Dr Ian Plummer
by Louis Nel
new version, April 2003
Introduction
The article previously posted here under the above title, dealt mainly with how the draw in a KO event influences the probability that a player will become the overall winner. In this new version, we take a broader view and look also at the probabilities that a player will reach the semi-final and other intermediate rounds.
For most players in the KO of a World Championship, reaching some round beyond the first is a more realistic aspiration than becoming the overall winner. Reaching the semi-final round leads to an automatic invitation to play in the next World Championship. So probabilities for reaching intermediate rounds certainly command interest and their inclusion brings better general understanding of the draws under consideration.
Notation for Draws
We assume a known ranking for the 32 players considered. For notational simplicity the pairings (matches) are described in terms of the rank positions e.g. 1_32 means the player ranked 1 is to play the player ranked 32. We assume also that the winner of the first pairing plays the winner of the second in the next round, the winner of the third pairing plays the winner of the fourth and so on. So the pairings of Round 1 imply the pairings for all subsequent rounds.
A Draw is a given list of pairings for Round 1. We begin by introducing names for the draws to be studied, for convenient reference.
Two Classical Draws
The Standard Draw is defined by the first round pairings
1_32 17_16 9_24 25_8 5_28 21_12 13_20 29_4 3_30 19_14 11_22 27_6 7_26 23_10 15_18 31_2
The Process Draw is defined by the first round pairings
1_17 25_9 13_29 21_5 7_23 31_15 11_27 19_3 4_20 28_12 16_32 24_8 6_22 30_14 10_26 18_2
The above two draws are long known for their role in the Seeded Draw and Process format.
Permutation Derived Draws
A given draw can be used to create a new one by permutation of a sublist of players. For example, the permutation
(5,6,7,8) --> (7,6,8,5)
applied to the Standard Draw, yields the new draw
1_32 17_16 9_24 25_5 7_28 21_12 13_20 29_4 3_30 19_14 11_22 27_6 8_26 23_10 15_18 31_2
So where 5 appeared in the Standard Draw we now have 7, where 6 appeared we still have 6, where 7 appeared we have 8 and where 8 appeared we have 5. An interchange of two players is a simple special case of a permutation of a sublist. A practical way to obtain a random permutation of the sublist (5,6,7,8) is to place tokens numbered 5,6,7,8 in a bag and draw them out one by one without peeking. The order in which they are drawn out e.g. 6,8,7,5 is then a random permutation of the sublist 5,6,7,8. In this way a random permutation of any sublist can be obtained.
The need for permutations arise where first round pairing of players from the same block or country is to be avoided; also to discourage jockeying for KO position during block play. A random permutation is one way of doing that. We study two permutation-derived draws.
An Standard-Random Draw (SR) is derived from the Standard Draw by successively applying random permutations to the sublists
(17,18,...,32),(9,...,16), (5,6,7,8) and (3,4).
Here follows an illustrative example:
This kind of draw has been used in the British Open.1_17 28_15 11_27 22_8 6_25 19_14 10_32 29_4 3_24 26_13 12_30 21_7 5_20 18_16 9_23 31_2
An Near Standard Draw (NS) is derived from the Standard Draw by successively applying random permutations to the sublists
(25,26,...,32) and (17,18,...,24).
So the players ranked (1,2,...,8) will have their oppponents randomly drawn from the sublist (25,...,32) while those ranked (9,...,16) will have their opponents randomly drawn from the sublist (17,...,24). The top 16 remain seeded relative to each other exactly as they are in the Standard Draw.
Numerical Examples
We consider a population of idealized players, each playing consistently at a certain skill level, reflected by a Grade on the World Ranking System. No real player ever plays consistently according to his Grade, but does so approximately. So the numerical studies to follow will apply at least approximately to real players, thus suggesting what might be expected about the probabilities studied.
For idealized players A and B with known Grades, the probability p(A,B) that A will beat B in the next game they play, can readily be calculated. One can then use these pairwise winning probabilities p(A,B) to calculate the probabililty for each player to win the tournament. The general reader need not be concerned with these computer executed calculations, but those interested will find explanations in the documents Winning Percentages associated with Grade Differences and Winning probabilities in knock-out events. . The probability of reaching the semi-final round is equal to the probability of winning the subtournament formed by the relevant 8 players. A similar consideration applies to the probability of reaching any other round. These remarks indicate how the tables to follow are arrived at.
