by Frank Duckworth & Tony Lewis
In the light of recent online discussion the FAQs on the Duckworth/Lewis method have been updated and have been grouped into several sections reflecting the various focuses of questioners recently and over the years.
Note that in all discussions the side batting first is called Team 1 and the side batting second is called Team 2.
We would be the first to agree that the best solution to interrupted matches is to resume when possible later and to play the match to its natural conclusion – or declare ‘no result’. The latter option is quite common even now in round-robin or league formats. But knock-out competitions require a result.
If sufficient overs can be fitted into a day then the industry prefers a result to be obtained in most forms of limited overs matches. Even in those competitions which do have a reserve day, if a result can be obtained in the one day using a ‘rain rule’ then this is by far the preferred option. [In the CWC2007 final which Australia won, the match officials forgot this arrangement and we had the farcical situation of Sri Lanka, who knew they had no chance before and after an interruption and the application of the D/L method, choosing to receive their remaining overs from slow bowlers in virtual darkness to avoid the need to return the next day]
The reasons for making it truly a one-day match are practical and pragmatic. Players (and officials) need to move on to their next matches for which transport (and hotel) arrangements will have been made and are often difficult to rearrange at such short notice. TV companies covering the matches would prefer to avoid upsetting their schedules both for their viewers and their outside broadcast crews. The ground’s management would need to employ all their staff again incurring extra expense. Not least, spectators would prefer to see the result on the day – after all, it’s billed as ‘one-day cricket’.
So the game is quite prepared to see a shortened match – provided a fair method for adjusting targets can be found. In the past that has been the problem.
At the start of one-day cricket the Average Run Rate (ARR) method was used to set the target. For example, if Team 1 made 250 in their 50 overs, an ARR of 5 runs per over, and Team 2’s innings was reduced to 25 overs, the score to beat was 5x25=125. Because Team 2 still had all 10 wickets this was a somewhat easier task than 250 in 50 overs and so, whenever rain was around, captains winning the toss usually chose to bat second to take advantage of this situation. For many years it was known to be unfair (and in several other ways too) but the method’s simplicity, and the lack of any viable alternative, meant that ARR was used in most one-day matches until the early 1990s.
The Australians devised the ‘Most Productive Overs’ (MPO) method whereby Team 2’s target for a reduced number of overs was based on the runs scored in the most productive of that many overs scored by Team 1. In other words, the target was reduced by the number of runs scored in the number of least productive overs equal to the number lost.
However, the rationale of this method was based on stoppages occurring during the interval, so that if Team 2 lost say 25 overs, then their target was derived from the 25 most productive overs in Team 1’s innings, irrespective of when the overs were lost. So if the stoppage was in the latter part of Team 2’s innings it often gave a grossly unfair target.
This deficiency was highlighted in the CWC1992 semi-final in Sydney when South Africa were reduced from requiring 22 runs in 13 balls to 22 runs in one ball when two overs were lost. This arose because the two least productive overs of England’s innings were in fact two maidens. A match with a very close outcome had been reduced to a certainty for England with much annoyance to players and spectators alike and much embarrassment to cricket’s authorities. (See also Q26)
The D/L method was developed during the mid 1990s and was the first to take account of the state of the match when overs were lost, i.e. both the number of overs which had been bowled and the number of wickets that were down. It came into operation at the start of 1997.
Whereas the Average Run Rate method set the target in proportion to the overs available to the two teams, the D/L method adjusts the score in proportion to the run-scoring resources available to the two teams . These are a given number of overs and ten wickets. The measure of resources is based on what teams have in an uninterrupted ODI innings so the resources at the start of such an innings, that is 10 wickets and 50 overs, are designated as 100%. In matches that have fewer overs per side resources are still based on the 50-over match. For example in a 25-overs per side match teams have half the overs of an ODI but still have all 10 wickets and so have more than half the resources of a 50-over innings – our tables calculate it to be about two thirds, i.e. around 67%.
As teams receive overs and lose their wickets they consume their resources. If rain interrupts play then the loss of overs reduces the resources of the batting side according to overs left and wickets lost at the stoppage and their interaction. Overs left are not worth much unless wickets are available to use them. Conversely, having plenty of wickets in hand is of little value unless there are sufficient overs with which to use them. So the overs and wickets resources change in their value according the state of the innings.
