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Hyunsuk Yang, Wone Keun Han, & Dongwook Park*

College of Engineering, Hongik University, Seoul, Korea

Abstract

A model based on the probability of the server winning the rally was employed to evaluate the

influence of the newly proposed scoring system, the best of five games of 11 points scoring system,

being experimented by the Badminton World Federation on singles badminton matches. The model, based

on the assumption of statistical independence on each point’s outcome, was used to generate predictions

ranging from the game- and match-winning probabilities to game- and match-length statistics for matches

under both the new and the current scoring system, the best of three games of 21 points. Validity of

these results was checked against tournament data, four sets each for the two scoring systems, as well as

previously published results, with satisfactory agreement in most cases. The results show that duration of

singles matches would be reduced noticeably under the new scoring system without affecting the match

outcome of the current scoring system.

Key words: Badminton, scoring system, singles, probability model

Badminton is a sport enjoyed by millions around the

world, and over the years, it has gone through several

changes in scoring system regarding how a game or a

match is to be won. In recent years, the changes were

implemented, first under the auspices of the International

Badminton Federation (IBF), and then later Badminton

World Federation (BWF), in order to adapt the game’s

characteristics to fit in with the changing times (Wikipedia,

2015). Most recent major overhaul of the rules was

implemented in 2005 by the BWF in the form of a best of

three games of 21 points scoring system (3⨯21 format). In

an attempt to make the playing time even shorter and more

predictable (BWF, 2014a), the BWF decided in 2014 to

Submitted : 10 October 2016, revised: 24 November 2016

accepted : 29 November 2016

Correspondence : dwpark@hongik.ac.kr

test a new rule, instating it in the BWF law of badminton

(BWF, 2015) as one of the alternative scoring systems.

Under this rule, a match would be decided by a best of

five games of 11 points scoring system (5⨯11 format),

with each game concluding at 11 points without a deuce or

setting feature (BWF, 2014b). There have even been

tournaments held under the experimental rule (BWF,

2014b) and some feedback from the players, trainers and

fans in general on the new rule (Badzine, 2014).

A number of research have been performed in the past

regarding the overall effects of rule changes in badminton

and also other sports. Arias, Argudo, and Alonso (2011)

presented a review of 139 studies dealing with rule modification

in various sports, with an emphasis on classification of the

studies. Wright (2014) gave a survey and an analysis of

numerous articles covering competition rules in 21 different

sports, focusing on the analytical methods employed and

International Journal of Applied Sports Science

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ISSN 2233-7946 (Online)

2016, Vol. 28, No. 2, 226-234. ISSN 1598-2939 (Print)

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Analysis of the best of five games of 11 points scoring system in singles badminton matches 227

on issues of fairness and consequences of rule changes.

There have also been published results on more specific

influences of badminton rules changes on the temporal and

notational aspects. Ming, Keong, and Ghosh (2008), for

example, performed a time motion and notational analysis

of 21-point and 15-point badminton match play for singles

and found that the total number of shots and rallies in a

match were significantly affected. Laffaye, Phomsoupha,

and Dor (2015) analyzed the characteristics of six Olympic

badminton men’s singles finals that were played under

three different scoring systems through a longitudinal study

and found a significant increase in the shot frequency but

a substantial decrease in the effective play time and work

density over the years. Some of the noteworthy work in

the past involving modelling of a match with a

probabilistic approach have focused on topics such as the

game-winning probability based on combinatorics under

the 21-point-game-with-setting rule (Hsi & Burych, 1971),

advantages/disadvantages of serving first in men’s doubles

under the ‘old’ rule (15-point game with side-out) (Marcotte,

1989), optimization of service type over the course of a

singles match under the 3⨯21 format (Bedford, Barnett, &

Ladds, 2010), and a simple analysis of the 3⨯21 format

without including the serve effect (BadmintonCentral.com,

2010). Perhaps the most comprehensive work to this date

is the article by Percy which addressed various issues

pertaining to the rule change from the ‘old’ (3⨯15 format

with side-out) to ‘current’ (3⨯21 format without side-out)

systems (Percy, 2008).