Example 1 (uniform Grade distribution)
Format: KO, best-of-3 before semifinal, then best-of-5
Gradefile = reggrds.in
Draws studied:
Std :
1_32 17_16 9_24 25_8 5_28 21_12 13_20 29_4
3_30 19_14 11_22 27_6 7_26 23_10 15_18 31_2
Proc :
1_17 25_9 13_29 21_5 7_23 31_15 11_27 19_3
4_20 28_12 16_32 24_8 6_22 30_14 10_26 18_2
SR1 :
1_29 24_16 11_32 23_8 6_18 22_9 15_27 28_3
4_25 26_10 12_30 20_5 7_31 21_13 14_17 19_2
NS1 :
1_29 24_16 9_22 27_8 5_30 23_12 13_17 28_4
3_26 18_14 11_20 25_6 7_31 19_10 15_21 32_2
Column headers:
P2 = Probability % of Reaching 2nd round
PQ = Probability % of Reaching Quarterfinal
PS = Probability % of Reaching Semifinal
PC = Probability % of Winning the Championship
Rk Grade Std Proc SR1 NS1
P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC
1 2800 99 89 67 28 91 70 49 22 99 90 69 29 99 90 67 28
2 2780 99 87 61 22 91 70 49 20 92 80 57 21 99 88 62 22
3 2760 98 84 55 16 91 70 49 15 98 84 56 16 97 82 54 16
4 2740 98 80 47 11 91 70 49 13 96 69 42 11 97 79 47 11
5 2720 97 74 39 8 91 70 32 8 90 67 35 8 98 74 40 8
6 2700 96 68 31 5 91 70 32 7 85 54 25 4 94 67 31 5
7 2680 94 61 24 4 91 70 32 5 97 72 30 5 97 64 25 4
8 2660 92 53 17 2 91 70 32 4 90 55 18 2 94 55 18 2
9 2640 90 45 13 1 91 27 12 2 86 39 14 1 86 43 13 1
10 2620 86 37 11 1 91 27 12 1 91 30 13 1 78 32 10 1
11 2600 82 30 9 1 91 27 12 1 96 43 10 1 78 28 9 1
12 2580 78 23 7 0 91 27 12 1 93 30 9 0 82 24 8 0
13 2560 73 17 5 0 91 27 6 0 75 24 6 0 64 15 5 0
14 2540 67 12 4 0 91 27 6 0 60 12 4 0 64 12 4 0
15 2520 60 9 3 0 91 27 6 0 85 14 4 0 70 10 3 0
16 2500 53 6 2 0 91 27 6 0 75 9 3 0 75 9 2 0
17 2480 47 5 1 0 9 3 1 0 40 6 1 0 36 6 1 0
18 2460 40 4 1 0 9 3 1 0 15 4 1 0 36 5 1 0
19 2440 33 3 1 0 9 3 1 0 8 3 1 0 22 4 0 0
20 2420 27 3 0 0 9 3 1 0 10 3 0 0 22 3 0 0
21 2400 22 2 0 0 9 3 0 0 25 3 0 0 30 2 0 0
22 2380 18 2 0 0 9 3 0 0 14 2 0 0 14 2 0 0
23 2360 14 2 0 0 9 3 0 0 10 2 0 0 18 1 0 0
24 2340 10 1 0 0 9 3 0 0 25 1 0 0 25 1 0 0
25 2320 8 1 0 0 9 0 0 0 4 1 0 0 6 1 0 0
26 2300 6 1 0 0 9 0 0 0 9 1 0 0 3 1 0 0
27 2280 4 1 0 0 9 0 0 0 15 1 0 0 6 1 0 0
28 2260 3 0 0 0 9 0 0 0 2 0 0 0 3 0 0 0
29 2240 2 0 0 0 9 0 0 0 1 0 0 0 1 0 0 0
30 2220 2 0 0 0 9 0 0 0 7 0 0 0 2 0 0 0
31 2200 1 0 0 0 9 0 0 0 3 0 0 0 3 0 0 0
32 2180 1 0 0 0 9 0 0 0 4 0 0 0 1 0 0 0
While numerical examples such as the above one provide a useful aid in the study of draw attributes, they need to be used cautiously. The Grade distribution has a strong influence. It varies beyond control from one event to the next. The uniform grade distribution used above is artificial and will never arise in practice. Despite this, it provides a valuable neutral testing ground to reveal general draw behavior. By contrast, any historic set of Grades will introduce its own peculiar bias, never to be encountered again. The numerical example to follow uses the actual Grades of the most recent World Championship, kindly supplied to me by Chris Williams. It will illustrate, among other things, the peculiar bias arising when three players are ranked one above another on the basis of very small Grade differences -- a situation frequently arising in real life situations. The reader needs to remain alert about such peculiarities, or misleading impressions can arise when examples with historic Grades are studied.
