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