The D/L calculations take account of the combined resources available to the two teams and adjust the target accordingly – see Q31-32 for information in books and websites with more details of the process. .
When the interruption occurs during the first innings, so that the match is shortened to one of fewer overs per side than it was at its start, Team 1 are usually more disadvantaged than Team 2. Before the stoppage they had been pacing their innings in the expectation of receiving say 50 overs but would have taken more risks to score faster had they known their innings was to be shortened. Team 2, on the other hand, know from the start of their innings that they have the reduced number of overs and can pace their entire innings accordingly Team 2 are set a higher target to compensate Team 1 for this disadvantage.
Consider, for example, when Team 1 have batted for 40 of an intended 50-over innings and then rain washes out the rest of their innings and there is just time for Team 2 to receive 40 overs. If they had wickets in hand, Team 1 might have expected to make around 70 or 80 runs in those final 10 overs. But Team 2 know they have only 40 overs to receive from the moment they start their innings. The average score in a 40-over innings is only 15 to 20 less than that made in 50 overs, so Team 1's loss is typically 55-60 runs greater than Team 2's. Hence the target is raised by this amount.
The necessity to set a higher target for Team 2 arises from the regulations for most competitions that require that lost overs, where possible, be divided equally between the two sides. It would be possible to compensate Team 1 for their disadvantage by allowing them to face more overs than Team 2 and in this way the latter need not be set an enhanced target, but this would require a more complicated calculation and, more importantly, it would reduce the scope for accommodating further stoppages. Because of these disadvantages, cricket authorities have preferred to stay with the present regulations and accept enhanced targets where these arise.
In limited-overs cricket no distinction is made between the two ways in which an innings is closed, using up all the overs or losing all ten wickets. In both cases the team have used up all the resources of their innings. In an uninterrupted innings, there is no difference between Team 1's score of 250, for instance, whether it were 250 for 3 wickets in 50 overs or whether it were 250 all out in 47 overs. Similarly in an interrupted innings, the method of target revision cannot and should not distinguish between whether Team 1's innings were terminated by being all out or by using up their allocation of overs.
When Team 1’s innings is shortened it usually means that Team 1 were disadvantaged and so an enhanced target is set to neutralise this disadvantage. But it is possible that the loss of overs might in fact have advantaged them, as in the following situation.
Suppose Team 1 started well but the wheels fell off and they were 150/9 in only 30 of their 50 overs. On average they would be all out shortly, leaving Team 2 to score at a rate of around 3 per over for their full 50 overs. But if rain interrupted play at this point and 19 overs were lost per side, then on the resumption Team 1 would have only one further over to survive and their run rate would then be close to 5 per over. By all the 'old' methods, for 31 overs also, Team 2 would have to score around 150, around 5 per over, to win - in other words Team 1 would have been greatly advantaged by the rain interruption changing a required scoring rate of 3 per over to 5 per over for Team 2. By the D/L method this advantage to Team 1 would be neutralised so that the target for Team 2 would be well below 150 in this circumstance, and fairly so, which maintains the advantage Team 2 had earned before the stoppage. In other words, and quite logically, Team 2 have to get fewer runs than Team 1 scored to win in the same number of overs.
At the top level of the game, the Professional Edition of the D/L method is now used. This requires use of a computer program. At lower levels of the game, where use of a computer cannot always be guaranteed, the Standard Edition is used. This is the method which was used universally before 2004; it is operated manually using the published tables of resource percentages.
In the Standard Edition, enhanced targets are obtained by applying the excess resource that Team 2 have over Team 1 to the quantity G50, which is the average runs scored in an uninterrupted 50-over innings. As resources are measured in terms of a full 50-over innings, then the same value of G50 is used for all lengths of match, e.g. in 40 over/side and Twenty20 matches. The current recommended value of G50 for all top level matches is 245.
G50 is not used in the Professional Edition, which employs a different approach to calculating enhanced targets.
