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Custom Knockout Tournament Seeding
Theory
There are various papers regarding the optimal seeding of single elimination (KO) tournaments.
Groh et al. (2008) lists the following fairness criteria:
The method should meet the following three axioms according to Schwenk (2000):
These criteria are in a tradeoff relation with each other by the designers perspective, so the optimal decision is not trivial. The tournament organizer may have good a priori knowledge of the win probability of one team versus another which leads to a different approach. There are methods which claim to maintain fairness even in the case of dropouts.
Here in this writing, I assume that the organizer have no quantifiable knowledge of the win probability matrix of the players. The players are, however, ordinally ranked. This means that they can be aligned by their strengthness. Maybe they have ELO ratings, or more commonly they just finished a regular season which determined their order.
Figure 1. Standard Method for seeding a tournament with 32 or fewer teams.
Figure 1. shows the standard method of seeding which is commonly used by most sports events. The seed algorithm is this: Start with {1-2} (e.g. the two highest ranked player). Repetitively create a new layer and fill the pairings in reversed order until all players get their slots. This makes {1-4,2-3} for the second layer and {1-8,4-5,2-7,3-6} for the third one, and so on. Notice, that the top rank gets 8th rank, the 2nd highest rank gets 7th rank and so on.
Groh et al. (2008) shows that this method maximizes the win probability of the strongest player, and is the unique one with the property that strictly stronger players have a strictly higher probability of winning (criteria 3 and 4). On the other hand, seeding {1-3,2-4} maximizes both total effort across the tournament and the probability of a final among the two top players (criteria 1 and 2).
Schwenk (2000) shows that the standard method violates axiom 2. The problem with the standard schedule is that {1} is assured a moderately tough opponent in the third round while {2} benefits from a significantly possible upset by {6} over {3}. This makes {2} more likely to win the tournament. Schwenk proposes cohort randomized seeding (see later).
Figure 2. FUMBBL Method for seeding a tournament with 32 or fewer teams.
Figure 2. shows the way seeding is done by FUMBBL. I did not find this method documented in any other place. I can not tell whether this method was designed by intention, or it is a result of a bug in the seeding algorithm. To be sure, I filled a bug report of this issue.
FUMBBL's algorithm does not fill the pairings by ranking, instead it fills them by position. This makes the seeding identical for the final and semi-finals but differs for the other rounds since it yields {1-8,4-7,2-6,3-5} for the third layer. Notice the 8, 7, 6, 5 series here. It goes by position.
Without doing some math I can't say for sure how the criteria and the axioms are satisfied by this method. According to my feelings, it is inferior to the standard method by criteria 1 and superior to it by criteria 2 and axiom 2. You can see that {2} got a much harder way to the final. {2}'s branch have the stronger half of players as opponenets in every layer.
Figure 3. Cohort Randomized Seeding for 16 or fewer teams.
Figure 3. shows the cohort randomized seeding where cohorts are numbered with roman numerals and each of them have 2^c players with c increasing from 2. The actual seed numbers are drawn randomly for each cohort without replacement. This makes cohort II cup (3, 4), cohort III cup (5, 6, 7, 8) and so on.
Schwenk (2000) claims that this method satisfies all three axioms. It also performs well in the analysis of Glickman (2008).
Practice
Now I will show how the above seeding methods can be applied to a FUMBBL Knockout tournament:
Table 1. Custom Group Scores to set up a KO tournament for 8 teams by method.
Table 2. Custom Group Scores to set up a KO tournament for 16 teams by method.
Table 3. Custom Group Scores to set up a KO tournament for 32 teams by method.
Unfortunately teams with identical scores are seeded in a weird but fixed way (I was unable to figure out the method) by FUMBBL. Ideally, FUMBBL should automatically seed each set of teams with identical scores randomly. That would make a 16 team Cohort Randomized Seeding setup more straightforward. 4 score to {1}, 3 to {2}, 2 to {3, 4}, 1 to {5..8}, and 0 to {9..16}. Instead, currently the random draws of the cohorts must be made by the organizer.
References
Glickman, Mark E. (2008): ”Bayesian locally-optimal design of knockout tournaments”, Journal of Statistical Planning and Inference, 138, 2117-2127.
Groh et al. (2008): ”Optimal Seedings in Elimination Tournaments”
Schwenk, A. (2000): ”What is the Correct Way to Seed a Knockout Tournament”, American Mathematical Monthly 107, 140-150.
