The Duckworth–Lewis method (often written as D/L method) is a mathematical formulation designed to calculate the target score for the team batting second in a limited overs cricket match interrupted by weather or other circumstances. It is generally accepted to be the most accurate method of setting a target score. The D/L method was devised by two English statisticians, Frank Duckworth and Tony Lewis.
The basic principle is that each team in a limited-overs match has two resources available with which to score runs: wickets remaining, and overs to play. Where overs are lost, setting an adjusted target is not as simple as to reduce the batting team's run target proportionally, because a team batting second with ten wickets in hand and 25 overs to play can be expected to play more aggressively than one with ten wickets and a full 50 overs, and can consequently achieve a higher run rate. The Duckworth–Lewis method is an attempt to set a statistically fair target for the second team's innings, based on the score achieved by the first team, taking their wickets lost and overs played into account.
- 1 Examples
- 2 Theory
- 3 Application
- 4 Duckworth−Lewis calculations
- 5 Example calculations for Duckworth−Lewis Standard Edition
- 6 History and creation
- 7 Updates
- 8 Criticism
- 9 Cultural influence
- 10 References
- 11 Further reading
- 12 External links
Stoppage in first innings
In the 4th India – England ODI in the 2008 series, the first innings was interrupted by rain on two occasions, resulting in the match being reduced to 22 overs a side. India (batting first) made 166/4. England's target was therefore set by the D/L method at 198 from 22 overs.
During the fifth ODI between India and South Africa in January 2011, rain halted play twice during the first innings. The match was reduced to 46 overs and South Africa scored 250/9. The D/L method was applied which adjusted the target to 268. As the number of overs was reduced in between South Africa's innings, this method takes into account what South Africa would have scored before the first interruption.
Both examples illustrate how the D/L method is applied. In the case of the first match, as England knew they had only 22 overs the expectation is that they will be able to score more runs from those overs than India had from their (interrupted) innings. England made 178/8 from 22 overs, and so the match was listed as "India won by 19 runs (D/L method)".
Stoppage in second innings
A simple example of the D/L method being applied was the first One Day International (ODI) between India and Pakistan in their 2006 ODI series. India batted first, and were all out in the 49th over for 328. Pakistan, batting second, were 7 wickets down for 311 when bad light stopped play after the 47th over.
In this example, Pakistan's target, had the match continued, was 18 runs in as many balls, with three wickets in hand. Considering the overall scoring rate throughout the match, this is a target most teams would be favoured to achieve. And indeed, application of the D/L method resulted in a target score of 304 at the end of the 47th over, with the officially listed result as "Pakistan won by 7 runs (D/L Method)".
Examples in T20
The D/L method was used in the group stage match between Sri Lanka and Zimbabwe at the 20/20 World Cup in 2010; Sri Lanka won the match by 14 runs according to the D/L method. Sri Lanka scored 140/7 in 20 overs batting first, and in reply Zimbabwe were 40/0 in 7 overs when rain interrupted play.
During the 2012/13 KFC Big Bash League, the D/L Method was used in the 2nd Semi Final played between the Melbourne Stars and the Perth Scorchers at the WACA Ground. Melbourne scored 183/2 from 18 overs after rain delayed the start of the match. Following a further rain delay, Perth returned to the field to face 13 overs, with a revised target of 139. Perth won the game by 8 wickets following a boundary off the final delivery.
The D/L method was used in the first semi-final match of 5th edition of ICC T20 WorldCup between Sri Lanka and West Indies, whereby Sri Lanka won by 27 runs. The match was stopped due to heavy rain and hailstorm.
The essence of the D/L method is 'resources'. Each team is taken to have two 'resources' to use to make as many runs as possible: the number of overs they have to receive; and the number of wickets they have in hand. At any point in any innings, a team's ability to score more runs depends on the combination of these two resources. Looking at historical scores, there is a very close correspondence between the availability of these resources and a team's final score, a correspondence which D/L exploits.
