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1 Swinburne Research Bank Fair measures of performance: the World Cup of Cricket. S R Clarke and P Allsopp. Journal of the Operational Research Society 52(4) 2001: pp (9) 2001 Operational Research Society Ltd. All rights reserved. This is the author s version of the work. It is posted here with permission of the publisher for your personal use. No further distribution is permitted. If your Library has a subscription to this journal, you may also be able to access the published version via the library catalogue.

2 Stephen R. Clarke and Paul Allsopp Swinburne University of Technology Luck can play a big part in tournament success, and progress is not necessarily the best measure of performance. A Linear model is used to fit least squares ratings to margins of victory in the World Cup of cricket. The Duckworth/Lewis rain interruption rules are applied to second innings victories to create a margin of victory in runs, equivalent to margins of victory for the team batting first. Results show that while the better teams progressed through the first round of the competition, some injustices occurred in the super six round. This appears to be due to the double counting of selected matches. Ordering teams by average margin of victory gives similar results to the more complicated linear model, and its use as a tie breaker is suggested. Publication of the margin of victory as estimated by the Duckworth Lewis method for second innings victories in all one day matches would provide a common margin of victory suitable for analysis. Keywords: sports, cricket, linear models, rain interruption Introduction While five-day Test Cricket has been played for over a century, because of constraints associated with time, location and the standardising of conditions, no method currently exists to determine a world champion of Test Cricket. The nine test match countries play irregular series of between one and six test matches. Wisden 1 publishes a world ranking based on two points for a series win and one point for a draw, but the system does not take into account home advantage or margin of victory. In practice, a country has to dominate test cricket for many years before they are generally recognised as the World champions. The West Indies were probably the last side to achieve this by remaining unbeaten in a test series from 1981 to The first one-day game of cricket was played at the international level in 1971, and it has quickly become the most popular form of the game. Its short duration allows for traditional round robin or knockout tournaments between several countries. First played in 1975, the World Cup of cricket is now played every four years, with competing teams playing the one-day version of the game. It allows all teams to compete in the same time frame and in similar locations and conditions, and is now arguably the single most prestigious event in cricket, the winners being recognised as World champions for the next four years. The 1999 World Cup was contested by 12 teams comprising the nine test match playing countries plus Scotland, Kenya and Bangladesh. These were seeded and divided into two groups. Group A comprised Sri Lanka, India, South Africa, England, Zimbabwe and Kenya. Group B comprised Australia, West Indies, Pakistan, New Zealand, Bangladesh and Scotland. The first phase of the tournament was the round robin Group Matches. Two points were allocated for each win, one point for a tie or no result and no points for a loss. The top three teams from Group A (South Africa, India and Zimbabwe) and group B (Pakistan, Australia and New Zealand) progressed to the second phase of the tournament,

3 Table 1 Results of the 1999 World Cup First Innings Second Innings Match Type Game Team Wickets Overs Runs Team Wickets Overs Runs Adjusted Runs Group-A Margin of victory 1 Sri Lanka ENGLAND* India* SOUTH AFRICA Kenya ZIMBABWE* Kenya ENGLAND* ZIMBABWE India* SOUTH AFRICA Sri Lanka* SOUTH AFRICA England* Zimbabwe SRI LANKA* INDIA Kenya* Zimbabwe ENGLAND* Kenya SOUTH AFRICA* INDIA Sri Lanka* INDIA England* ZIMBABWE* South Africa SRI LANKA Kenya* Group-B 16 Scotland AUSTRALIA* PAKISTAN* West Indies Bangladesh NEW ZEALAND* Australia* NEW ZEALAND PAKISTAN Scotland* Bangladesh* WEST INDIES PAKISTAN Australia* BANGLADESH Scotland* New Zealand WEST INDIES* Bangladesh AUSTRALIA* Scotland* WEST INDIES PAKISTAN New Zealand* West Indies AUSTRALIA* BANGLADESH Pakistan* Scotland NEW ZEALAND*