For a full description of the operation of the Standard Edition see http://www.icc-cricket.com/rules_and_regulations.php
As can be seen in our academic papers (see Q30) the background mathematical formula to the method is indeed exponential – this captures the concept that more and more overs do not mean more and more runs indefinitely; the limit of 10 wickets per side regardless of overs has the effect of flattening the curve. What the curve represents is the average total of runs in the overs available (based on copious match data). The curve does not represent the rate of scoring in any particular overs. Experience over the 15 years usage to date shows that this curve fits the average totals for overs left very well, and similarly for the corresponding curves for average runs scored in remaining overs for different numbers of wickets down.
Suppose Team 1 score 250 in 50 overs. What’s the target for Team 2? It’s 251 of course and cricketers and spectators don’t need this stated on the board, just the 250. If Team 1 restrict Team 2 to 249 in their 50 overs then everyone knows Team 1 have won – by 1 run. In other words the 250 is the ‘balance’ of the match. – it’s what Team 2 have to beat and Team 1 have to keep Team 2 below.
Similarly, the par scores displayed on scoreboards, and given on a sheet to officials, team camps and media, represent the balance of the match. As the innings progresses Team 1 are trying to keep Team 2 below the par score by economical bowling and taking wickets, and Team 2 are trying to get ahead of it – in case the match is abandoned at that point. Adding one run to the par score displayed would distort the neutrality of that balance.
In that 2003 match South Africa did indeed mistake the par score for the score they needed to win – they thought the par of 229 at the end of the 45th over was enough to win on abandonment and having reached 229/6 in 44.5 overs spurned an opportunity off the 6th ball to go for an extra run and the match was tied. But it was a misreading of the sheet – which clearly stated “if match abandoned par score shown in table below is that needed to TIE”.
Whereas the furore has been over the South African camp’s mistake and their resulting elimination from the competition, the Sri Lankan captain Sanath Jayasuriya read the same table correctly and knew the result was a tie when the match was abandoned.
This is a simple idea but unfortunately it would create many difficulties and problems over implementation. First is how to apportion wickets deducted for overs lost bearing in mind not only the rate of deduction (which might result in a fraction of a wicket!), but also the fact that the earlier wickets are usually more valuable than the later wickets. Second is the problem of deciding which batsmen shall not be allowed to bat. This could cause dissatisfaction not only to the batsmen excluded but also to the spectators who may have come to see particular players bat.
Because of such problems cricketing authorities have always regarded the idea of deducting wickets as an unacceptable option.
The problem with maintaining Team 2's probability of achieving their target across a stoppage is that it would mean that the target depended upon how many runs they had scored at the point of interruption. The more runs they had scored the more they would need, and the less they had scored the less they would need.
For instance, suppose that in three parallel matches, Team 1 score 250 in their 50 overs and Team 2's innings is interrupted after 20 overs with 10 overs lost in each case but with the scores at 60/2, 100/2 and 140/2. In all three cases the resources remaining were reduced from 67.3% to 52.4%, a loss of 14.9%, and so the target would be reduced by 14.9% of 250 to 213 (calculations based on the D/L tables for the Standard Edition). If one set the revised target by scaling the runs still required by the resources remaining after and before the stoppage, which would maintain an equal probability of achieving the target, the targets would be different in the three cases, at 208, 217 and 226 respectively. It is surely unjust for a team to have to face a higher target because they had scored more runs. And an absurdity in the comparative results would be quite possible. Suppose, for instance, that the final scores of Team 2 in the three matches above are respectively 210, 216 and 225. The team scoring the most (225) would have lost the match and the team scoring the least (210) would have won.
The perceived problem with the way the revised target is set only arises when Team 2 are well ahead, or well behind, their par score. For instance, if they were 30 runs behind par at a stoppage and afterwards there was only time for a very few overs, they would still be 30 runs behind par and would have these few overs to make up the deficit, so their task may become virtually impossible. (If the match were washed out completely, they would have lost by 30 runs; nobody would dispute this.) It is Team 2's obligation to remain close to par to avoid losing if the match were terminated or their task being made more difficult if the innings were to be shortened. (See also Q27.)
