UPA Manual of Tournament Formats

Basic Theories of Tournament Scheduling

Copyright (c) The Ultimate Players Association, 1994

Please direct comments on this document to Eric Simon (70540.1522@compuserve.com).

2. Basic Theories of Tournament Scheduling

There are a couple of basic rules that should be applied to tournament scheduling, each of which is elaborated below. They are:
Every team must get an equal chance and there should be as much independence from seeding as possible.

That is, all teams should have a fair shot at winning the tournament (and/or qualifying to the next level). No team should have to do something different in the schedule from the other teams because of how they are seeded. Now it may be that some pools are of different size than other pools (there just isn't any number that divides into thirteen!), but in this case an entire pool is playing more games than other pools. No team in the larger pool is doing anything different than any other team in that pool.

There have been some awfully "funky" formats for sectional tournaments before. I can recall one in which the top seeded team completely bypassed the first day of the tourney, and were automatically placed in a "winners" pool of the next day (i.e., different "divisions.") Such tournaments have their place, and may even be desirable in certain circumstances, but not for UPA Series events (unless teams have a choice of the division in which they would like to be placed).

Each team should get a good number of games.

Teams that travel from far away should be able to go home after a tournament feeling like they played their fair share of games, no matter where they finished. This means that teams should be eliminated from the tournament as late as possible, and that if they are eliminated early, there are 'Division II' options available to them. On the other hand,

Teams should not have to play more than four games per day.

Four games per day, games to 15, is about the most any team should be asked to play. (Limited exceptions for five games to 13). People who are into triathlons and marathons can go compete in those other competitions. Furthermore, these games should be scheduled in two-hour time blocks (with a cap of "plus two" after 1 hour and 40 minutes). Time caps are needed, not only for scheduling purposes, but so that a team does not have to play more than eight hours of Ultimate in a day. In those games where the wind is strong, both teams have good defenses, or neither team has a good offense, games to 15 can drag out for a long time. A succession of long, killer games often results in unwarranted injuries.

There are exceptions to this rule, but they should be limited to Regional and post-Regional play. For example:

Again, these exceptions were Regionals or Easterns, where, presumably, the teams are stronger than they are at Sectionals.

Another set of exceptions come into play when a tournament is faced with an undesirable number of teams. For example, thirteen teams is just a difficult number to work with. Many of these situations are dealt with by eliminating the number of teams to eight teams after the first day, and then a double elimination format the second day. The double elimination format, however, results in five games, and so games can be no longer than to the score of thirteen.

The order of the games should be fair.

Should this be in here? What is fair? Can anybody help out at all here?

2.1. Seeding

"Seeding" a tournament is a way of ranking the teams and, assuming there is pool play, distributing the teams into the pools in such a way as to be fair to the teams involved, and to avoid matching up the best teams in the early rounds. For example, suppose, at your sectional, you were trying to choose the top three teams to qualify for regionals. The last situation you want is where a team gets beaten by the number one and number two teams and never has a chance to compete against the number three team. Suppose you had ten teams divided into two pools of five. You would try your best not to put the two best teams in the same pool. If this is difficult to determine (and it sometimes is in those sections where teams don't play each other often), you would try to make sure that when pool play is done, the top three teams in each pool have not yet been eliminated.

Ideally, it shouldn't matter where or how teams are seeded. Indeed, when the entire tournament is in one round-robin pool, it doesn't matter how the teams are seeded. Attempts have been made, in every format in this manual, to minimize the effect of seeding. A team might get eliminated earlier or later because of where they were seeded, but whether or not the team qualifies for the next level tournament should not depend on seeding.

2.2. How to seed

Ranking the teams is your business. It is useful to get as much information as possible beforehand; i.e., which teams have beaten other teams, etc. As general rules, one should take into account the entire season, giving more weight to games and tournaments that were played more recently. (In college situations, the "Top 20" rankings service can be used as a guide; the rankings should not necessarily be controlling).

Anybody want to elaborate on 'general principles' of seeding here?

2.3. A One-Day or Two-Day Format?

For Regional Championships:
Even though there may be as few as four teams, two days of play are expected for teams that will be competing for spots at nationals. There should be no exceptions (except in the case where there are three teams or less competing).
For Sectional Championships:
Sectionals should be a two-day tournament. If there are 7 or more teams competing, then there are no exceptions to this rule (except as noted in the format descriptions). If there are 6 or less teams, there are limited circumstances where a oneday format is acceptable. Whether or not there are two days of play depends on the characteristics of the section. Where most, sometimes all, of the teams advance, and none of the teams are traveling far, a one-day tournament might be acceptable. The sectional coordinator should ask the teams in the section if they prefer a one- or two-day tournament. Note: a greater weight should be given to the preferences of the teams that are travelling far, and, secondarily, to the weaker teams. If the weaker teams want to play more games, the better teams should accommodate them. This is even more important if the weaker teams face elimination at sectionals. The UPA should afford them the opportunity to play more games before their season ends.
The round-robin schedules should be modified:
The round-robin schedules were computer-generated. Such things like equity regarding which fields are played on have not been taken into account. Neither has the order of the games. So, for example, if you are trying to choose one winner from the tournament, you will want to schedule the top two seeds to play each other the last round. Suppose, however, you had ten teams in two pools of five, and you were going to choose the top two teams from each pool to go into the semifinals. In that case, you would schedule seeds 2 and 3 to play each other the last round. That is the game that will mean the most, not 1 v 2 if both teams have already essentially qualified.