There are various papers regarding the optimal seeding of single elimination (KO) tournaments.
Groh et al. (2008) lists the following fairness criteria:
- Find the seeding(s) that maximizes the probability of a final among the two highest ranked players.
- Find the seeding(s) that maximizes the win probability of the highest ranked player.
- Find the seeding(s) that maximizes total expected effort in the tournament.
- Find the seeding(s) with the property that higher ranked players have a higher probability of winning the tournament.
The method should meet the following three axioms according to Schwenk (2000):
- Delayed Confrontation: Two teams rated among the top 2^j shall never meet until the field has been reduced to 2^j or fewer teams.
- Sincerity Rewarded: A higher-seeded team should never be penalized by being given a schedule more difficult than that of any lower seed.
- Favoritism Minimized: The schedule should minimize favoritism to any particular seed. (Or, if you prefer, Fairness Maximized.)
These criteria are in a tradeoff relation with each other by the designers perspective, so the optimal decision is not trivial. The tournament organizer may have good a priori knowledge of the win probability of one team versus another which leads to a different approach. There are methods which claim to maintain fairness even in the case of dropouts.
Here in this writing, I assume that the organizer have no quantifiable knowledge of the win probability matrix of the players. The players are, however, ordinally ranked. This means that they can be aligned by their strengthness. Maybe they have ELO ratings, or more commonly they just finished a regular season which determined their order.
Figure 1. Standard Method for seeding a tournament with 32 or fewer teams.
Figure 1. shows the standard method of seeding which is commonly used by most sports events. The seed algorithm is this: Start with {1-2} (e.g. the two highest ranked player). Repetitively create a new layer and fill the pairings in reversed order until all players get their slots. This makes {1-4,2-3} for the second layer and {1-8,4-5,2-7,3-6} for the third one, and so on. Notice, that the top rank gets 8th rank, the 2nd highest rank gets 7th rank and so on.
Groh et al. (2008) shows that this method maximizes the win probability of the strongest player, and is the unique one with the property that strictly stronger players have a strictly higher probability of winning (criteria 3 and 4). On the other hand, seeding {1-3,2-4} maximizes both total effort across the tournament and the probability of a final among the two top players (criteria 1 and 2).
Schwenk (2000) shows that the standard method violates axiom 2. The problem with the standard schedule is that {1} is assured a moderately tough opponent in the third round while {2} benefits from a significantly possible upset by {6} over {3}. This makes {2} more likely to win the tournament. Schwenk proposes cohort randomized seeding (see later).
Figure 2. FUMBBL Method for seeding a tournament with 32 or fewer teams.
Figure 2. shows the way seeding is done by FUMBBL. I did not find this method documented in any other place. I can not tell whether this method was designed by intention, or it is a result of a bug in the seeding algorithm. To be sure, I filled a bug report of this issue.
FUMBBL's algorithm does not fill the pairings by ranking, instead it fills them by position. This makes the seeding identical for the final and semi-finals but differs for the other rounds since it yields {1-8,4-7,2-6,3-5} for the third layer. Notice the 8, 7, 6, 5 series here. It goes by position.
Without doing some math I can't say for sure how the criteria and the axioms are satisfied by this method. According to my feelings, it is inferior to the standard method by criteria 1 and superior to it by criteria 2 and axiom 2. You can see that {2} got a much harder way to the final. {2}'s branch have the stronger half of players as opponenets in every layer.
Figure 3. Cohort Randomized Seeding for 16 or fewer teams.
Figure 3. shows the cohort randomized seeding where cohorts are numbered with roman numerals and each of them have 2^c players with c increasing from 2. The actual seed numbers are drawn randomly for each cohort without replacement. This makes cohort II cup (3, 4), cohort III cup (5, 6, 7, 8) and so on.
Schwenk (2000) claims that this method satisfies all three axioms. It also performs well in the analysis of Glickman (2008).
Practice
Now I will show how the above seeding methods can be applied to a FUMBBL Knockout tournament:
- More > Edit Group Info. Enable Score.
- Teams > (Edit Scores). Edit the scores of the qualified teams according to Tables 1-3. below.
- Tournaments > (Create new Tournament). Create a Knockut tournament.
- (Start Tournament)
- Important! Seeding: Group Score
- Make sure that only qualified teams are enabled. Start.