Using a published table or computer which gives the percentage of these combined resources remaining for any number of overs (or, more accurately, balls) left and wickets lost, the target score can be adjusted up or down to reflect the loss of resources to one or both teams when a match is shortened one or more times. This percentage is then used to calculate a target (sometimes called a 'par score') that is usually a fractional number of runs. If the second team passes the target, then the second team is taken to have won the match; if the match ends when the second team has exactly met (but not passed) the target (rounded down to the next integer) then the match is taken to be a tie.
An example of such a tie was found in the one day international between England and India on 11 September 2011. This match was frequently interrupted by rain in the final overs, and a ball-by-ball calculation of the Duckworth-Lewis 'par' score played a key role in the tactical decisions made during those overs. At one point, India were ahead according to this calculation, during one rain delay (and would have won if play was unable to be resumed). At a second rain interval, England, who had scored some quick runs (precisely because they were aware of the need to get ahead in terms of D/L) would correspondingly have won if play hadn't resumed. Play was finally called off with just 7 balls of the match remaining and England's score equal to the Duckworth-Lewis 'par' score, therefore resulting in a tied match.
This example does show how crucial (and difficult) the decisions of the umpires can be, in terms of assessing at exactly what point the rain is heavy enough to justify ceasing play. If they had done so one ball earlier, England would have been ahead on D/L, and so would have won the match (equally, if play had stopped one ball later, without England scoring off that ball, India would have won the match – indicating how finely-tuned D/L calculations can be in such situations).
The Duckworth–Lewis Standard Edition is fairly simple to apply, requiring a published reference table of total resources remaining for all possible combinations of overs and wickets, and some simple mathematical calculations. The Professional Edition, which has been in use in all international one-day cricket matches since early 2004, uses substantially more sophisticated statistical modelling, and requires the use of a computer instead of a published reference table.
As with most non-trivial statistical derivations, the D/L method can produce results that are somewhat counter intuitive, and the announcement of the derived target score can provoke a good deal of second-guessing and discussion amongst the crowd at the cricket ground. This can also be seen as one of the method's successes, adding interest to a "slow" rain-affected day of play.
For 50-over matches, each team must face at least 20 overs before D/L can decide the game, unless one or both sides have been bowled out in less than 20 overs and/or the team batting second has reached its target in less than 20 overs. For Twenty20 games, each side must face at least five overs before D/L can decide the game, unless one or both sides have been bowled out in less than five overs and/or the team batting second has reached its target in less than five overs.
If these prerequisites are not met, the match is declared a no result.
The team that bats first is called Team 1, their final score is called S, the total resources available to Team 1 for their innings is called R1, the team that bats second is called Team 2, and the total resources available to Team 2 for their innings is called R2.
|Overs remaining||Wickets in hand|
For each reduction in overs, the loss in total resources available to the batting team is found using a published reference table, then Team 2's target score is changed as follows:
- If R2<R1, reduce Team 2's target score in proportion to the reduction in total resources, ie. S x R2/R1.
- If R2=R1, no adjustment to Team 2's target score is needed.
- If R2>R1, increase Team 2's target score by the extra runs that could be expected to be scored on average with the extra total resource, ie. S + G50 x (R2–R1)/100, where G50 is the average 50-over total. Team 2's target score isn't simply increased in proportion to the increase in total resources, ie. S x R2/R1, as this 'could lead to some unrealistically high targets if Team 1 had achieved an early high rate of scoring [in the powerplay overs] and rain caused a drastic reduction in the overs for the match.' Instead, D/L Standard Edition requires average performance for Team 2's additional resource over Team 1.
G50 is the average score expected from the team batting first in an uninterrupted 50 overs-per-innings match. This will vary with the level of competition and over time. The annual ICC Playing Handbook gives the values of G50 to be used each year when the D/L Standard Edition is applied:
|ICC Playing Handbook edition||Under-19 internationals||Under-15 internationals||Matches between ICC associate member nations||Women’s ODIs|
Duckworth and Lewis write, 'We accept that the value of G50, perhaps, should be different for each country, or even for each ground, and there is no reason why any cricket authority may not choose the value it believes to be the most appropriate. In fact it would be possible for the two captains to agree a value of G50 before the start of each match, taking account of all relevant factors. However, we do not believe that something that is only invoked if rain interferes with the game should impose itself on every game in this way. In any case, it should be realised that the value of G50 usually has very little effect on the revised target. If 250 were used, for instance, instead of 235, it is unlikely that the target would be more than two or three runs different.'