4 Table 1 (Continued) Results of the 1999 World Cup First Innings Second Innings Match Type Game Team Wickets Overs Runs Team Wickets Overs Runs Adjusted Runs Super-Six Margin of victory 31 AUSTRALIA India* Pakistan* SOUTH AFRICA Zimbabwe* New Zealand INDIA* Pakistan AUSTRALIA Zimbabwe* SOUTH AFRICA* New Zealand PAKISTAN* Zimbabwe India* NEW ZEALAND South Africa* AUSTRALIA New Zealand* PAKISTAN Semifinals 41 Australia South Africa* Final 42 Pakistan* AUSTRALIA Average Standard Deviation Note: 1) Winning team in upper case, *: denotes team winning the toss 2) Game 33 classified a No result because rain had stopped play 3) Game 41 Australia advanced to final on superior run rate. 4) Adjusted runs is the expected score, according to the D/L rain interruption tables, if the innings of the team batting second was completed 3

5 the Super-Six,where each of the three teams from Group A played the three teams from Group B. Each team in the Super-Six also carried forward any points scored against the other qualifying teams in their group matches, which ensured the Super-Six round was effectively a round robin between the six teams. The top four teams at the end of the Super-Six phase were Pakistan, Australia, South Africa and New Zealand. These teams played a seeded knockout tournament, which ultimately saw Australia defeat Pakistan in the final. Table 1 gives a listing of all match results, with the winner shown in upper case. This is a seemingly ad hoc set of 42 matches, and one could ask many questions about its structure. Why are 30 matches invested in finding the best six out of 12 teams, some of which are clearly not in the same class as the best? Another nine matches are needed to eliminate a further two teams, yet only three matches invested in reducing the top four to a single winner. In order to give the greatest chance to the better teams, fewer matches would be used in the early stages when teams are widely different in ability, and more matches in the later stages when teams would be expected to be more equally matched. Doubtless many other considerations come into play, such as giving all teams a minimum number of matches and giving the weaker countries experience against the stronger teams. Did this structure compromise what should be the primary goal of crowning the best side as World Champions? In any tournament, there is a set of rules under which the tournament operates, and a team can do no more than win under the prescribed conditions. In one sense it can be argued that provided all teams know the rules at the outset, and there is no obvious bias against particular teams, the competition is fair and the winner is the best team. On the other hand, it should be recognised that there is a large element of luck in cricket, particularly in the one-day form. Essentially random events can decide the winners of close games. In one semi-final Australia tied with South Africa. In any objective analysis, this result tells you that on the day, the two teams were equal. However under the rules, Australia advanced and South Africa was eliminated. In addition, quirks of the rules can produce effects that were unintended and results unfair to some teams. Two examples will illustrate. New Zealand and Zimbabwe played in the only game abandoned due to rain. As a result, no team was declared the winner and each team received a point. This advantaged Zimbabwe who was in the weaker position at the time the match was called off. However it also made Australia's task of getting to the semi-finals much harder, as both teams it was competing against received a point, ensuring Australia had to win its final Super-Six match. Secondly, the structure of the tournament meant that some games were double-counted. Had South Africa defeated Australia in the last Super-Six match, Australia would have been eliminated after finishing second in their Group and winning two out of their three Super-Six games for five wins out of eight matches overall. On the other hand, Zimbabwe, who only finished third in their Group and won none of their Super-Six rounds would have progressed to the final four with three wins and one draw out of eight matches. This arose because Zimbabwe's wins against South Africa and India in the Group matches were counted again in the Super-Six, whereas Australia's losses to New Zealand and Pakistan were counted twice. Most unbiased observers would recognise this as unjust, and a case of the weaker performing team advancing. 4