Example 2 (Grades of WCC, Dec 2002)
All input data other than Grades are identical to those of Example 1.
Rk Grade Std Proc SR1 NS1
P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC
1 2874 100 95 87 67 96 88 80 63 99 95 87 68 99 95 87 67
2 2740 97 85 69 18 89 75 61 18 90 77 63 17 99 87 70 19
3 2656 94 72 51 5 82 63 46 5 93 75 52 5 91 70 50 5
4 2641 93 70 45 4 82 61 43 5 90 66 43 4 92 69 44 4
5 2595 89 61 31 2 78 54 10 2 77 52 26 2 91 63 32 2
6 2568 87 56 24 1 74 48 16 1 70 41 18 1 84 53 24 1
7 2558 84 51 15 1 79 54 24 1 89 56 18 1 89 54 15 1
8 2557 82 47 6 1 82 54 25 1 79 48 6 1 86 49 6 1
9 2551 81 44 5 1 82 9 5 1 72 38 16 1 72 39 5 1
10 2527 76 38 9 0 80 19 9 0 80 29 14 0 65 32 8 0
11 2516 67 31 11 0 82 28 14 0 92 46 4 0 65 31 11 0
12 2513 66 27 10 0 82 29 15 0 84 36 13 0 74 30 11 0
13 2508 64 21 9 0 83 35 4 0 65 30 8 0 59 19 8 0
14 2507 62 18 8 0 83 39 10 0 59 12 6 0 60 18 9 0
15 2482 56 9 4 0 82 36 12 0 78 21 9 0 61 9 4 0
16 2479 54 3 1 0 90 39 13 0 72 4 2 0 72 4 2 0
17 2454 46 2 1 0 4 2 1 0 41 6 3 0 41 10 4 0
18 2448 44 6 2 0 11 5 2 0 30 12 3 0 40 9 3 0
19 2438 38 8 3 0 18 8 3 0 10 4 2 0 35 12 2 0
20 2425 36 8 2 0 18 7 3 0 23 10 2 0 35 12 3 0
21 2417 34 9 2 0 22 9 1 0 35 11 2 0 39 4 1 0
22 2415 33 10 2 0 26 10 2 0 28 9 2 0 28 9 0 0
23 2365 24 6 1 0 21 8 1 0 21 6 0 0 26 6 1 0
24 2343 19 4 0 0 18 6 1 0 28 1 0 0 28 1 0 0
25 2338 18 4 0 0 18 1 0 0 10 3 1 0 16 4 1 0
26 2327 16 4 0 0 20 1 0 0 20 3 1 0 9 2 0 0
27 2302 13 3 0 0 18 2 0 0 22 2 0 0 14 3 0 0
28 2301 11 2 0 0 18 2 0 0 7 2 0 0 8 2 0 0
29 2283 7 1 0 0 17 2 0 0 1 0 0 0 1 0 0 0
30 2278 6 1 0 0 17 3 0 0 16 2 0 0 9 2 0 0
31 2268 3 1 0 0 18 3 0 0 11 2 0 0 11 2 0 0
32 2172 0 0 0 0 10 1 0 0 8 1 0 0 1 0 0 0
The above two examples suggest that the Process Draw gives a sharp drop between ranks 16 and 17 in the P2 column and a corresponding sharp drop between 8 and 9 in the PQ column, which carries over to some extent in the PS column.
Let us look more closely at how the top 5 players are handled in the above two examples respectively (separated by the dotted line):
Rk Grade Std Proc SR1 NS1
P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC
1 2800 99 89 67 28 91 70 49 22 99 90 69 29 99 90 67 28
2 2780 99 87 61 22 91 70 49 20 92 80 57 21 99 88 62 22
3 2760 98 84 55 16 91 70 49 15 98 84 56 16 97 82 54 16
4 2740 98 80 47 11 91 70 49 13 96 69 42 11 97 79 47 11
5 2720 97 74 39 8 91 70 32 8 90 67 35 8 98 74 40 8
--------------------------------------------------------------
1 2874 100 95 87 67 96 88 80 63 99 95 87 68 99 95 87 67
2 2740 97 85 69 18 89 75 61 18 90 77 63 17 99 87 70 19
3 2656 94 72 51 5 82 63 46 5 93 75 52 5 91 70 50 5
4 2641 93 70 45 4 82 61 43 5 90 66 43 4 92 69 44 4
5 2595 89 61 31 2 78 54 10 2 77 52 26 2 91 63 32 2
|
In the historic Grades, the top two were well above the others while that is not the case in the Uniform Grade distribution. The two examples show how winning expectations are influenced by that. The expectation of player 5 to reach the semi-final (column PS) seems conspicuously lower under Proc than under the other draws, in both examples. Other than that, all four the draws appear to handle the top 5 players fairly well.