For each reduction in overs, the loss in total resources available to the batting team is found using a computer, then Team 2's target score is changed as follows:
- If R2<R1, reduce Team 2's target score in proportion to the reduction in total resources, ie. S x R2/R1.
- If R2=R1, no adjustment to Team 2's target score is needed.
- If R2>R1, increase Team 2's target score in proportion to the increase in total resources, ie. S x R2/R1. The problem of early high scoring rates potentially producing anomalously high targets has been overcome in the Professional Edition, which is essentially 'a different table of resource percentages for every total score in the Team 1 innings.' Therefore Team 2's target score can be simply increased in proportion to the increase in total resources when R2>R1, and there is no G50.
However, the resource percentages used in the Professional Edition are not publicly available, so a computer must be used which has the software loaded.
Example calculations for Duckworth−Lewis Standard Edition
As the resource percentages used in the Professional Edition are not publicly available, it's difficult to give examples of the D/L calculation for the Professional Edition. Therefore examples are given from when the Standard Edition was widely used, which was up to early 2004.
Team 1's innings completed; Team 2's innings delayed
On 18 May 2003, Lancashire played Hampshire in the National League. Rain before play reduced the match to 30 overs each. Lancashire batted first and scored 231-4 from their 30 overs. Hampshire's innings was then further reduced to 28 overs.
|Total resources available to Lancashire (R1)||30 overs and 10 wickets||75.1%|
|Total resources available to Hampshire (R2)||28 overs and 10 wickets||71.8%|
|Hampshire's par score||231 x R2/R1 = 231 x 71.8/75.1||220.850 runs|
Hampshire's target was therefore 221 to win (in 28 overs), or 220 to tie. They were all out for 150, giving Lancashire victory by 220 − 150 = 70 runs.
Team 1's innings completed; Abandonment during Team 2's innings
On 3 March 2003, Sri Lanka played South Africa in the 2003 Cricket World Cup Pool B. Sri Lanka batted first and scored 268-9 from their 50 overs. South Africa had reached 229-6 from 45 overs when play was abandoned.
|Total resources available to Sri Lanka (R1)||50 overs and 10 wickets||100.0%|
|Total resources available to South Africa at the start of their innings||50 overs and 10 wickets||100.0%|
|Total resources remaining to South Africa when play abandoned||5 overs and 4 wickets||14.3%|
|Total resources available to South Africa (R2)||100.0% − 14.3%||85.7%|
|South Africa's par score||268 x R2/R1 = 268 x 85.7/100.0||229.676 runs|
Therefore, South Africa's retrospective target from their 45 overs was 230 runs to win, or 229 to tie. In the event, as they had scored exactly 229, the match was declared a tie.
South Africa scored no runs off the very last ball. If play had been abandoned without that ball having been bowled, the resource available to South Africa at the abandonment would have been 14.7%, giving them a par score of 228.6, and hence victory.
Team 1's innings completed; Team 2's innings interrupted
On 16 February 2003, New South Wales played South Australia in the ING Cup. New South Wales batted first and scored 273 all out (from 49.4 overs). Rain interrupted play when South Australia had reached 70-2 from 19 overs, and at the re-start their innings was reduced to 36 overs (ie. 17 remaining).
|Total resources available to New South Wales (R1)||50 overs and 10 wickets||100.0%|
|Total resources available to South Australia at the start of their innings||50 overs and 10 wickets||100.0%|
|Total resources remaining to South Australia at the interruption||31 overs and 8 wickets||68.6%|
|Total resources remaining to South Australia at the re-start||17 overs and 8 wickets||46.7%|
|Total resources lost to South Australia by the interruption||68.6% − 46.7%||21.9%|
|Total resources available to South Australia (R2)||100.0% − 21.9%||78.1%|
|South Australia's par score||273 x R2/R1 = 273 x 78.1/100.0||213.213 runs|
South Australia's target was therefore 214 to win (in 36 overs), or 213 to tie. In the event, they were all out for 174, so New South Wales won by 213 − 174 = 39 runs.