6 In this paper we will use linear modeling techniques to rate the teams. To reward teams for strength of victory we will use the winning margin as the dependant variable, rather than a 1-0 win loss variable. This requires rationalising the two forms of winning margin currently used in cricket. We do this using the Duckworth/Lewis (D/L) rain interruption rules. Winning margin using the Duckworth/Lewis rain interruption rules. In test cricket different margins of victory are quoted depending on whether the team batting last wins or loses. Although inappropriate, these are also used in one-day cricket. In a oneday cricket match the team batting first is declared the winner if they restrict the team batting second to a score less than that achieved in the first innings. The margin of victory is recorded in runs as the difference between the two scores. If the team batting second pass the first innings score, they are declared the winner and the margin of victory is recorded as the number of wickets in hand, irrespective of the number of balls left. The quoted margin gives only part of the story, and can be quite misleading as to the superiority of the victory. For example, in the Super-Six match between Australia and South Africa, Australia responded to South Africa s first innings score of 271 with a score of 272 for the loss of five wickets. Australia was declared the winner by five wickets. While this appears to be a relatively convincing win, Australia hit the winning runs with only two balls to spare, and so might be thought to be lucky to win one of the closest matches of the tournament. By contrast, in the England versus Zimbabwe Group A match, in reply to Zimbabwe's 167 runs, England scored 168 runs off overs for the loss of only three wickets. This seven wicket victory was a thorough thrashing, as England had overs in hand left to score many more runs. Clarke 2 suggested the D/L rain interruption rules could be used to provide a better method of declaring the winning margin in one-day matches. In one-day cricket, rain interruptions can occur at various stages. Various unscientific methods have been used in the past to adjust targets, but the statistically based method proposed by Duckworth and Lewis 3-6 is now almost universally accepted, and under World Cup rules was the method to be used in this tournament. (However, because an extra day was set aside to complete interrupted matches, it was not actually invoked during the tournament). Developed from past statistics, the method relies on formula based tables which show the available resources (balls to go and wickets in hand) a team has remaining at any stage of the game. Table 1 shows that all first innings teams used up all their resources, having either batted for 50 overs or lost 10 wickets. When applied to a rain interruption, the D/L method calculates new target scores depending on the amount of resources the team lost during the interruption. In an interruption to the first innings, the method can project the score to that which would have been achieved in a full fifty overs. In the case of a rain interruption to the second innings, it gives the par score, the score the team batting second would need to be in order to achieve the first innings score (i.e. the score necessary to win the match if rain stops play at that point). Unless they score the winning runs off the last available ball, winning second innings teams have some resources left when they pass the first innings score. Thus for our purposes, the D/L method can be used to project the score of the second innings at the time they passed the first innings score, to the score they could be expected to achieve if they completed the 50 overs. This projected score, and the resultant margin of victory, is greater the more wickets and overs a second innings team has in hand. The margin of victory for second innings can thus be stated as a number of runs, as with first innings wins. 5

7 In the above two matches, the D/L model predicts that Australia would have scored 275 runs off 50 overs and so, in effect, defeated South Africa by only 4 runs. On the other hand the D/L model predicts that England were on target to score 247 runs. Their margin of victory becomes 80 runs, and gives a true reflection of how dominant they were in winning the match. In abandoned matches, the margin of victory is recorded in runs as how much the score at the point of victory exceeds the par score, and Duckworth and Lewis 5 suggest extending this for all second innings wins. In many one-day matches the par score is now displayed on the scoreboard, so this suggested margin of victory is easily calculated. However we believe our measure is more appropriate for analysis, as it better represents the degree of superiority, and better equates first and second innings victories. For example, in the unlikely event a team dismisses their opponents in the first innings for only 20 runs, when they pass that target they must of necessity be less than 20 runs ahead of their par score. The comprehensive victory would only have a small margin of victory if the excess over par score method is used. Had the team batted first and dismissed their opponent for only 20 in the second innings, the margin of victory would probably be in excess of 150 runs. To be fair to teams batting second, we must allow for the runs they would have scored had their innings not been prematurely terminated. Here we apply to the truncated second innings the same method D/L use to project a first innings score prematurely terminated by rain. Allsopp and Clarke 7 explore this further by comparing the margins of victory given by different methods. The final two columns of Table 1 give the second innings scores projected using the D/L tables, and the resultant margin of victory of the team batting first. A negative margin indicates the team batting second won. The No result match, New Zealand versus Zimbabwe has also been included. Even with the extra day, the second innings of this match did not pass the 25 over mark as required under the rules of the competition to use the D/L method for determining the winner. However under normal one-day match regulations, the D/L method is considered valid provided 15 second innings overs having been bowled before the rain interruption, so this match just qualifies. The second innings are on average 6.5 overs shorter than the first innings, and this results in fewer wickets and runs. However the rate at which wickets fall and runs scored was remarkably similar: 0.16 wickets and 4.5 runs per over in the first innings and 0.15 wickets and 4.5 runs per over in the second innings. Note that the average of the second innings scores, when adjusted by the D/L method, is only two more than the average first innings score. Moreover, the mean margin of victory when the first team wins is 68, whereas when the second team wins it is 67. This demonstrates the suggested method of attributing a margin of victory in runs is fair to the team batting second. On the other hand, using the par score method would result in a mean margin of victory when the second team wins of only 46. This method would clearly disadvantage teams batting second. 6