When Team 2's innings is prematurely terminated by the weather the result is decided by comparing their actual score with their par score. Whether Team 2 have won or lost, the difference of their score from the par score is the best measure available of the margin of victory and so it has been decided that the result should be given in terms of this margin in all such cases.
Even when a game is not prematurely terminated it is still possible to describe a victory for Team 2 in terms of a margin of runs. When they hit the winning run their score will be ahead of par by a certain margin and there is a good case for expressing the result in terms of this margin of runs in all cases. For instance, if Team 2 score the winning run off the last ball available, to describe their victory in terms of the wickets they had in hand gives no indication of its narrowness.
The decision on which edition should be used is for the cricket authority which runs the particular competition. The Professional Edition can only be operated by running the computer software WinCODA.
Playing conditions for ODIs and for most countries’ national competitions require that the Professional Edition is used where a computer can be guaranteed to be available for all matches; otherwise, or in the unlikely event of the computer failing to be available and operable, the Standard Edition is used.
Some recreational club competitions use the Std Edn and some the Pro Edn depending on their opportunities to obtain the WinCODA software through their country’s cricket administration.
We must stress that the method does not attempt to predict the result at a stoppage. It adjusts the target according to resources available at the time of the interruption and resources lost from it. Whatever happens in a match after that is irrelevant. As is commonly stated, cricket is a gloriously uncertain game and much can, and does, happen to change fortunes in a match – if the weather permits it to continue.
For example in the U19 World Cup on 19 th August 2012 South Africa scored 244 in their 50 overs. After 27 overs England were progressing smoothly at 102/1 and were 12 runs ahead of par. Rain threatened and had they gone off the field at that point never to return then England would have won – by 12 runs; they were in the stronger position and based on what has happened on average that result would have fairly reflected that position. However, rain didn’t arrive and soon afterwards England slumped to 141 all out in 40.3 overs and lost by 103 runs. Here then was an example of the ‘glorious uncertainty’ of the game. The D/L method did not make a wrong ‘prediction’ at the possible termination; it would have been the weather, not D/L that had denied South Africa the chance of turning the match around.
First let us look at uninterrupted matches. Statistics on 50-overs per side matches show that on average Team 1 win about 52% of the time and Team 2 about 48% of the time. Ties are not uncommon but are sufficiently infrequent that they can be ignored.
From time to time we look at the overall performance of the D/L method. Although in a particular match bias cannot be determined, in the long run if there were any bias then the distribution of wins in matches where the D/L method has been used would be significantly different from the 52 to 48 ratio, In practice the ratio of Team 1 to Team 2 wins, to the nearest whole number, is 52% to 48%. In other words there is no evidence of bias to either side in interrupted matches by using the D/L method. Contrast that with the bias to Team 2 in the ARR method (see Q2)
We have recently analysed Twenty20 matches for the same bias – see Q19.
One of the great virtues of the D/L method is that it is ‘strategically neutral’. It takes full allowance of the state of the match at an interruption and sides should generally play as though the rain were not going to materialise.
An exception to this is if Team 2 are behind the D/L par score. If there is a danger of the match being terminated then they would want to increase their scoring rate to get up with par, even if this means taking greater risks of losing wickets and so making their task even more difficult – it would be their only chance. But if the interruption might not be terminal then taking too many risks and losing too many wickets could make their task impossible upon resumption. In such circumstances it might be best to play only slightly more aggressively to catch up on par and hope that the reduction in overs is not so great as to prevent them catching up on par on resumption of play. Similarly, if Team 2 are well ahead of par they might choose to play slightly more conservatively so as not to risk going behind par by losing wickets.
A myth which is still occasionally put out by broadcasters is that when Team 1’s innings is resumed after a stoppage with a reduction of the overs per side then Team 2’s (usually enhanced) target will depend on the number of wickets that that are lost in their remaining overs. This is simply not true. When Team 1 resume their innings, the target enhancement has already been established based on the loss of resources due to the stoppage and, just as in an uninterrupted innings, Team 1 must try and make as many runs as they can regardless of how many wickets they may lose in the process. Remember that when all the available overs have been bowled, wickets in hand have no value.
Playing conditions are continually changing and it is necessary to ensure that the method is keeping abreast of such changes. The parameters of the formula are reviewed when there is sufficient data to make such a review worthwhile. On average this occurs every two to three years.
