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Technical
Seeding a Knockout

Jenny Clarke ran simulations on the outcome of a knockout tournament when the starting order of the players was either determined by their grade, or as a result of winning previous block games. She concludes that seeding based on block results has potential to create a rather unfair draw - even for those who have performed well in their block.

I ran some simulations based on 16 players with the following grades:

2700, 2650, 2620, 2600, 2350, 2320, 2300, 2200, 2100, 2000, 1900, 1800, 1700, 1700, 1700, 1700

These "players" were split into 4 blocks of 4, and the results of the 4 blocks were hypothetically:

  Block A   Block B   Block C   Block D
1st
2700
2000
1900
2350
2nd
2100
2650
2320
1800
3rd
1700
2300
1700
1700
4th
2200
1700
2620
2600

From here I looked at TWO draws:

Scenario A:  16-player KO draw seeded entirely by grade (note that grade is NOT recalculated after each game in these simulations).

Scenario B: 16-player KO draw seeded by block results (A1 is the winner of block A, etc), to give the first 8 matches as:

A1  v  D4
B2  v  C3
C2  v  B3
A4  v  D1
C1  v  B4
D2  v  A3
A2  v  D3
B1  v  C4

I then ran a simulation over around 1500 iterations and looked at probabilities for each of the players to win the event for each scenario. The probabilities had settled down well before this number, but it didn't hurt to run a few hundred extra.

This produced the following results:

Initial
Seeding

Scenario A: Win %
for grade-based Draw

Scenario B: Win %
for block-based Draw

1
39.90
22.82
2
26.89
25.49
3
18.93
45.80
4
14.12
4.59
5
0.06
0.39
6
0.06
0.45
7
0.06
0.32
8
0.00
0.13
9
0.00
0.06
10
0.00
0.00
11
0.00
0.00
12
0.00
0.00
13
0.00
0.00
14
0.00
0.00
15
0.00
0.00
16
0.00
0.00

Key points in comparing grade-based seeding to seeding based on block results from the results shown above:

  • Seed 1 has won their block, yet had their chance of winning the event reduced from 40% to 23%, due to the other block results
  • Seed 3, having come 4th in their block, has actually increased their chance of winning the tournament, from 19% to 46%.
  • Seed 4, on the other hand, was 4th in their block also, has had their chance of winning the event change from 14% to 4.6%, as one might anticipate
  • The winner of block C, seeded number 11 on grade, had their chance of winning the event move from 0% to 0%.

The key observation here is that using block results for seeding, results in substantial random changes to the top few players' likelihood of winning, that bear no correlation to how they actually performed in their block. Further, it does not significantly enhance the probability of others winning.

I did try changing the input grades a bit, as clearly the top 4 are a much stronger than everyone else, however that just gave the next guys a wee chance to win, which for number 5 seed in the example I used was doubled, from 2 to 4% going to block results seeding, but it remained that the number 1 seed's prospects were dramatically reduced, having won their block, while the number 3 seed's chances were more than doubled, having come fourth in their block.

Hopefully this shows that seeding based on block results has potential to create a rather unfair draw - even for those who have performed well in their blocks!

Regards to all,

Jenny

Author: Dr Jenny Clarke
All rights reserved © 2017-2017


Updated 9.ii.17
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