination. This kind of tournament goes back to at least 1973 when the format was used for a chess club tournament at James Fowler High School in Calgary, Alberta, Canada then designed by Russell Irvin Johnston.
One method of arranging a double-elimination tournament is to break the competitors into two sets of brackets, the Winners Bracket and Losers Bracket (W and L Brackets for short; also sometimes Upper Bracket and Lower Bracket, respectively) after the first round. The first round winners proceeding into the W Bracket and the losers proceeding into the L Bracket. The W Bracket is conducted in the same manner as a single-elimination tournament, except of course that the losers of each round "drop down" into the L Bracket. Another method of double elimination tournament management is the Draw and Process.
As with single-elimination tournaments, most often the number of competitors is equal to a power of 2 (8, 16, 32, etc.) so that each round there are an even number of competitors and never any byes. The maximum number of games in a double elimination tournament is one less than twice the number of teams participating (e.g., 8 teams – 15 games). The minimum number is two less than twice the number of teams (e.g., 8 teams – 14 games).

Conducting the tournament
If the Bracket arrangement is being used then each round of the L Bracket is conducted in two stages; a minor stage followed by a major stage. Both contain the same number of matches (assuming there are no byes) which is the same again as the number of matches in the corresponding round of the W Bracket. If the minor stage of an L Bracket round contains N matches, it will produce N winners. Meanwhile, the Nmatches in the corresponding round of the W Bracket will produce N losers. These 2N competitors will then pair off in the matches of the corresponding major stage of the L Bracket.
For example, in an eight-competitor double-elimination tournament, the four losers of the first round, W Bracket quarter finals, pair off in the first stage of the L Bracket, the L Bracket minor semifinals. The two losers are eliminated, while the two winners proceed to the L Bracket major semifinals. Here they will contest the two losers of the W Bracket semifinals. The winners of the L Bracket major semifinals contest the L Bracket minor-final, with the winner playing the loser of the W Bracket final in the L Bracket major final.
The championship finals of a double elimination tournament is usually set up to be a possible two games. The rationale is that since the tournament is indeed double elimination, it is unfair to have the W Bracket champion eliminated with its first loss. Therefore, while the W Bracket champion needs to beat the L Bracket champion only once to win the tournament, the L Bracket champion must beat the Winners' Bracket champion twice.
A Draw and Process tournament requires less intervention by the manager. The competitors are allocated their first round positions on the competition grid and this is played as if it were a single elimination event. This grid is called the "Draw". A second compeition grid called the "Process" is then produced and again playes as a single elimination event. The fixed arrangement of the Process ensures that players who met in the first round of the Draw cannot meet until the final of the Process. Similarly, players who meet in the second round of the Draw cannot meet until the semi finals of the Process. If the same person wins both the Draw and Process then they are the overall winner and the losing finalists will play each other for second and third place. Otherwise the winners of the Draw and Process will play off to determine the winner.
Pros and cons
The double-elimination format has some advantages over the single-elimination format, most notably the fact that third and fourth places can be determined without the use of a consolation or "classification" match involving two contestants who have already been eliminated from winning the championship.
Some tournaments, such as in tennis, will use "seeding" to prevent the strongest contestants from meeting until the later round. However, in tournaments where contestants are placed randomly in the draw, or in situations where seeding is not available, it is possible for 2 of the strongest teams to meet in the early rounds rather than a final or semifinal as would be expected in a seeded draw. Double elimination overcomes this shortfall by allowing a strong team which loses early to work their way through the L Bracket and progress to the later rounds, despite meeting the strongest team in the early rounds of competition.
Another advantage of the double-elimination format is the fact that all competitors will play at least twice and approximately three quarters will play three games or more. In a single-elimination tournament with no byes, half of the competitors will be eliminated after their first game. This can be disappointing to those who had to travel to the tournament and were only able to play once.
A disadvantage compared to the single-elimination format is that at least twice the number of matches have to be conducted. Since each player has to lose twice and since the tournament ends when only one player remains, in a tournament for N competitors there will be either 2n-2 or 2n-1 games depending on whether or not the winner was undefeated during the tournament.
The fact that the final game in the tournament may be unnecessary is also a disadvantage, particularly if broadcasting and ticket sales companies have an interest in the tournament.
SCHEDULE:
G1 2 VS. 3
G2 4 VS. 5
G3 6 VS. 7
G4 8 VS. 9
G5 BYE VS. W1
G6 W5 VS. W2
G7 W3 VS. W4
G8 L1 VS. L2
G9 L3 VS. L4
G10 W8 VS. L6
G11 W9 VS. L7
G12 W10 VS. L5
G13 W6 VS. W7
G14 W12 VS. W11
G15 W14 VS. L13
G16 W15 VS. W13
G17 If necessary