8 Analysis of the Group Matches phase We model the margin of victory wij of team i batting first against team j batting second as wij = ui + h - uj + εij (1) where ui is a measure of the ability of team i, h is the advantage (or disadvantage, if negative) of the team batting first and εij is a zero mean random error. Since the ui are relative, we require an additional constraint, and here we choose they average 100. Various authors have applied similar models to predict sporting results, where h becomes a home field advantage In traditional cricket it is generally an advantage to bat first, since a pitch can deteriorate over the five days of a test match, and this is usually the choice of the captain winning the toss. This is not the case in one-day cricket. Based on a Dynamic Programming model of one-day cricket, Clarke 13 suggests bowling first is an advantage, but no statistical study has shown this to be the case in practice. In this World Cup, the winner of the toss elected to bowl 27 times and bat only 15 times. Such a discrepancy may be due to perceived strategic advantages in the tournament as a whole, rather than improving the chance of winning that particular match. Table 1 shows that teams batting first won 19 matches as against 21 for those batting second, but there may be a tendency for strong teams to send weaker teams in to bat first. By allowing a parameter for batting first, its advantage or otherwise can be estimated. There are other parameters that could be included in the model. Winning the toss is considered a big advantage in traditional cricket, but is not so important in the shorter game. de Silva BM and Swartz 14 in their analysis of 427 one-day international cricket matches did not find any evidence to suggest that a side winning the toss gained a significant advantage. In this tournament the winner of the toss won only 17 of the 40 decided matches, and including it in the model produced only small insignificant negative effects. Since it has never been suggested that winning the toss is a disadvantage, we report no further on this aspect. de Silva BM and Swartz 14 used a logistic regression model on a 1-0 win/loss margin. However the margin of victory as defined above is a more sensitive measure of a team's superiority. For example, in the Group A matches, South Africa convincingly defeated England by 122 runs, whereas Zimbabwe defeated India by only three runs. Clearly the margin of victory provides a more accurate measure of the relative performances of each team. By defining indicator variables, PROC REG from SAS 6.12 can be used to fit model (1) to the margins of victory for Groups A and B. Since the group matches are independent, the indicator variable matrices are unconnected, and we need two restriction equations for the ratings. Since the stated aim of the seeding process was to provide two even groups, we restrict the average ratings of the teams in each group to equal 100. We also assume any advantage in batting first is common to the two groups. The resulting ratings are given in Table 2. The model as a whole proved to be significant, with a coefficient of determination 7

9 of 57.7%. The common first innings advantage throughout the preliminary rounds was 10 runs, not statistically significant (p = 0.513). Table 2 Ratings obtained by fitting a linear model to the margin of victory in the Group Matches Group-A Group-B Team Ratings Overall rank estimated by the model Group rank estimated by the model Actual tournament rank South Africa A1 A1 India A2 A2 Zimbabwe 96 8 A4 A3 England A3 A4 Sri Lanka 68 9 A5 A5 Kenya A6 A6 Pakistan B4 B1 Australia B1 B2 New Zealand B3 B3 West Indies B2 B4 Bangladesh B5 B5 Scotland B6 B6 The analysis suggests that in the group matches phase of the tournament both England and the West Indies were unlucky to be eliminated. In Group A, England are rated a solid 14 runs ahead of Zimbabwe. Zimbabwe's three run win over India was the narrowest victory in the whole tournament and could easily have been a loss. Had head to head result been used to decide which of the two advanced to the Super-Six, England's 80 run victory over Zimbabwe would have carried them through. However the multiple tie for third place in the group meant that run rate was used and England was eliminated. In group B, the West Indies are rated marginally ahead of New Zealand and 15 runs ahead of Pakistan. Although here only ranked fourth in the Group, Pakistan ended up finishing first in group B. Possible reasons for the low rating of Pakistan are explored below. However, assuming each group is equal, the ratings also produce a ranking independent of group. This suggests four teams from Group B should have advanced, and only two from Group A. This would see the West Indies as the replacement for Zimbabwe in the Super-Six teams rather than England. There were two preliminary matches in Group B that warrant further examination. When the residuals to the model as fitted to the group matches are examined, one match has astudent t residual over 2.0. This is match 29, between Bangladesh and Pakistan, where Bangladesh beat the previously unbeaten Pakistan by 62 runs. Furthermore when a Cook's D statistic is calculated, this match has the highest influence of When this second last match of Group B occurred, Pakistan was assured of top place in Group A by virtue of its unbeaten record, and Bangladesh was guaranteed to finish fifth in the group. Thus the match was immaterial to any placings in the World Cup, and only team pride was at stake. There seem good reasons why the match result was not a true reflection of the team abilities and should 8