In this paper, we present a direct comparison between

the current 3⨯21 format and the experimental 5⨯11

format in regard to their influence on singles matches by

using a probability model taking into account the service

effect. Our analysis will focus on highlighting the game-

and match-winning probabilities and also game- and

match-lengths - as represented by the number of points

played - associated with singles matches under the two

respective scoring systems. It is hoped that the results of

our study would help shed some light on how the 5⨯11

scoring system fares against the 3⨯21 format in the

context of preservation of the match outcome characteristics

and also effects on match-lengths. The significance of such

an effort lies in that the BWF-sanctioned 5⨯11 format

may be able to offer a viable option for managing

badminton matches and tournaments, with the important

advantage of requiring less time.

We assume that the outcome of each point is a

statistically independent and identically distributed event,

which is determined solely by the point-winning

probability associated with the server, taken to be constant

throughout the course of a match. It is generally true that

the outcome of each point, game, and even matches can

depend on the previous outcomes as well as the current

score, thereby making these events correlated to varying

degrees in many cases. In spite of this, we have decided

to employ the statistical independence assumption because

incorporating such statistical dependences into the model is

a very complex task and we wanted to focus on the direct

consequences of the rule change instead. Thus, the model

for a singles match is completely characterized by two

parameters p1 and p2, each representing the probability that

Player A or Player B wins the corresponding rally point

when he or she has the serve. Let P(m,n,x) represent the

probability that the score of the on-going game is m:n

(Player A: Player B) with the ensuing serve belonging to

Player x (x = 1, 2 for Players A, B, respectively) in a

game first started with Player A’s serve. The recursive

relations appropriate for calculations of P(m, n, x) are as follows:

P(m,n,1) = P(m-1,n,1)*p1 + P(m-1,n,2)*(1-p2)(1)

P(m,n,2) = P(m,n-1,1)*(1-p1) + P(m,n-1,2)*p2(2)

where the initial conditions are such that P(0,0,1) = 1 and

P(0,0,2) = 0 for Player A having the first serve in the

game. This approach has also been used by several in the

past (Bedford, Barnett, & Ladds, 2010; Brown, Barnett, &

Pollard, 2008). Beginning with the first serve of the game,

various running-scores and the associated probabilities of

228 Hyunsuk Yang et al.

occurrence are generated in this manner as each additional

rally point is played.

For each game, there are four different possible cases to

consider depending on the first server and the game winner.

Let P1(2)game denote the probability that Player A(B) is the

winner, given that Player A starts the game, and Q1(2)game

the probability that Player A(B) is the winner, given that

Player B starts the game. Note that P2game = 1 - P1game and

Q₂game = 1- Q1game. Under the 5⨯11 format, the first player

to reach 11 points wins the given game regardless of the

final margin. Thus,

. Meanwhile, under

the 3⨯21 format, the first player to reach 21 points, with

the margin of at least two, wins the given game. In the

event that the score reaches a 20-all tie, the players play

by the deuce rule, in which the first to lead the opponent

by two points takes the game, unless the score becomes

29-all wherein the winner of the next point, i.e., the first

to reach the 30-point mark, wins the game (22:20, 23:21,

..., 30:28, 30:29). Therefore, the probability of Player A

winning the game can be calculated by summing over the

probabilities of all possible cases, namely P(21,n,1) (n =

0,1,…,19), P(m,m-2,1) (m = 22,23,…,30) and P(30,29,1).

Q1(2)game can be obtained in a similar fashion.