History and creation
The D/L method was devised by two British statisticians, Frank Duckworth and Tony Lewis, as a result of the outcome to the semi-final in the 1992 Cricket World Cup between England and South Africa, where the most productive overs method was used. Rain stopped play for 12 minutes with South Africa needing 22 runs from 13 balls chasing England's 252/6 off 45 overs. The revised target left South Africa needing 21 runs from one ball, which was a reduction of only one run compared to a reduction of two overs, and a preposterous target given that the maximum score from one ball is generally six runs. The D/L method avoids this flaw: in this match, the revised D/L target would have left South Africa four to tie or five to win from the final ball. Duckworth said, "I recall hearing Christopher Martin-Jenkins on radio saying 'surely someone, somewhere could come up with something better' and I soon realised that it was a mathematical problem that required a mathematical solution."
It was first used in international cricket in the second game of the 1996–97 Zimbabwe versus England One Day International series, which Zimbabwe won by seven runs, and was formally adopted by the International Cricket Council in 1999 as the standard method of calculating target scores in rain shortened one-day matches.
Various different methods had been used previously, with the most common being the average run-rate method, and the most productive overs method.
All of these methods have flaws that are easily exploitable:
- The average run-rate method takes no account of how many wickets the team batting second have lost, but simply reflect how quickly they were scoring when the match was interrupted, so if a team felt a rain stoppage was likely they could attempt to force the scoring rate without regard for the corresponding highly likely loss of wickets, skewing the comparison with the first team.
- The most productive overs method also takes no account of how many wickets the team batting second have lost, and also has the further effect of penalizing the team batting second for good bowling, as their best overs are ignored in setting the revised target.
An example of this is in the 1988/89 Benson and Hedges World Series Cup, where the average run rate method was used: in the third final between Australia and the West Indies, rain stopped play for one hour and 25 minutes with the West Indies needing 180 off 31.2 overs chasing Australia's 226/4 off 38 overs. The revised target left the West Indies needing 61 off the 11.2 overs that remained, and the West Indies won the match and the competition with 4.4 overs remaining and eight wickets in hand after Desmond Haynes hit a Steve Waugh full toss for six. Australian fans loudly booed this unsatisfactory conclusion, and criticism from the media and both captains led to the average run rate method being replaced by the most productive overs method for setting revised targets in interrupted matches. In this match, the D/L method would have increased the West Indies target to 232 to take into account a two-hour rain delay during Australia's innings, and then revised the target to 139 after the second interruption.
The published table that underpins the D/L method is regularly updated, using source data from more recent matches. From the 1999 Cricket World Cup match in Bristol between India and Kenya, Tony Lewis noticed that there was an inherent weakness in the formula used at the time that would give a noticeable advantage to the side chasing a total in excess of 350. A correction was very soon built into the formula and the software to correct this, by including a 'match' factor. However, this minor correction was not fully adopted by users until the 2004 update. Updating the source data in its own right would reflect the overall trend that one-day matches were achieving significantly higher scores than in previous decades, affecting the historical relationship between resources and runs.
At the same time as this update, the D/L method was also split into a Professional Edition and a Standard Edition. The main difference is that while the Standard Edition preserves the use of a single table and simple calculation – suitable for use in any one-day cricket match at any level – the Professional Edition uses substantially more sophisticated statistical modelling, and requires the use of a computer. The Professional Edition has been in use in all international one-day cricket matches since early 2004.
In June 2009, it was reported that the D/L method would be reviewed for the Twenty20 format after its appropriateness was questioned in the quickest version of the game. Lewis was quoted admitting that "Certainly, people have suggested that we need to look very carefully and see whether in fact the numbers in our formula are totally appropriate for the Twenty20 game."