10 be ignored. In the Australia versus West Indies match it was reported in The Age (1/6/99) that Australia blatantly manipulated the match and purposely adopted a go-slow approach to ensure that the West Indies, rather than New Zealand qualified for the Super-Six. This would have advantaged Australia since they would have carried the two points for beating the West Indies forward into the Super-Six. The go-slow approach began in the 28 th over of the run chase when Australia were four wickets for 92 runs and needed only 19 runs to win. Using the D/L model to determine what Australia would have achieved if they had not adopted a go-slow approach and assuming there were 22 overs to be bowled we arrive at a score of 201 runs. If we regenerate the regression model with the adjustment to the margin of victory in the Australia versus West Indies match and the omission of the Pakistan versus Bangladesh match we arrive at the ratings displayed in Table 3. Table 3 Team ratings obtained after Group Matches with two results adjusted Group-A Group-B Team Ratings Overall rank estimated by the model Group rank estimated by the model Actual tournament rank India A1 A1 South Africa A2 A2 Zimbabwe A3 A3 England 99 8 A4 A4 Sri Lanka 66 9 A5 A5 Kenya A6 A6 Pakistan B1 B1 Australia B2 B2 New Zealand B3 B3 West Indies B4 B4 Bangladesh B5 B5 Scotland B6 B6 As expected, the ratings of Pakistan Bangladesh altered dramatically, and the West Indies dropped about 10 points. Curiously, Australia's rating dropped by a couple of points. Furthermore, some teams in Group A changed considerably, even though the adjusted matches occured in the other group. In particular England's rating dropped by about 10 points, and are now rated behind Zimbabwe. These apparent anomalies occur due to the connection of the group results through the common parameter for batting first. The two changes both reduced the advantage experienced by teams batting first, and in the new fit this is now estimated at a disadvantage of two runs. Thus the ratings of teams which mainly batted second (England all five matches, Australia four out of five) will also be reduced, as they no longer have such a big handicap to overcome. Significantly, the within group rankings produced from the ratings in Table 3 and the actual ratings from the tournament rules are now equivalent. This suggests that the deserved teams moved through to the next round of the tournament, and the points system adopted by the organisers adequately reflected the overall relative abilities of the teams. However if we remove the restriction that three teams from each group must advance, the West Indies replaces Zimbabwe in the Super- Six round. 9

11 Super-Six and finals matches Before the Super-Six phase of the tournament we have two groups operating independently of each other. However, the matches between teams from different groups provide connectivity during the Super-Six phase, and the requirement of an average rating of 100 for each group translates into a single restriction for the whole group. Subsequently fitting a regression model to the margins of victory and rating relative ability we arrive at the results outlined in Table 4. All teams and matches up to this stage have been included in the analysis. As the Australian go-slow was well documented in the press and admitted by the Australian captain, the result of the Australia - West Indies match has been adjusted as outlined above. As reasons for Pakistan's poor showing against Bangladesh is conjecture, this result has been allowed to stand. However, it could be argued that in the following analysis Pakistan's true ability is somewhat higher than the actual rating. Note that as more matches are included, the effects of any single match are reduced. Team Table 4 Ratings obtained after completion of the Super-Six Matches Team rating in rank order Overall rank estimated by the model Actual rank after Super- Six Australia South Africa West Indies New Zealand Pakistan India England Zimbabwe Bangladesh Sri Lanka Kenya Scotland The tournament results and the model are in fairly close agreement, with three exceptions. The model rates Pakistan and Zimbabwe much lower than the tournament, and the West Indies higher. However of the six teams that were still left in the competition at this stage, the model selects the same four teams to progress as the tournament rules, although the model would have seeded Australia and South Africa to meet in the final. Before the tournament began there was some discussion that although the favourites Australia and South Africa were in different groups, Group A was stronger than Group B. This may have been due to the presence in Group B of two non-test playing countries, and the presence of current World champions Sri Lanka in Group A. As it turned out, Sri Lanka performed badly, and three of the semi finalists and both finalists came from Group B. If we sum the ratings after completion of the Super-Six, the Group-A and B totals are 543 and 658 respectively, with the mean difference between the groups being 19 rating points. This suggests that the Group-B teams were able to produce better performances in the tournament. 10