Meanwhile, once the probability distribution of the

game-ending scores is obtained as described above, it is

straightforward to calculate the probability distribution of

the total number of points (N = m + n) played in the

game. The mean and standard deviation of N, which are

related to the average and fluctuation of the game duration,

respectively, can then be calculated in a simple manner,

In both formats, the initial server of the first game is

randomly determined, but thereafter, the winner of the

previous game is awarded the first serve of following

game. Here, it shall be assumed that each game is

statistically independent, aside from the initial serve

consideration just mentioned.

Let us first take the 3⨯21 format, in which the

match-ending score must be either 2-0 (two-games-to-zero)

or 2-1 (two-games-to-one) in favor of the winning player.

The winning player must take the first two games in

straight fashion in the 2-0 scenario (WW) and win one of

the first two games and the last game, i.e., the third, in the

2-1 scenario (LWW or WLW). Thus, the probability of

Player A winning by the first scenario is either P1game ×

P1game (when A serves first) or Q1game× P1game (when B

serves first), depending on the first server. The probability

of Player A winning via the second scenario is P1game ×

P2game × Q1game + P2game × Q1game × P1game (when A serves first)

or Q1game × P2game × Q1game + Q₂game × Q1game × P1game (when

B serves first). The possible match-winning scores under

the 5⨯11 format are 3-0, 3-1 and 3-2, with one (WWW),

three (WWLW, WLWW, LWWW) and six sub-scenari os

(WWLLW, WLWLW, WLLWW, LWWLW, LWLWW,

LLWWW) in each respective case. The match-winning

probability associated with each (sub-)scenario can be

computed in a manner similar to that of the 3⨯21 format,

with the exception of having to use this time the

game-winning probabilities P1game, P2game, Q1game, and Q₂game

derived earlier for the 5⨯11 format. The overall

match-winning probability for a given player can then be

determined simply by adding the probabilities for all

possible sub-scenarios in each format.

In the process of evaluating the match-winning

probability as described above, it is possible to obtain the

mean and variance of N, the total number of points in a

match, associated with a particular match-winning scenario.

By taking a weighted average of these quantities over all

possible sub-scenarios, the overall expected values are

derived.

Analysis of the best of five games of 11 points scoring system in singles badminton matches 229

Three sample values of p1 (= 0.4, 0.5, 0.6) were chosen

to demonstrate and distinguish the characteristics of the

game under the 3⨯21 and 5⨯11 formats, while the value

of p2, the corresponding probability of the opponent, was

allowed to vary between 0 and 1 to account for

possibilities of encountering an opponent of all skill levels.

Before proceeding with presentation of the calculation

results, it should be pointed out that the results of the

previous works for the 3⨯21 format (Bedford, Barnett, &

Ladds, 2010; BadmintonCentral.com, 2010; Percy, 2008)

were reproduced exactly with our model when a direct

comparison was possible. Figure 1 displays the game-winning

probability of Player A, under the 5⨯11 and 3⨯21

formats, respectively, for the three different values of p1

(= 0.4, 0.5, 0.6). Clearly, the general trends between the

two sets of curves are quite similar. As the value of p2 is

increased from 0 toward p1, P1game decreases from 1, slowly

at first and then more rapidly to 0.5. P1game continues to

decrease toward 0 as p2 approaches 1, showing a

saturation-like behavior near the end region. This general

trend is more pronounced for the cases of the 3⨯21

format compared to those of the 5⨯11 format, which can

be attributed to the fact that longer games of the former

format enhance the skill-level discrepancy between the two

players.

Figure 2 shows the match-winning probability of Player

A under the 5⨯11 and 3⨯21 formats for p1 = 0.4, 0.5,

0.6. The match-winning probability curves are more

saturated, i.e., flatter, in the end regions and steeper near

the p1 = p2 location compared to the game-winning

probability counterparts. The most important development

is that the difference of match-winning probability under

two different game formats has been drastically reduced

across the entire range of p2, compared to the game-winning

probability curves of Figure 1. Apparently, over the course

Figure 1.