G= 9
N-1(2)+1 9-1(2)+1
8(2)+1
16+1 games

DIAGRAM L5 1 Bye G12 L1 2 G1 G5 W5 W12 G8 3 W1 G6 W6 W10 4 G2 G10 G14 L2 5 G15 W2 G13 W15 L3 6 G3 W3 W13 * G9 7 G7 W11 8 G4 G11 W7 L4 9 W4 G16 G17

H=8
N-1(2)+1 8-1(2)+1
7(2)+1
14+1 games
Diagram

L1 1 G1 W1 W7 G7 2 G5 W5 3 G2 G9 G12 L2 4G13 W2 G11 W13 L3 5 G3 W3 W11 * G8 6 G6 W8 7 G4 G10 W6 L4 8 W4 G14 G15

G8 L3 VS. L4
G9 W7 VS. L5
G10 W8 VS. L6
G11 W5 VS. W6
G12 W7 VS. W8
G13 W12 VS. L11
G14 W13 VS. W11
G15 If necessary

Schedule:
G1 1 VS. 2
G2 3 VS.4
G3 5 VS. 6
G4 7 VS. 8
G5 W1 VS. W2
G6 W3 VS. W4
G7 L1 VS. L2

ROUND ROBIN
A round-robin tournament (or all-play-all tournament) is a competition "in which each contestant meets all other contestants in turn".
Terminology
The term round-robin is derived from the term ruban, meaning "ribbon". Over a long period of time, the term was corrupted and idiomized to robin.
In a single round-robin schedule, each participant plays every other participant once. If each participant plays all others twice, this is frequently called a double round-robin. The term is rarely used when all participants play one another more than twice,] and is never used when one participant plays others an unequal number of times (as is the case in almost all of the major United States professional sports leagues – see AFL (1940–41) and All-America Football Conference for exceptions).
In the United Kingdom, a round-robin tournament is often called an American tournament in sports such as tennis or billiards which usually have knockout tournaments. In Italian it is called girone all'italiana (literally "Italian-style circuit"). In Serbian it is called the Berger system (Бергеров систем, Bergerov sistem), after chess player Johann Berger. A round-robin tournament with four players is sometimes called a "quad".
Use
In sports with a large number of competitive matches per season, double round-robins are common. Most association football leagues in the world are organized on a double round-robin basis, in which every team plays all others in its league once at home and once away. This system is also used during qualification for major tournaments such as the FIFA World Cup and the respective continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup, etc.). There are also round-robin chess, go, and Scrabble tournaments. The World Chess Championship decided in 2005 and in 2007 on an eight-player double round-robin tournament where each player faces every other player once as white and once as black.
Group tournaments rankings usually go by number of matches won and drawn, with any of a variety of tiebreaker criteria.
Frequently, pool stages within a wider tournament are conducted on a round-robin basis. Examples with pure round-robin scheduling include the FIFA World Cup, UEFA European Football Championship and UEFA Cup (2004–2005) in football, Super Rugby (rugby union) in the Southern Hemisphere during its past incarnations as Super 12 and Super 14 (but not in the current 15-team format), the Cricket World Cup, Indian Premier League Twenty-20 Cricket and many American Football college conferences, such as the Big 12 (which currently has 10 members). The group phases of the UEFA Champions League and Copa Libertadores de América are contested as a double round-robin, as are most basketball leagues outside the United States, including the regular-season and Top 16 phases of the Euroleague; the United Football League has used a double round-robin for both its 2009 and 2010 seasons.