12 In the group matches, the mean first innings scores for Groups A and B were 235 and 185 runs respectively (p = 0.03), which certainly suggests the Groups were different. However it is difficult to attribute this to any one cause. In the Group A matches, the bowlers obtained a wicket every 6.5 overs at 4.7 runs per overs, while in Group B a wicket fell every 5.4 overs for only 3.9 runs per overs. Either Group A had the better batsmen or group B had the better bowlers. There is a school of thought in test cricket that bowlers win matches, and the fact that Group B teams were more successful suggests this might also be true in one-day cricket. It might be possible to pursue this by using a model that incorporates both attacking and defensive components for each team. The semi-finals saw Pakistan defeat New Zealand, and Australia tie with South Africa. Australia then convincingly defeated Pakistan in the final. Incorporating these results gives the ratings shown in Table 5. Overall, the team batting first gained an advantage of 9 runs, which is not statistically significant (p = 0.48). This causes little change from the previous table, the major change being that Pakistan move further down the table due to their poor final performance. While the tournament has ultimately given the World Title to the team the model rates the best, only two of the top four teams made the semi finals, and the second best team failed to reach the final. Many of the differences in the two results can be traced back to the carrying forward of match results from the group matches to the Super-Six rounds. The two teams most disadvantaged by this were India and Australia, who carried no points through (or alternatively, their only two losses were counted twice). While Australia overcame this handicap, they were on the point of elimination throughout the super-six rounds. The teams most advantaged were Pakistan and Zimbabwe, who both carried two wins through to the Super-Six. During the Super-Six Zimbabwe had no wins, and Pakistan's only win was over Zimbabwe. Team Table 5 Team ratings after completion of all matches Number of games played Team rating Mean margin of victory Rank order by the model Actual final rank Australia South Africa West Indies India New Zealand Pakistan England Zimbabwe Bangladesh Sri Lanka Kenya Scotland

13 Simpler methods Although Model 1 correctly allows for quality of opponent in rating performance, in many tournaments this averages out. In these cases an average margin of victory may be an acceptable measure of rating teams. Table 5 also gives the average margins of victory for each team. While we would expect the better sides to generally play higher quality opposition (through their participation in the Super-six and finals) the subsequent rankings are in exact agreement with those given by the linear model. This suggests cricket followers could arrive at reasonable conclusions by working with the average (unsigned) margin of victory, and that margin of victory could be used as a tie breaker. For example, the average margin of victory for group A matches was 66, Group B 77, the Super Six and finals 53. This shows that Group B matches were more one sided (the teams more variable) and that as expected matches became closer as the weaker sides were eliminated. In the Group A matches, India, England and Zimbabwe were tied on 6 points each at the completion of the preliminary rounds. The average margin of victory for the teams was 57.8, 1.2 and 0.4 respectively. On this measure India was clearly the best side, but a better run rate saw Zimbabwe go through in the actual tournament. In Group B, the three way tie would have seen either the West Indies or New Zealand eliminated, depending on whether the adjustment for Australia s slow batting is made or not. Margin of victory is a natural measure, similar to goal difference used in soccer, and is easy to manipulate. However its use requires the official bodies to publish the calculated D/L figure in second innings victories. Conclusions It is apparent the D/L method is useful not just for dealing with rain interrupted matches but for predicting a second innings team s score when they have won a match in less than the fifty allotted overs and thus have both wickets and overs in hand. In particular, this could be used to give a margin of victory in runs that is equivalent to the quoted margin for a first innings victory. Were this figure quoted regularly, fans could better judge the winners superiority. When investigating topics such as cricket ratings, home advantage, advantage of winning the toss, etc it provides a more sensitive measure than win or loss. We have demonstrated here its application by fitting a least squares model to the resultant margins. The model passes the necessary tests. For example, an Anderson-Darling Test for the normality of the residuals for the model fitted to all the data (Table 5) gave p = Moreover, the ratings obtained seem reasonable, and the fit correctly predicts 74% of the individual match results. Nevertheless there is room for possible improvements. A model which includes batting and bowling ratings could be investigated. We also assume team ratings are constant throughout the tournament. Models or methods which allow ratings to alter might be tried. While it is not suggested this method be adopted in order to determine the winner in a oneday tournament, it can assist in gaining insight into how well a team has performed, and to what degree their final placing depends on luck. It might well be used by national administrators or coaches to measure progress. 12