No! The D/L method is neither a document (which can be copyrighted) nor a material invention (which can be patented.) So we don’t receive royalties and hence have no reason to be especially pleased when it rains! We have, however, negotiated payment from the cricket authorities for monitoring the operation of the method and maintaining it through changes in the game. More details are given in our book (see Q31).
Many commentators and players believe that the D/L method is biased towards Team 2 in T20 games. Is their any evidence for this perceived bias? Up to end of August 2012, the D/L method has been used in 189 T20 matches, of which 181 have produced a result. Of these, 90 have been won by Team 1, 84 by Team 2 and 7 have been tied. Apportioning the ties equally between the two teams Team 2 have won 48.3% of matches, slightly less than half. How does this compare with the percentage of wins in uninterrupted matches? A trawl of major competitions such as T20Is, IPL and Big Bash Australia reveals the interesting statistic that Team 2 win about 51.1% of matches. So the results for D/L affected Twenty20 matches are little different from those in uninterrupted matches. This comparison certainly provides no evidence that the D/L method is biased towards Team 2.
Experience from extensive data over recent years shows that the same formula very closely fits both 50-over and T20 average totals and also other lengths of matches in between (eg South Africa’s and ECB’s 40-over competitions). Consequently, the same single formula satisfactorily covers all the various formats of the limited overs game.
ODIs a have minimum overs rule of 20 overs per side for a match to be valid. Some countries such as the ECB lower the requirement to 10 overs per side in domestic matches. The objective is clearly to give a reasonable chance of getting a result and, more importantly, for spectators to get the chance to see some cricket on a rain-reduced day.
When Twenty20 emerged through the ECB a minimum of 5 overs per side was stipulated and this has been copied by ICC and other countries into all T20 matches. Occasionally a team feels hard done by, as illustrated in Q27. The focus becomes whether a 5 over minimum per side is too small for a “fair” result to be obtained; a poor over (or two) has little chance of being retrieved and maybe the minimum could be 8 overs, the same percentage as the minimum in ODIs. ICC’s Cricket Committee reviewed this rule in 2010 but concluded that the regulation should remain in place for T20Is in order to maximise the chances of a resul t.
The purpose of the PowerPlay overs is to encourage attacking batting. But in addition it forces attacking field placing, so whereas runs do tend to be scored faster there is a corresponding increase in the wickets lost. The statistics collected over many years show that the runs scored per resources consumed in the PowerPlay overs are closely consistent with the same measure in the non-PowerPlay overs. We have therefore concluded that the D/L method does not need separate allowance for the PowerPlay overs. This conclusion has been found to be true for all the various forms of fielding restrictions from the 15-over rule to the present PowerPlay regulations. In practice, this is a fortunate conclusion as it would be extremely inconvenient for scorers to have to enter details of the number of PowerPlay overs that have already been bowled at a stoppage.
Although it is quite true that the extent to which the effective resources of the batting and bowling sides are depleted by a stoppage depends on the identities of the individual players affected, there is no way in which such factors could be incorporated into an objective rule for revising targets. It would require both teams to identify, before every match, the way the total quality of their sides, in respect of both batting and bowling, is divided between the individual team members. Furthermore, it would be necessary to input details of who was still to bowl and to bat and to perform the calculation based on this before a revised target could be computed. As well as leading to contention, such a procedure would be quite impractical to implement.
It would be extremely difficult, and highly inconvenient to the scorers, to make an allowance for the previously trialled ‘SuperSub’ rule (whereby the 11 payers in a side of 12 that start the match must be declared before the toss) in carrying out D/L calculations. But as the rule was only experimental, no attempt was made to consider any modification. In fact, ICC decided in March 2006 that the rule would be discontinued.
Games between teams of 12 (and in some cases 13) are played in some competitions. Under this rule, 11 of the 12 (or 13) are nominated to bat and 11 are nominated to bowl. This affects the relative strengths of the ten partnerships in an innings. If this rule were more widely adopted, the D/L computer program WinCODA would be adjusted so that due allowance for this effect could be made.