The first server’s game-winning probability unde

r

two different scoring systems (the solid lines are

for the 5

⨯

11 format and the broken lines are

for the 3

⨯

21 format). The dependence on the

opponent’s point-winning probability p

2

is shown

for three different values of the first server’

s

p

robabilit

y

p

1

.

Figure 2.

The first server’s match-winning probability unde

r

two different scoring systems (the solid lines are

for the 5

⨯

11 format and the broken lines are for the

3

⨯

21 format). The dependence on the o

p

ponent’

s

point-winning probability p

2

is shown for three

different values of the first server’s

p

robabilit

y

p

1

.

of a match spanning more than one game, the differences

due to the different scoring systems largely disappear,

causing the match-winning probabilities for the two scoring

230 Hyunsuk Yang et al.

systems to converge toward each other.

In the upper portion of Figures 3(a) and 3(b), the

average number of points in a game is displayed for three

different values of p1 under the two formats, again

assuming that Player A serves first. In all figures, the

average number of points in a game peaks in the vicinity

of p1 = p2 - due to the evenly-contested nature of play

(leading to longer games) - and falls off - albeit in an

asymmetric manner - as the disparity between the two

players grows. Also, note that both the peak and mean

(averaged over p2) values of the average number of points

are slightly higher for the p1 = 0.4 cases. This is so

because when both p1 and p2 are low, e.g., in the case of

p1 = 0.4 and p2 = 0.38 ~ 0.39, serves are expected to

change hands more frequently as the server is more likely

to lose the point than win it, thereby leading to a longer,

see-saw type of game. If both p1 and p2 are high, on the

other hand, as in the case of p1 = 0.6 and p2 = 0.59 for

example, servers are expected to retain their serve longer,

making it more likely to be able to string together

consecutive points and leading to shorter games. The

standard deviation of the number of points played in a

game is displayed in the lower portion of Figures 3(a) and

3(b), respectively, for the three p1 values. Note that the

standard deviation curve does not fluctuate much with p2

for all three p1 cases under both formats, remaining in the

2 ~ 3 and 4 ~ 5 ranges, respectively, for the most part,

which represent only a small fraction of the average

number of points in a game.

Next, the behavior of the average number of points in

a match is displayed in the upper portion of Figures 4(a)

and 4(b) for various combinations of p1 value and scoring

format. It is seen that the average number of points per

match is considerably lower under the 5⨯11 format

compared to the 3⨯21 results across the board. The

average number of points for a match ranges from 62 ~ 78

for the 5⨯11 format and from 75 ~ 97 for the 3⨯21

format, considering only the 0.4 ~ 0.6 range for p2.

Finally, the standard deviation curves are shown in the

lower portion of Figures 4(a) and 4(b). The curves exhibit

a broad peak region in the 15 ~ 16 range for the 5⨯11

format and a substantially narrower peak region -

indicating less sensitivity to p2 variation - in the 19 ~ 20

range for the 3⨯21 format, respectively.

Figure 3.

The average and standard deviation of the numbe

r

points played in a game (a) under the 5

⨯

11

format and (b) under the 3

⨯

21 format. They are

shown as a function of the opponent’s point-winnin

g

probability p

2

for three choices of p

1

(solid lines

are for p

1

= 0.4, dashed lines for p

1

= 0.5,

dotted lines for

p

1

= 0.6

)

.

Analysis of the best of five games of 11 points scoring system in singles badminton matches 231

Figure 4.

The average and standard deviation of the numbe

r

points played in a match (a) under the 5

⨯

11

format and (b) under the 3

⨯

21 format. They are

shown as a function of the opponent’s point-winnin

g

probability p

2

for three choices of p

1

(solid lines

are for p

1

= 0.4, dashed lines for p

1

= 0.5,

dotted lines for

p

1

= 0.6

)

.

