Winning the Coin Toss and the Home Team Advantage in

y International Cricket Matches OneDa



Basil M de Silva and Tim B Swartz

Abstract

This pap er provides a statistical analysis of oneday international cricket matches played

during the s Two general conclusions are obtained contrary to widespread opinion

winning the coin toss at the outset of a match provides no comp etitive advantage and the

advantage of playing on ones home eld increases the logo dds of the probability of winning

by approximately

Keywords ODI cricket matches statistical mo delling



B de Silva is Senior Lecturer Department of Statistics and Op erations Research RMIT University GPO Box

V Melb ourne Victoria Australia T Swartz is Asso ciate Professor Department of Mathematics and

Statistics Simon F raser University Burnaby BC Canada VAS T Swartz was partially supp orted by a grant

from the Natural Sciences and Engineering Research Council of Canada The authors wish to thank all referees

for insightful comments that led to an improved pap er

INTRODUCTION

The game of cricket has worldwide app eal and is enjoyed by p eople of all ages Despite this

fact and despite the intense b o okkeeping and scrutin y that the game has received there has not

b een an overwhelming amount written on the sp ort from a statistical p ersp ective Hop efully a

we and Middledorp Kumar Gane close to complete list of pap ers includes Cro

salingam Kumar and Ganeshanandam Kimb er Clarke Danaher

Chapman Burrows and Talb ot Croucher Sumner and Mobley

P ollard Pollard Benjamin and Reep Elderton and Wo o d

In this pap er we consider a statistical analysis of two issues asso ciated with oneday cricket

matches In rstclass cricket or more days winning the coin toss is undoubtedly regarded

as a great advantage Since the oneday match is the mo dern ospring of the rstclass game it

might casually b e assumed that winning the coin toss in the oneday game is also an advantage

We investigate this b elief We also investigate the eect of the hometeam advantage in oneday

matches Both of these topics are controversial and we b elieve that data can shed some light on

their resolution

Our data consists of oneday international ODI matches involving games b etween the

nations b elonging to the International Cricket Council ICC These games represent the

game of cricket played at the highest level with relative stability amongst the teams We have

up until the Asia Cup concluding collected data on the matches played during the s

in July This time p erio d captures the mo dern game of cricket where the rules have

b een relatively uniform It is also the case that recent data is more extensive and reliable To

keep strategies constant we have limited the data to full over matches and have ignored

matches decided by run rates The data was collected from the comprehensive CricInfo web

page httpwwwcric ketorg

In Section we lo ok at data involving a teams choice to bat rst up on winning the coin

toss at the outset of a match We come to the surprising conclusion that winning the coin toss

of the existence of provides no comp etitive advantage In Section we consider the question

a home team advantage We conclude that such an advantage do es exist and we quantify the

advantage In Section we provide some concluding remarks

BATTING ORDER STRATEGIES

At the b eginning of a match a coin is tossed and the team that wins the toss is granted

the choice of batting rst or second Some p eople b elieve that a team should bat rst establish

a numb er of runs and pro duce a psychological hurdle for the second team to overcome Other

p eople b elieve that there is an advantage in batting second as this team knows what score its

opp onent has pro duced This additional information allows the team batting second to adjust

their strategy accordingly Still others feel that the choice b etween batting rst or second should

dep end on auxilliary and sub jective variables such as the weather the pitch ie eld conditions

the teams health the teams morale the opp onent whether the team will bat in daylight or

under o o dlights etc Clearly this is a topic of considerable interest

As a preliminary study Table provides summary data on the ODI matches involving

the ICC nations We see from column B that there is great disparity amongst the various

1

teams with resp ect to their decision to bat either rst or second F or example up on winning

the coin toss Australia cho oses to bat rst of the time whereas Sri Lanka cho oses to bat

rst only of the time

oposed strategies However in most cases such Ideally we would like to compare various pr

strategies would dep end on variables eg team morale coaching whims pitch conditions that

can not b e easily extracted from past data Another dicult y in comparing prop osed strategies is

that matches have not b een randomized according to these strategies Therefore p erceived eects

due to prop osed strategies may actually b e attributed to an unknown confounding variable For