- Once the tournament has been started, you may freely clear and disable scores (1 and 2).
Rank | Standard | FUMBBL | Cohort |
1 | 7 | 7 | 7 |
2 | 6 | 6 | 6 |
3 | 5 | 5 | II: ?{4, 5} |
4 | 4 | 4 | II: ?{4, 5} |
5 | 1 | 3 | III: ?{0..3} |
6 | 3 | 2 | III: ?{0..3} |
7 | 2 | 1 | III: ?{0..3} |
8 | 0 | 0 | III: ?{0..3} |
Table 1. Custom Group Scores to set up a KO tournament for 8 teams by method.
Rank | Standard | FUMBBL | Cohort |
1 | 15 | 15 | 15 |
2 | 14 | 14 | 14 |
3 | 13 | 13 | II: ?{12, 13} |
4 | 12 | 12 | II: ?{12, 13} |
5 | 9 | 11 | III: ?{8..11} |
6 | 11 | 10 | III: ?{8..11} |
7 | 10 | 9 | III: ?{8..11} |
8 | 8 | 8 | III: ?{8..11} |
9 | 1 | 7 | IV: ?{0..7} |
10 | 5 | 6 | IV: ?{0..7} |
11 | 7 | 5 | IV: ?{0..7} |
12 | 3 | 4 | IV: ?{0..7} |
13 | 2 | 3 | IV: ?{0..7} |
14 | 6 | 2 | IV: ?{0..7} |
15 | 4 | 1 | IV: ?{0..7} |
16 | 0 | 0 | IV: ?{0..7} |
Table 2. Custom Group Scores to set up a KO tournament for 16 teams by method.
Rank | Standard | FUMBBL | Cohort |
1 | 31 | 31 | 31 |
2 | 30 | 30 | 30 |
3 | 29 | 29 | II: ?{28, 29} |
4 | 28 | 28 | II: ?{28, 29} |
5 | 25 | 27 | III: ?{24..27} |
6 | 27 | 26 | III: ?{24..27} |
7 | 26 | 25 | III: ?{24..27} |
8 | 24 | 24 | III: ?{24..27} |
9 | 17 | 23 | IV: ?{16..23} |
10 | 21 | 22 | IV: ?{16..23} |
11 | 23 | 21 | IV: ?{16..23} |
12 | 19 | 20 | IV: ?{16..23} |
13 | 18 | 19 | IV: ?{16..23} |
14 | 22 | 18 | IV: ?{16..23} |
15 | 20 | 17 | IV: ?{16..23} |
16 | 16 | 16 | IV: ?{16..23} |
17 | 1 | 15 | V: ?{0..15} |
18 | 9 | 14 | V: ?{0..15} |
19 | 13 | 13 | V: ?{0..15} |
20 | 5 | 12 | V: ?{0..15} |
21 | 7 | 11 | V: ?{0..15} |
22 | 15 | 10 | V: ?{0..15} |
23 | 11 | 9 | V: ?{0..15} |
24 | 3 | 8 | V: ?{0..15} |
25 | 2 | 7 | V: ?{0..15} |
26 | 10 | 6 | V: ?{0..15} |
27 | 14 | 5 | V: ?{0..15} |
28 | 6 | 4 | V: ?{0..15} |
29 | 4 | 3 | V: ?{0..15} |
30 | 12 | 2 | V: ?{0..15} |
31 | 8 | 1 | V: ?{0..15} |
32 | 0 | 0 | V: ?{0..15} |
Table 3. Custom Group Scores to set up a KO tournament for 32 teams by method.
Unfortunately teams with identical scores are seeded in a weird but fixed way (I was unable to figure out the method) by FUMBBL. Ideally, FUMBBL should automatically seed each set of teams with identical scores randomly. That would make a 16 team Cohort Randomized Seeding setup more straightforward. 4 score to {1}, 3 to {2}, 2 to {3, 4}, 1 to {5..8}, and 0 to {9..16}. Instead, currently the random draws of the cohorts must be made by the organizer.
References
Glickman, Mark E. (2008): ”Bayesian locally-optimal design of knockout tournaments”, Journal of Statistical Planning and Inference, 138, 2117-2127.
Groh et al. (2008): ”Optimal Seedings in Elimination Tournaments”
Schwenk, A. (2000): ”What is the Correct Way to Seed a Knockout Tournament”, American Mathematical Monthly 107, 140-150.