In this match South Africa had bowled only 45 of their 50 overs, in which England had scored 252/6, when the time ran out. South Africa suffered financial penalties for this slowness but there were no cricketing penalties so they were chasing 253 to win in 45 overs as though this had been the match length at the start. They had reached 231/6 in 42.5 overs when play was suspended due to a rain shower. At that point they required 22 runs to win with 13 balls remaining. The match was very evenly balanced. Two overs were then deducted due to the rain so they came back onto the field with only one ball remaining. The Most Productive Overs method which was in use required that the target be reduced by the number of runs scored in the two least productive overs of England’s innings. Since there had been two maiden overs, no reduction in target was made, leaving South Africa still requiring 22 runs but with only one ball to get them from.
Using the current Professional Edition procedure for ODIs, South Africa would have needed four runs to win from that final ball – an eminently achievable and fair target.
The D/L method attracted a significant amount of criticism from this match with Paul Collingwood, England’s captain, at the forefront with his post-match comment "There's a major problem with Duckworth-Lewis in this form of the game". It was in the group stages of the competition, on 3rd May 2010 in Guyana in which England had scored 191 in their 20 overs including 60/1 in the 6 PowerPlay overs. As Collingwood claimed, you would expect to win the match 95% of the time with such a total. West Indies, however, through Chris Gayle and partly due to some poor England bowling, got off to a flying start and were 30/0 in 2.2 PowerPlay overs. Rain intervened and 14 overs were lost. The revised target was 60 in 6 overs leaving WI with “only” 30 to get in 3.4, but now non-PowerPlay, overs. “Too easy” was the cry.
Some sober reflection, however, shows that the WI’s overall requirement was exactly what England had scored in their 6 PowerPlay overs (with which they were very pleased) and WI only had 2.2 PowerPlay overs. So the target was actually very reasonable.
The issue arises because D/L’s methodology is to maintain the margin of advantage across a stoppage, not the probability, (see Q3 and Q11). That flying start put WI 11 runs ahead of the par of 19 for 2.2 overs with no wicket lost and so they returned to the field still 11 runs ahead of par. Had England held WI to par then the requirement would have been 41 in 22 balls, a suitably more challenging one.
In the event WI won with just one ball to spare. This was helped by more indifferent bowling, England giving away no fewer than 8 runs to wides in their 6 overs.
This match brought into focus the matter of whether a 5 over minimum per side (6 in this case) is too small for a “fair” result to be obtained; a poor over (or two) has little chance of being retrieved (see Q21).
The over-by-over tables for use with the Standard Edition may be found on the ICC website (see http://www.icc-cricket.com/rules_and_regulations.php; go to ‘Duckworth Lewis ’and click on Duckworth/Lewis Standard Edition Table for a pdf file.) The full tables are available to all cricket authorities. The general public may obtain these by purchase of the booklet or our book (see Q31).
This software is not yet available for sale to the general public; when it is details will be available on the ICC website (http://www.icc-cricket.com/).
There are two academic papers that explain the mathematics of the method. Whereas they provide the necessary theory, due to commercial sensitivity some fine details are omitted.
Students of mathematics may be able to obtain the papers from their campus libraries, or from other sources. Otherwise the papers can be obtained from the journal’s publishers Palgrave Macmillan.
Duckworth, F. C. and Lewis, A. J. (1998). A fair method of resetting the target in interrupted one-day cricket matches. Journal of the Operational Research Society, Vol. 49, No. 3, pp. 220-227.
Duckworth, F. C. and Lewis, A. J. (2004). A successful Operational Research intervention in one-day cricket. Journal of the Operational Research Society, Vol. 55, No. 7, pp. 749-759.
This booklet gives a very thorough understanding of the workings of the method. “Your Comprehensive Guide to the D/L Method…”, which is available in electronic or hard-copy form from Acumen Books, tel: +44 (0) 1782 720753 or www.acumenbooks.co.uk
Our book, “Duckworth Lewis: The method and the men behind it” , published by SportsBooks http://www.sportsbooks.ltd.uk , gives the full story of how we became involved and how we developed the method. It also includes the full tables for the Standard Edition.
Cricinfo and ICC provide the best sources.