Zealand should always bat rst Even example consider the very simple strategy that New

though New Zealand has chosen to bat rst in roughly half of its matches it may b e the case

that they did so only in matches that they thought they should win In this case batting rst

would b e confounded with games involving weaker opp osition

For these reasons we only investigate observe d strategies of the ICC teams In this context

we would like to compare a teams strategy against its opp osite strategy However this to o can

not b e tested The reason is that data on the opp osite strategy is unavailable since we do no

know what strategy is b eing incorp orated when a team loses the coin toss Therefore we are

left with comparing a teams strategy versus its opp onents opp osite strategies More precisely

we lo ok at a teams p erformance when it wins the coin toss versus its p erformance when it loses

the coin toss

i Consider the data x y n where n is the numb er of games played by the team

i i

th

of interest x if the team winsloses the coin toss in the i game and y if

i i

the team winsloses the game We have the statistical mo del y j and x Bernoulli p

i i i

y j x Bernoulli q where P x P x for i n We are therefore

i i i i i

P P

n n

q We note that if the p versus the strategy interested in comparing the strategy

i i

i=1 i=1

P P

n n

opp onents of a team collectively have the teams opp osite strategy then p q Of

i i

i=1 i=1

the matches in the data set games resulted in ties We exclude these matches from the

analysis

The diculty with the ab ove mo del is that we have n parameters ie p s q s and only

i i

P

n

n observed resp onses ie y s We therefore consider several metho ds to compare p with

i i

i=1

P

n

q The rst metho d requires the assumption that p p and q q for all i n

i i i

i=1

This simplistic assumption is clearly unrealistic as it assumes that the opp onents are all of equal

strength and that the playing conditions are constant over time However it is a go o d starting

p oint and the test of H p q versus H p q is easily carried out using a twosample

0 1

Binomial test The rst binomial variable is the numb er of wins having won the coin toss and

the second binomial variable is the numb er of wins having lost the coin toss The pvalues for

to b e each of the ICC nations are given in column of Table These pvalues are very high

signicant pvalues must b e small typically Therefore under this mo del we observe

no evidence of successful strategies for any of the ICC teams

Our second metho d allows for opp onents of varying strengths However this approach

is also unrealistic since it assumes that teams have constant strength over time We mo del

logitp where x if the ith game is against opp onent k k and

i i i k i

P P P P P

8 n n n n

q p q versus H p The pvalues for testing H

i i i 1 i k 0

i=1 i=1 i=1 i=1 k =1

for each of the ICC nations are based on logistic regression and are given in column of Table

Venables The pvalues were obtained using the function glm binomial family in SPLUS

and Ripley Based on tests at level signicance none of the teams exhibit eective

strategies The team which comes closest to signicance is India pvalue We note that

India cho oses to bat rst in approximately half of its matches

Our third metho d is based on a Bayesian approach Allowing exibility for the strengths of

the dierent teams and allowing for varying strengths over time we express the likeliho o d as

n

Y

y (1x )

x y

y )x i i (1y )(1x ) (1

i i

i i i i

q q y x j p q p p

i i

i i

i=1

Intro ducing indep endent priors p Betaa b and q Betac d the Bayesian paradigm

i i i i i i

immediately gives indep endent marginal p osterior distributions p j x y Betaa y x b

i i i i i

y x and q j x y Betac y x d y x The p osterior probability of

i i i i i i i i i

P P

n n

H is then P H j x y P p q j x y which is obtained by simulating from the

0 0 i i

i=1 i=1

P P

n n

marginal p osterior distributions and calculating the prop ortion of time that p q

i i

i=1 i=1

The tradeo for the exibility of the Bayesian mo del is the need to sp ecify prior information We

consider Beta priors ie Uniform priors which are standard default priors and are

P P

n n

q reduce p and often used to represent ignorance Here the p osterior distributions of

