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Technical
How the Draw Influences Outcome in Knock-Out Events

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:

   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
This kind of draw has been used in the British Open.

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.

Author: Louis Nel
All rights reserved © 2003


Updated 28.i.16
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