Our model, as mentioned earlier, is very simple, and as

such, it is imperative to check the predicted results against

actual data of relevance in order to gain a measure of

validation. In order to compare our calculation results

against actual badminton matches, we analyzed a sample

of tournament data available from a website (Tournament

Software (http://www.tournament software.com)). While

the tournament data cannot provide a direct means of

verifying the predictions of game- and match-winning

probabilities calculated from our model, it may still be

possible to make some comparisons regarding the game-

and match-length statistics under a reasonable set of

assumptions.

Over the period of August 2014 to November 2014, 21

international tournaments (including 12 junior events)

sanctioned by BWF were held using the 5⨯11 format

(BWF, 2014b). Of these, men’s singles (MS) and women’s

singles (WS) match results from four selected events

representing various competition levels, geographical areas

and calendar dates were used as sample data. The data

from 264 MS and 163 WS match results were used for

comparison against the results from our calculation. We

calculated the average and the standard deviation for the

number of games per match, number of points per match,

and the match duration (in minutes). For comparison, we

also selected four tournaments with similar attributes that

employed the conventional 3⨯21 format, and the statistics

of 273 MS and 138 WS matches played in those

tournaments were analyzed.

The statistics regarding the match duration in minutes

cannot be directly compared against our model, as the

number of strokes or the time required to complete a rally

is not incorporated in the model. However, it may still be

possible to make a comparison between the tournament

and model data regarding the statistics of the number of

games per match and the number of points played per

match under the following assumptions. One, the skill

levels of the players that participated in the aforementioned

tournaments span a certain range, which corresponds to the

point-winning probability range of 0.3 ~ 0.7 in our model,

and two, the probability distribution is uniformly

distributed within that range. The range of 0.3 ~ 0.7 seems

reasonable since it is expected that completely lop-sided

matches are rather unlikely in competitive international

matches. Under these assumptions, calculations can be

232 Hyunsuk Yang et al.

made within our model for all cases within the probability

grid of 0.3 < p1, p2 < 0.7 and the results averaged, which

can then be compared against the tournament statistics.

The results for the 5⨯11 and 3⨯21 formats are

summarized in Tables 1 and 2, respectively. Under the

5⨯11 format, the average number of games per match

shows a range of 3.61 ~ 3.74 for MS matches and 3.29 ~

3.57 for WS matches (with the standard deviation in the

0.63 ~ 0.75 range) compared to 3.78 ± 0.72 for our

calculation result. The corresponding ranges are 2.15 ~

2.35 for MS matches and 2.30 ~ 2.38 for WS matches

(with the standard deviation in the 0.36 ~ 0.49 range)

under the 3⨯21 format, respectively, compared to 2.30 ±

0.42 of our calculation. The average number of points per

match spans the range of 62.20 ~ 66.79 points for MS

matches and 54.42 ~ 64.26 for WS matches (with the

standard deviation in the 13.12 ~ 16.90 range) under the

5⨯11 format and 73.64 ~ 83.14 for MS matches and

79.06 ~ 86.04 for WS matches (with the standard deviation

in the 16.60 ~ 21.44 range) under 3⨯21 format. The

corresponding theoretical results, on the other hand, are

67.11 ± 14.22 and 81.95 ± 16.23, respectively. Thus, it

appears that there is a fairly good agreement between the

tournament and the analysis data. Also, note that the

average match time is considerably shorter - by more than

several minutes in most cases - for the 5⨯11 tournaments

for both MS and WS matches, whereas the difference in

the match time fluctuation between the two sets of

tournaments is generally only a couple of minutes or less.

The fact that the match-durations are significantly shorter

in the 5⨯11 tournaments may be of paramount interest

from a tournament organizer standpoint as there should be

more flexibility and margin of error in scheduling the

matches and managing the tournament.