i i

i=1 i=1

to the sums of indep endent uniform and triangular distributions Column of Table gives

the p osterior probabilities for each of the ICC teams Again we see no evidence of successful

strategies

As a fourth metho d it was suggested by a referee that we try an empirical Bayes approach

where the prior parameters a b c d i n are determined by the data Using the

i i i i

observed winning prop ortions W from Table we set W a a b i n where

o o i i i

th

a a b is the mean prior probability of winning the i match given a winning coin toss

i i i

Similarly we set W c c d i n It is not so obvious how one might determine the

o i i i

2

second parameter in the prior Beta distributions We decided on setting min W W

o o

equal to the prior variances We reasoned that standard deviations would bring us to the

closest endp oint ie either or and would therefore provide prior distributions that ought to

capture the true parameter values Column of Table gives the p osterior probabilities based

on this empirical Bayes approach Here there is even less evidence that any of the observed

strategies are successful The p osterior probabilities using metho ds and were obtained using

a Fortran program The IMSL subroutine rnb et was used to generate the Beta variates

In summary under all metho ds of analysis we have observed that the team strategies

oer no advantage in deciding the outcome of matches Although the sets of strategies do not

include all p ossible strategies we suggest that it would b e unlikely for someone to discover a

b etter strategy We therefore conclude that winning the coin toss has no impact on the outcome

of matches

THE HOME TEAM ADVANTAGE

The existence of a home team advantage in ODI cricket matches is generally accepted We

provide some statistical evidence for this p oint of view and then investigate whether the home

team advantage can b e quantied

In many physical sp orts eg American fo otball so ccer basketball it is well accepted that

a home team advantage exists see Stern and Pollard Perhaps in these physical

sp orts the supp ort of the home crowd is such that a physical lift is often provided which carries

the home team to victory For example in the season of the National Basketball

Asso ciation the home teams won games while losing games for a home team winning

p ercentage of

In sp orts that are not as physical a home team advantage may still exist due to familiarity

with lo cal conditions biased ociating etc The game of cricket may fall into this class It

is interesting to note from columns W and W of Table that every ICC nation has a higher

h o

winning p ercentage during home games Using the sign test this is convincing evidence of the

9

existence of a home team advantage ie pvalue

To investigate the eect of the home team advantage we mo dify our notation and let p

ij k

b e the probability that team i defeats team j at site k where i j k and in addition

where k denotes a nonICC site We intro duce the mo del logitp

i j ij k ij k

P

9

and

i

i=1

if team i is the home team

ij k

if the game is played on a neutral site

if team j is the home team

This is a parameter mo del where is a measure of the dierential strength of team i There

i

fore the oset represents the advantage in logo dds that team i has over team j The

i j

mo del also assumes that the home team advantage is constant over all ICC teams Note that

the logit transformation of p is natural in two resp ects Firstly for teams of equal strength

ij k

that play on a neutral site we have logit p which implies p Therefore there is

ij k ij k

no need for an intercept term Secondly it is sensible to quantify the home team advantage on

the logo dds scale since we should exp ect small relative improvements for strong teams that win

most of their games Conversely we should exp ect large relative improvements for weak teams

that lose most of their games

Again we use the function glm family binomial in SPLUS to carry out the logistic regres

with sion and again we exclude the tied games from the matches We obtain

standard error To put this quantity in p ersp ective a team with a winning p ercentage of

would increase its winning p ercentage to when playing at home Therefore the home

team advantage provides a considerable edge to the home team

We now consider a further mo dication to the mo delling pro cedure It is generally b elieved

that worldwide Pakistan and the West Indies are p opular teams This may b e due to their

amb oyant style West Indies their frequency of travel and their high success rates during the

late s and early s For this reason we consider these nations to have a home eld

advantage ie for matches actually played on neutral sites Similarly we consider Sri Lanka

and New Zealand to b e less p opular teams This may b e due to the fact that these nations have

small emmigrant p opulations worldwide We therefore assign a home team disadvantage ie

to Sri Lanka and New Zealand for matches played on neutral sites With these mo dications

we obtain a comparable with a slightly smaller standard error

CONCLUDING REMARKS

Given the increasing worldwide p opularity of sp ort and the increasing access to sp orts

related data it seems that serious statistical investigations of issues in sp ort are b ecoming more

common In the American Statistical Asso ciation formed the Section on Statistics in