Let us now look closely at how players ranked 6,7,8,9,10 compare in the above two examples. In Example 2 the Grades of 7,8,9 are virtually identical, so their relative rank positions are extremely chancy. Yet, their expectations differ considerably under the various draws. In both examples, Proc gives a notable drop in the PS column from position 8 to 9.
Rk Grade Std Proc SR1 NS1
P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC P2 PQ PS PC
6 2700 96 68 31 5 91 70 32 7 85 54 25 4 94 67 31 5
7 2680 94 61 24 4 91 70 32 5 97 72 30 5 97 64 25 4
8 2660 92 53 17 2 91 70 32 4 90 55 18 2 94 55 18 2
9 2640 90 45 13 1 91 27 12 2 86 39 14 1 86 43 13 1
10 2620 86 37 11 1 91 27 12 1 91 30 13 1 78 32 10 1
-----------------------------------------------------------
6 2568 87 56 24 1 74 48 16 1 70 41 18 1 84 53 24 1
7 2558 84 51 15 1 79 54 24 1 89 56 18 1 89 54 15 1
8 2557 82 47 6 1 82 54 25 1 79 48 6 1 86 49 6 1
9 2551 81 44 5 1 82 9 5 1 72 38 16 1 72 39 5 1
10 2527 76 38 9 0 80 19 9 0 80 29 14 0 65 32 8 0
|
It seems that as far as correspondence between rank position and winning probabilities are concerned, the Process Draw lags behind the other three. It is ahead of them as regards avoidance of lobsided games, an attribute not reflected in the above numerical studies.
The relative importance of various draw attributes and the extent to which they may be regarded as favorable or unfavorable is a matter of judgment, ultimately to be made by those in charge. This article aims merely at promoting better informed decisions.
Smaller Events
The foregoing study of 32-player events should give an idea of what to expect in smaller events and also in events with single-game matches, after certain adaptations are made.
The given tables for the Standard and Process draws apply directly to the 8-player best-of-three events formed by the sublists (1,...,8), (9,...,16), (17,...,24), (25,...,32) provided that the PRS column is interpreted as the "overall winner" column.
In a single game match, the winning probability is smaller than for a best-of-three match between the same players. The same winning probability would result from a larger Grade difference between the players. The following table quantifies this. It lists the winning probability percentages for best-of-one, best-of-three and best-of-five matches in case of a given Grade difference Gdif between the players. The table shows, for example, that bo3 winning percentage for Gdif = 20 equals bo1 winning percentage for Gdif = 30. In other words, in a population where the typical Grade difference between successively ranked players is 30, best-of-one matches will be just as effective as best-of-three matches are in a population with typical successive Grade differences of 20. In particular, if the Grades column in Example 1 above is replaced by one with increments of 30 instead of 20, then the listed winning percentages apply to best-of-one matches instead of best-of-three matches. Also, if the bottom half of a KO population is much weaker than the top half, it may be worth considering best-of-one matches in the first round followed by best-of-three in the later rounds.
Gdif bo1 bo3 bo5 10 51 52 52 20 52 53 54 30 53 55 56 40 55 57 59 50 56 59 61 60 57 60 63 70 58 62 65 80 59 64 67 90 60 65 69 100 61 67 71 110 62 68 72 120 63 70 74 130 65 71 76 140 66 73 77 150 67 74 79 160 68 75 80 170 69 77 82 180 70 78 83 190 71 79 84 200 72 80 86 210 72 81 87 220 73 82 88 230 74 84 89 240 75 85 90 250 76 85 91 260 77 86 91 270 78 87 92 280 78 88 93 290 79 89 94 300 80 90 94 310 81 90 95 320 81 91 95 330 82 91 96 340 83 92 96 350 83 93 96 360 84 93 97 370 85 94 97 380 85 94 97 390 86 94 98 400 86 95 98 |
For 16-player events, an SR draw can be obtained by permutation of the sublists (9,...,16), (5,6,7,8) and (3,4), while for an NS draw the sublists are (13,14,15,16) and (9,10,11,12). For 8-player events, the SR sublists are (5,6,7,8) and (3,4) while the NS sublists are (7,8) and (5,6). This is merely an indication about formatting. While the above numerical examples can be applied to the Standard and Process draws in 16-player or 8-player events (as indicated), they cannot be applied to SR or NS draws in these smaller events. Separate tables will need to be calculated. The 32-player case will nevertheless give a general idea of what to expect for smaller events.