2.4. How to read these schedules

Below is a sample six-team round-robin schedule. First look at the schedule "matrix." The team numbers go down the side, the round numbers go across the top. So, for example, in this matrix, team number 3 plays team number 6 in the first found.
              Six Teams
                1  2  3  4  5
                -  -  -  -  -
            1 - 5  4  3  2  6
            2 - 4  3  6  1  5
            3 - 6  2  1  5  4
            4 - 2  1  5  6  3
            5 - 1  6  4  3  2
            6 - 3  5  2  4  1
In the schedule below, each row is a round, and each column can be a field. So, in the first round, 1 plays 5 on field one, etc.
            1 v 5   2 v 4   3 v 6
            1 v 4   2 v 3   5 v 6
            1 v 3   2 v 6   4 v 5
            1 v 2   3 v 5   4 v 6
            1 v 6   2 v 5   3 v 4

2.5. Tie breakers for round-robins

The worst part of a round-robin is the possibility of a three-way tie between teams that have all beaten each other. (E.g., A beats B, B beats C, and C beats A). One unlucky team will have to finish 3rd when they only lost one game. This is the very reason why the UPA tries to avoid pool play when the third place team can get eliminated. So, for example, in a tournament that is organized into 4 pools of 5, where the top 2 teams advance, a team can lose one game, to the best team, and be denied the chance to finish in the top eight.

Nevertheless, while the UPA tries to minimize such situations, they all can't be avoided. The following are the UPA Standard set of round-robin tie breaker rules.

Rule 1. A given tie-breaker rule applies equally to all the teams that are tied.
For an example of this application, see Example 3.1.

Rule 1a. If, after the application of a given rule, all of the teams are still tied, go to the next rule. For an example of this application, see Example 3.3.

Rule 1b. If not all teams, but one or more subgroups of the teams are tied, separate these teams into groups and go back to rule 2 with each of the groups individually. For an example of this application, see Example 3.2.

Rule 2. Won-loss records, counting only games between the teams that are tied.
Example 2.1. A and B are tied for third place at 4-2, and during the tournament, A has beaten B. Then, A gets third place and B gets fourth place. When only two teams are involved, this rule is commonly called "head-to-head."

Example 2.2. A, B, and C, are tied for first place; they are all 3-2 after the six team round-robin. A has beaten both B and C, while B has beaten C. The records among the three teams only are: A is 2-0, B is 1-1, and C is 0-2. A finishes first, B finishes second, and C finishes third.

Example 2.3. A, B, and C are in a three-way tie. A has beaten B, B has beaten C, C has beaten A. The relevant records for all three teams are 1-1. This tie-breaker won't work, and you must go on to tie-breaker #3.

Rule 3. Point differentials, counting only games between the teams that are tied.
Example 3.1. A, B, C are in a three-way tie for first place. A has beaten B 15-10, B has beaten C 15-12, and C has beaten A, 15-13. A's point differential, then, is +5 and -2, which equals +3. B's is -2 and C's is -1. A finishes first, C finishes second, and B finishes third. Note that the three point differentials, in this case, +3,-2,-1, must always add up to zero. Note also that we do not use the point differential to choose the winner and then go "head to head" to choose the other two. This would be a violation of Rule #1, which say that we must apply a tie-breaker rule equally to all the teams that are tied.

Example 3.2. A, B, C are in a three-way tie for first place. A has beaten B 15-11, B has beaten C 15-12, and C has beaten A, 15-13. A's point differential, then, is +4 and -2, which equals +2. B's is -1 and C's is -1. A takes first place. B and C are still tied. When, after the application of a rule, there are still teams that are tied, we go back to rule 2. Since B beat C, B takes 2nd place, and C takes 3rd. At this point we do not go onto rule 4.

Example 3.3. A, B, C are in a three-way tie for first place. A has beaten B 15-13, B has beaten C 16-14, and C has beaten A, 15-13. A's point differential, then, is +2 and -2, which equals +0. B's is 0 and C's is 0. This tie-breaker can not be applied, go on to tie-breaker rule 4, unless there are only three teams in the pool to begin with, in which case you should have played an extra point. See the discussion on three-team pools in Section 3.1.