The D/L method has been criticized on the grounds that wickets are a much more heavily weighted resource than overs, leading to the suggestion that if teams are chasing big targets, and there is the prospect of rain, a winning strategy could be to not lose wickets and score at what would seem to be a "losing" rate (e.g. if the required rate was 6.1, it could be enough to score at 4.75 an over for the first 20–25 overs).
Another criticism is that the D/L method does not account for changes in proportion of the innings for which field restrictions are in place compared to a completed match.
More common informal criticism from cricket fans and journalists of the D/L method is that it is overly complex and can be misunderstood. For example, in a one-day match against England on 20 March 2009, the West Indies coach (John Dyson) called his players in for bad light, believing that his team would win by one run under the D/L method, but not realizing that the loss of a wicket with the last ball had altered the Duckworth-Lewis score. In fact Javagal Srinath, the match referee, confirmed that the West Indies were two runs short of their target, giving the victory to England.
More recently, concerns have been raised as to its suitability for Twenty20 matches, where a high scoring over can drastically alter the situation of the game and variability of the run-rate is higher over matches with a shorter number of overs.
- "A Decade of Duckworth-Lewis". BBC. 1 January 2007. Retrieved 2009-03-21.
- Scorecard for the rain-affected 4th ODI between India and England on 23 November 2008, from Cricinfo.
- Data Analysis Australia's detailed mathematical analysis of the Duckworth-Lewis Method daa.com.au.
- Duckworth/Lewis Method of Re-calculating the Target Score in an Interrupted Match
- espncricinfo D/L FAQ's Q4
- ICC Playing Handbook
- ICC Playing Handbook 2006-07
- espncricinfo D/L FAQ's Q6
- espncricinfo D/L FAQ's Q13
- espncricinfo D/L FAQ's Q15
- "22 off one ball – A farcical rain rule leaves everyone bewildered", from Cricinfo.
- "Stump the Bearded Wonder", Bill Frindall explains how D/L would apply to 1992 WC semi-final
- Scorecard of the 2nd ODI between England and Zimbabwe, 1 January 1997, from Cricinfo.
- 3rd Final, 1988/89 Benson and Hedges World Series Cup
- Rain affected rules from Cricinfo.
- Duckworth-Lewis to review their formula for T20 matches
- Bhogle, Srinivas, The Duck worth/Lewis Factor, Rediff.com.
- Booth, Shane, quoted in For a Fair Formula, The Hindu.
- Varma, Amit, Simple and subjective? Or complex and objective?, ESPNcricinfo
- Charlie Brooker, AV campaigners have created a stupidity whirlpool that engulfs any loose molecules of logic, dismisses the claim of the simplicity by citing the method's formula, The Guardian, 25 April 2011. Retrieved 2011-04-28
- The anomalous contraction of the Duckworth-Lewis method
- BBC news interview with The Duckworth Lewis Method
- Interview with band
- Duckworth, FC & Lewis, AJ "Your Comprehensive Guide to The Duckworth Lewis Method for Resetting Targets in One-day Cricket", Acumen Books, 2004. ISBN 0-9548718-0-4
- Duckworth, F "A Role for Statistics in International Cricket" Teaching Statistics, (June 2001) Volume 23, No. 2 pp 38–44
- Duckworth, FC & Lewis, AJ "A fair method for resetting the target in interrupted one-day cricket matches" Journal of the Operational Research Society, (Mar 1998) Volume 49, No. 3 pp 220–227 JSTOR 3010471
- ICC's D/L method FAQ
- Cricinfo's D/L method FAQ
- ICC's D/L method (standard edition) table of resource percentages
- Cricinfo's explanation of the D/L method
- ICC's explanation of the D/L method
- BBC Sport's explanation of the D/L method
- Web based Calculator for the Standard Edition of the Duckworth Lewis method
- Calculator for the Professional Edition of the D/L method
- Alternatives to D/L