Evaluation
In theory a round robin tournament is the fairest way to determine a champion among a known and fixed number of participants. Each player or team has an equal chance against all other participants. The element of luck is seen to be reduced as compared to a knockout system since a few bad performances need not cripple a competitor's chances of ultimate victory. A participant's final record is thus seen to be more accurately represented in the results since it was arrived at over a prolonged period against equal competition. This can also be used to determine which teams are the poorest performers and thus subject to relegation if the format is used in a multi-tiered league. In English football, the Football League, the (round-robin) League champions are generally regarded as the "best" team in the land, rather than the (knockout) FA Cup winners.
The primary disadvantage to a round robin tournament is the time needed to complete it. Unlike a knockout tournament where half of the participants are eliminated after each round, a round robin requires one round less than the number of participants if the number of participants is even, and as many rounds as participants if the number of participants is odd. For instance, a 32 team tournament can be completed in just 5 rounds in a knockout format. However if the same teams are put through a round robin it would require 31 rounds to finish. Other issues stem from the difference between the theoretical fairness of the round robin format and practice in a real event. Since the victor is gradually arrived at through multiple rounds of play, teams who perform poorly can be eliminated from title contention rather early on, yet they are forced to play out their remaining games. Thus games occur late in competition between competitors with no remaining chance of success. Moreover, some later matches will pair one competitor who has something left to play for against another who does not. This asymmetry means that playing the same opponents is not necessarily equitable: the same opponents in a different order may play harder or easier matches. Teams may also suffer injuries to their star players during competition and thus a match-up may have a completely different complexion than it would have if the order of play was different. There is also no showcase final match.
Further issues arise where a round-robin is used as a qualifying round within a larger tournament. A competitor already qualified for the next stage before its last game may either not try hard (in order to conserve resources for the next phase) or even deliberately lose (if the scheduled next-phase opponent for a lower-placed qualifier is perceived to be easier than for a higher-placed one). Four pairs in the 2012 Olympics Women's doubles badminton having qualified for the next round, were disqualified for attempting to lose in the round robin stage to avoid compatriots and better ranked opponents. The round robin stage at the Olympics was a new introduction and potential problems were readily known prior to the tournament.
Swiss system tournaments attempt to combine elements of the round-robin and elimination formats, to provide a reliable champion using fewer rounds than a round-robin, while allowing draws and losses. Also if the tournament is not held at a true neutral location and is instead at a team's home field or away the system of Double Round Robin is an effective equalizer. In this format each team plays each other twice, once away and once at home, in an effort to account for meetings of teams where home field could sway the results.

Scheduling algorithm
If N is the number of competitors, a pure round robin tournament requires n/2 (n-1) games. If N is even, then in each of (n-1) rounds, n/2 games can be run in parallel, provided there exist sufficient resources (e.g. courts for a tennis tournament). If N is odd, there will be N rounds, each with n-1/2 games, and one competitor having no game in that round.
G=9
N (N-1)/2
9(9-1)/2
9(8)/2
= 36 games

R1 | R2 | R3 | R4 | A VS. Bye | A VS. I | A VS. H | A VS. G | B VS. I | Bye VS. H | I VS. G | H VS. F | C VS. H | B VS. G | Bye VS. F | I VS. E | D VS. G | C VS.F | B VS. E | Bye VS. D | E VS. F | D VS. E | C VS. D | B VS. C | R5 | R6 | R7 | R8 | A VS. F | A VS. E | A VS. D | A VS. C | G VS. E | F VS. D | E VS. C | D VS. B | H VS. D | G VS. C | F VS. B | E VS. Bye | I VS. C | H VS.B | G VS. Bye | F VS. I | Bye VS. B | I VS. Bye | H VS. I | G VS. H |

R9 | A VS. B | C VS. Bye | D VS. I | E VS. H | F VS. G |

H=8
8(8-1)/2
8(7)/2
= 28 gaMES

R1 | R2 | R3 | R4 | A VS H. | A VS. G | A VS. F | A VS. E | B VS.G | H VS. F | G VS. E | F VS. D | C VS. F | B VS. E | H VS. D | G VS. C | D VS. E | C VS. D | B VS. C | H VS. B |

R5 | R6 | R7 | R8 | A VS. D | A VS. C | A VS. B | A VS. H | E VS. C | D VS. B | C VS. H | B VS.G | F VS. B | E VS. H | D VS. G | C VS. F | G VS. H | F VS. G | E VS. F | D Vs. e |