14 While the method demonstrated would probably be too much of a 'black box' for cricket administrators, reasonable results might be obtained by using average margin of victory, a concept easily understood by cricket followers. While such a method downgrades the importance of winning, it would always encourage attacking play by both teams, as they are rewarded for runs and wickets. To maximise their rating, a team would always have to attempt to win a match by as large a margin as possible. One-day cricket is designed to be entertaining, and methods that encourage bright cricket are to be encouraged. While probably too radical a measure to be used alone, the use of average margin of victory should at least be investigated for breaking ties on the number of wins. It is a simple statistic that is easily calculated and understood provided the relevant margin of victory is published for each match. In a tournaments such as this it is important that factors completely out of the participants control do not have a great effect on the outcome. There was no evidence to suggest the team winning the toss had an advantage. While the majority of teams winning the toss elected to bowl first, the estimated advantage in batting first was a statistically insignificant 9 runs. One day tournament organisers can take solace from the fact that no real advantage appears to come to the team that wins the toss or bats first. While the format adopted by the World Cup organisers was successful in crowning the top team, there were some anomalies in the final rankings. South Africa was unfortunate not to be in the final, Pakistan and Zimbabwe were ranked too high, and the West Indies and India too low. Some of these injustices can be traced to the practice of carrying some points from the Group Matches through to the Super-Six round. Others are due to a culture that doesn t reward a loss, no matter how meritorious and the random occurrences that happen in any sporting contest. A dropped catch taken here, a different umpiring decision there, and two other teams could have been competing in the final. While these contribute to the excitement of sport, they should not deter us from recognising that the rules chosen by tournament organisers are not the only means of measuring achievement. Other more objective and fairer measures of achievement may be appropriate. References 1. Engel M (ed) (1997). Wisden Cricketers' Almanack. Vol. 134 John Wisden: Guilford. 2. Clarke SR (1998). Test Statistics. In Bennett J (ed). Statistics in Sport. Arnold: London, pp Duckworth F and Lewis T (1998). A fair method for resetting the target in interrupted one-day cricket matches. J Opl Res Soc 49: Duckworth F and Lewis T (1996). A fair method for resetting the target in interrupted one-day cricket matches. In: de Mestre N (ed). Mathematics and Computers in Sport. Bond University: Gold Coast, Qld. pp Duckworth F and Lewis T (1999). Your comprehensive guide to the Duckworth/Lewis method for resetting targets in one-day cricket: University of the West of England: Bristol. 13

15 6 Lewis T and Duckworth F (1998). Developments in the Duckworth-Lewis method of target resetting in one-day cricket matches. In: de Mestre N and Kumar K, (eds).fourth Conference on Mathematics and Computers in Sport. Bond University: Gold Coast, Qld. pp Allsopp P and Clarke SR (1999). Measuring the performance of teams in one-day cricket, In Bulmer M, McCrae B and Stacey K, (ed). Proceedings of the Mathematics 2000 Festival. Australian Association of Mathematics Teachers: Melbourne, pp Stefani RT (1980). Improved least squares football, basketball and soccer predictions. IEEE Trans Systems, Man and Cybernetics 10: Harville DA (1980). Predictions for National Football League games via linear-model methodology. J Am Stat Ass 75: Stefani RT and Clarke SR (1992). Predictions and home advantage for Australian rules football. J Appl Stats 19: Harville DA and Smith MH (1994). The home-court advantage: How large is it, and does it vary from team to team? The Am Statistician. 48: Clarke SR and Norman JM (1995). Home ground advantage of individual clubs in English soccer. The Statistician 44: Clarke SR (1988). Dynamic Programming in one day cricket - Optimal scoring rates. J Opl Res Soc 39: de Silva BM and Swartz TB (1997). Winning the coin toss and the home team advantage in one-day international cricket matches. The New Zealand Statistician. 32 (2):