Yonex Dutch Open

(Grand Prix)

Brazil International

Badminton Cup

(Int. Challenge)

Fernbaby Auckland

International

(Int. Series)

Bulgaria Eurasia Open

(Future Series) Our

Calculation

<MS > < WS >< MS >< WS >< MS >< WS >< MS >< WS >

No. of games 3.73

±0.75 3.57

±0.74 3.72

±0.73 3.29

±0.64 3.61

±0.72 3.52

±0.68 3.74

±0.75 3.38

±0.63 3.78

±0.72

No. of points 66.45

±15.55 64.26

±14.97 64.99

±16.27 54.42

±15.92 62.20

±16.52 59.00

±16.90 66.79

±15.56 59.78

±13.12 67.11

±14.22

Match length

(min.) 30.52

±9.96 29.40

±10.14 34.10

±11.64 27.04

±10.72 26.82

±10.89 24.10

±10.21 29.62

±9.78 26.20

±8.13

Table 1.

Summary of match length statistics under the 5

⨉

11 format

Scottish Open

(Grand Prix) Lagos International

(Int. Challenge)

OUE Singapore

International Series

(Int. Series)

Yonex Riga

International

(Future Series)

Our

Calculation

< MS >< WS >< MS >< WS >< MS >< WS >< MS >< WS >

No. of games 2.35

±0.48 2.32

±0.47 2.15

±0.36 2.30

±0.47 2.26

±0.44 2.30

±0.46 2.29

±0.46 2.38

±0.49 2.30

±0.42

No. of points 83.14

±20.08 82.27

±18.96 73.64

±16.60 79.06

±21.44 78.35

±20.22 79.60

±19.67 79.96

±20.36 86.04

±20.83 81.95

±16.23

Match length

(min.) 38.61

±14.00 36.12

±10.84 30.63

±13.02 27.78

±10.15 32.67

±11.95 33.58

±12.35 35.98

±11.10 36.43

±9.76

Table 2.

Summary of match length statistics under the 3

⨉

21 format

Analysis of the best of five games of 11 points scoring system in singles badminton matches 233

At this point, we would like to address the issue of the

‘first-serve’ effect, namely whether and if so when it is

advantageous to be the first server in the game. Although

the results are not included here, it can be shown that the

game-winning probability is greater for the first server,

provided pj > 1- pi, where pi and pj are the point-winning

probabilities of the first server and first receiver,

respectively. The first server is also at an advantage when

two players of identical skill-levels with p1 (= p2) > 0.5 are

facing each other, which might require a counter-measure

of some sort to offset such bias. These first-serve effects,

though inherent in both scoring formats on game and

match levels, are generally more pronounced for the 5⨯11

cases due to the fewer number of points involved.

The experimental 5⨯11 scoring system appears to offer

an attractive alternative to the current 3⨯21 counterpart

based on the analysis presented in this paper. Results were

generated based on the probabilistic analysis of a

two-parameter model and partially validated by a statistical

analysis of data from eight tournament results. The

findings suggest that singles matches would tend to be

completed in less number of points and time under the

5⨯11 format, with the characteristics of the match

outcome hardly changed from those of the current 3⨯21

format. To wit, the new rule is more forgiving - less

sensitive - to the player’s skill-level difference insofar as a

single game’s outcome is concerned, but the eventual

match winner is highly unlikely to change, even under the

new rule. The variation of the match length, on the other

hand, is not expected to decrease noticeably under the

5⨯11 system. At least from the quantitative standpoint

then, these preliminary findings imply that singles matches

in the trial format might perhaps be more exciting to watch

from the viewers' perspective and more appealing to

tournament organizers and broadcasting partners. Shorter

matches would mean that the viewers would be able to

concentrate on the matches more while the tournament

organizers would be able to schedule more matches in a

given time duration and/or with more room to cope with

late-running matches. We conclude with a remark that the

analysis presented here represents only a first attempt at

predicting the potential consequences of the scoring system

change, and possible impact on other aspects of the game

need to be examined as well for a more complete

assessment.

This work was supported by the Hongik University

Research Fund.

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