Rule 4. Point differentials, counting games against all common opponents.
Example 4.1. As in example 3, above, A, B, C are in a three-way tie for first place in a four team pool. A has beaten B 15-13, B has beaten C 16-14, and C has beaten A, 15-13. All three point differentials are 0. Suppose all of them have played D; A beat D 15-9, B beat D 15-7, and C beat D 15-12. B takes first place (because their point differential against the common opponent, D, was +8), A takes second (by beating D by six goals), and C takes third (beating D by three goals).

Rule 4a. Multiple games against common opponents are averaged.

Example 4.2. Assume all of example 1, but that, for some reason, B beat D twice, 15-7 and 15-12. Take the average of the scores and only count it once, thus, we would calculate the point differential as though B beat D once, by a score of 15-9.5. Then A takes first (point differential of +6), B takes second (we count, as a point differential, the score of 15-9.5, which is 5.5), and C takes third (beating D by three goals).

Rule 5. Point differentials, counting games against all common opponents, excluding each team's best and worst differentials.
This is about the same as Rule 4, except that the best and worst scores are eliminated. This eliminates "blowouts" with which the teams in question might have been involved.

Rule 6. Points scored, counting only games among the teams that are tied.
Note to UPA officials: I thought long and hard about whether we should count most points scored, or least points allowed. There is really no good reason why one should count more than the other. The reason why I chose points scored, is that it, at least, provides an incentive for the game to start on time, and to minimize all the things that delay games (extra long time outs, excessive foul calling, procrastinating at the beginning of games and at the beginning of the second half, stalling in the hope that the game is capped, stalling because the team is tired, etc.).

Rule 7. Points scored, counting games against all common opponents.

Rule 8. Points scored, counting games against all common opponents, excluding each team's best and worst scores.

Rule 9. Sum of the square roots of the absolute values of the point differentials, counting games against all common opponents.
This has the effect of minimizing the effect of blowouts without totally excluding them from consideration. Thus, as described in the example below, a team that wins its games by the scores of 15-8 and 15-10 will have a better differential that a team that wins 15-12 and 15-4. (One might assume that at a certain point, the losing team in the 15-4 game gave up since points became somewhat irrelevant to them.) It also has the added benefit of making it almost mathematically impossible to still be tied after this rule, unless all the scores of the teams involved are identical.

Strictly speaking, mathematically, the computation regarding losses should be subtracted. Here's an example:

            A beats W 15-12         B beats W 15-8
            A beats X 15-4          B beats X 15-8
            A lost to Y 15-8        B lost to Y 15-12
            A lost to Z 15-8        B lost to Z 15-4
Suppose, further, that these were the only games involving common opponents of A and B, and that all the tie-breakers are the same through the first eight rules. Then, applying this rule, A's differentials are 3, 11, -7 and -7. The tie-breakers, then, are
            A's tie breakers = sqrt(3) + sqrt(11) - sqrt(7) - sqrt(7)
            B's tie breakers = sqrt(7) + sqrt(7) - sqrt(11) - sqrt(3)
A's tie breakers are approximately -0.2 while B's tie breakers are approximately +0.2.

Rule 10. Flip a disc.
As a last resort, discs should be flipped. Two discs should be flipped with one team calling "odd" or "even."

2.6. Tie Breakers for double elimination tournaments

A double-elimination tournament can leave an ambiguity for third place under certain conditions. This is patently unfair for those regions that are picking three teams fron Regionals to qualify for Nationals. The following are the rules to break that tie, or order of precedence:
Rule 1. Won-loss records, counting only games between the teams that are tied.
Example 1. A and B are tied for third place at 4-2, and during the tournament, A has beaten B. Then, A gets third place and B gets fourth place. When only two teams are involved, this rule is commonly called "head-to-head."

Rule 2. Number of wins against the top two teams.
Example 2. C and D have finished 3rd and 4th respectively, but both teams have lost only to the top two teams, A and B. During the tournament, however, C actually played B twice, once winning, and once losing. D played B once. Both teams lost one game to A. C, by having one win against B finishes ahead of D, who has no wins against B.

Rule 3. The point differentials of the losses of common opponents (use the average if a team has lost twice to the same team).
Example 3.1. C and D come in third and fourth, respectively, in their double-elimination, but each team lost only to A and B, and they have not played each other. If neither team has any wins against A and B, use the point differentials of C's and D's games against A and B only. C lost to A 19-13 and to B, 19-11. D lost games to A 19-9, and to B 19-18. C's loss differential is -14, D's is -11. D takes third place.

Example 3.2. The same as above except that D's two losses were to B, 19-9 and 17-14, and D never played A. Averaging the scores would give a score of 18-11.5, or a point differential of 6.5. C's point differential is -6, and so C takes third place.

Example 3.3. The same as above except that D's two losses were to B and to E. The loss to B was 19-9, the loss to E was 17-14.

Should it be relevant if C has played E ? I don't know the answer, and so I am not finishing this example.

Rule 4. Point differentials of all common opponents.
This really can't be done until the issues in Example 3.3 are decided.