Sp orts and have sp onsored b oth publications and sessions In the same year Bond University

Australia b egan biennial meetings on Mathematics and Computers in Sp ort with Statistics in

Sp ort as one of the stated themes We also note that since the statistics magazine Chanc e

has devoted considerable attention to sp orts related topics

In this pap er we have added to the growing interest in sp orts and statistics by considering

questions involving ODI cricket matches Surprisingly we have found that winning the coin toss

at the outset of a match provides no comp etitive advantage We have also estimated that the

logo dds of winning a match increases by approximately when a team plays at home This

second result may have implications for b etting

REFERENCES

Burrows BL and Talb ot RF Boycott Botham and the exp onential distribution

Teaching Statistics

Chapman B An analysis of the determinants of EnglishAustralian test cricket

crowds estimating the value of Don Bradman Technical Rep ort Centre for Economic

Policy Research Australian National University

Clarke SR Consistency in sp ort with particular reference to cricket NZOR Pro

ceedings th Annual Conference Victoria University of Wellington

Croucher JS AngloAustralian test cricket dismissals Bul letin of Applied

Statistics

Crowe SM and Middeldorp J A comparision of leg b efore wicket rates b etween

Australians and their visiting teams for test cricket series played in Australia

The Statistician

Danaher PJ Estimating a cricketers batting average using the pro duct limit esti

Zealand Statistician mator New

Elderton WP Cricket scores and some skew correlation distribution Journal of the

Royal Statistical Society Series A

Ganesalingam S Kumar K and Ganeshanandam S A statistical lo ok at cricket

data in Mathematics and Computers in Sport Bond University Australia

Kimb er A A graphical display for comparing b owlers in cricket Teaching Statistics

in Mathematics and Computers in Sport Kumar K Is cricket really by chance

Bond University Australia

Pollard R Cricket and statistics in Optimal Strategies in Sports SP Ladany and

RE Machol editors North Holland Amsterdam

Pollard R Home advantage in so ccer a retrosp ective analysis Journal of Sports

Science

Pollard R Bejamin B and Reep C Sp ort and the negative binomial distribution

in Optimal Strategies in Sports SP Ladany and RE Machol editors North Holland

Amsterdam

Sumner J and Mobley M Are cricket umpires biased New Scientist no

Stern HS A statistician reads the sp orts pages Chance

Venables WN and Ripley BD Modern Applied Statistics with Splus SpringerVerlag

Wo o d GH Cricket scores and geometrical progression Journal of the Royal Statis

tical Society Series A

Table Summary data for the ICC nations Here B is the prop ortion of time that a team

1

cho oses to bat rst up on winning the coin toss W is the overall winning prop ortion W is

o c

the winning prop ortion in games that a team has won the coin toss and W is the winning

h

prop ortion in games played on a home eld The quantities in parentheses are the numb er of

cases

Nation B W W W

1 o c h

Australia

England

India

New Zealand

Pakistan

South Africa

Sri Lanka

West Indies

Zimbabwe

Table Pvalues and p osterior probabilities used to assess the ecacy of strategies involving

the decision to bat rst Metho d requires that the probability of winning a match is constant

over opp onents and time metho d allows for the varying strengths of opp onents but requires

constant strength over time metho d is based on a Bayesian mo del with indep endent uniform

priors and metho d is based on an empirical Bayes approach

Nation Metho d Metho d Metho d Metho d

1 2 3 4

Australia

England

India

New Zealand

Pakistan

South Africa

Sri Lanka

West Indies

Zimbabwe