D/L method: answers to frequently asked questions (updated December 2008)
by Frank Duckworth & Tony Lewis
(Note: throughout the following, the side which bats first is called Team 1 and the side batting second is called Team 2.)
1. What is the difference between the Standard Edition and the Professional Edition?
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.
2. Why should Team 2 sometimes be set the task of scoring more runs than were made by Team 1 when they have the same number of overs to face?
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 and would not have taken the risks of scoring as fast as they would have done 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 60 or 70 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 20 to 25 less than that made in 50 overs, so Team 1's loss is typically 40-45 runs greater than Team 2's and the target is raised by about 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 complicated calculation and would reduce the scope for accommodating further stoppages. Because of these disadvantages, cricket authorities have preferred to stay with the present regulations.
3. Why should this apply when Team 1 have been bowled out?
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.
4. When Team 2 have more resources than Team 1, why do you not simply scale up the target by the ratio of resources?
In the Professional Edition, which is now used in most top-level matches including ODIs, the problem of early high scoring rates producing anomalously high targets has been overcome, and so direct scaling is employed. So this question only relates to the Standard Edition.
In the Standard Edition, to scale up a target by the ratio of resources could lead to some unrealistically high targets if Team 1 had achieved an early high rate of scoring and rain caused a drastic reduction in the overs for the match; see Q10, for instance). We have preferred, therefore, to assume average performance for Team 1's additional loss of resource over Team 2.
5. But why should the target score sometimes go down if there is an interruption in the first innings and teams have the same number of overs?
In interruptions to the first innings the D/L method makes appropriate allowance for the comparative resources lost by the stoppage.
Consider the following situation. Suppose Team 1 started well in the style of the renowned Sri Lankan 1996 World Cup winning team but the wheels fell off and they were 150/9 in 30 of the 50 overs. On average Team 1 would be all out shortly, leaving Team 2 to score at the rate of around 3 per over for their full 50 overs. If rain interrupted play at this point and 19 overs were lost per side, then on the resumption Team 1 would have only one 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.
6. When Team 2 have the more resource, you increase the target by applying the excess resource to the quantity known as G50, which is the average score for a 50-over innings. Why do you not use a different value of G50 according to ground conditions on the day?
The quantity G50 is not used in the Professional Edition as used from the start of 2004, enhanced targets being calculated by scaling Team 1’s score in the direct ratio of the resources available to Team 2 and Team 1, so this question only applies to the Standard Edition.
The key is simplicity. 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 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.
In the Professional Edition, the option has been retained for an average 50-over score to be input for the purposes of predicting Team 1’s eventual score from any point of their innings. This facility is purely for media or spectator interest and is not a part of the target calculation. Rather than use the default value of G50, commentators have the option of entering a ‘best guess score’ for 50 overs before the match starts, this taking account of ground conditions.
7. Why don’t you take away wickets as well as overs to balance up teams’ resources?
This is a simple idea but unfortunately it creates 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.
8. When Team 2's innings is interrupted, why do you not set a target that maintains the probability of achieving the target across the stoppage?
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) have lost the match and the team scoring the least (210) 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.
9. How can Team 2 win by a number of runs?
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.
10. Suppose we are playing a 50-overs-per-side game where only 10 overs per side are needed for the match to count. Team 1 send in pinch hitters and get off to a wonderful start making 100 for no wicket after 10 overs. There is then a prolonged stoppage and when play can resume Team 1's innings is closed and there is only just time for Team 2 to face the minimum 10 overs. The D/L calculation (Standard Edition) gives Team 2's target as 151 in 10 overs. How can this practically impossible target be justified?
11 Same playing regulations as in Q10. Team 1 make the excellent score of 350 in their 50 overs and Team 2 start their reply cautiously and reach 40/0 in 10 overs. The heavens now open (or the floodlights fail) and further play is ruled impossible. Under the Standard Edition of the D/L system Team 2 are declared the winners by 3 runs. They were clearly already falling behind the run rate they needed even allowing for the fact that they had all their wickets intact, so how can this result be justified?
The above represent the two worst-case scenarios for treatment by the Standard Edition of the D/L method. They could only give such extreme consequences with playing regulations that allow a minimum of 10 overs per side for the match to count. But a similar, though less exaggerated, injustice could still arise even with a minimum of 20 overs per side required.
The Standard D/L method was devised so that anyone could perform the calculations with nothing more than the single table of resource percentages and a pocket calculator. This was regarded as an essential requirement for the method. It was considered that to be totally dependent on a computer would mean that the method could not be used universally, it would be vulnerable to computer failure and it would be more difficult to explain how the targets were calculated.
The use of the simplifying single table of resource percentages meant that actual performance must necessarily be assumed to be proportional to average performance. In 95% of cases this assumption is valid, but the assumption breaks down when an actual performance is far above the average, as is the case in the scenarios of Q10 and Q11 and in the record-breaking match between South Africa and Australia (March 2006) in which South Africa scored 438/9 to beat Australia’s 434 in 50 overs.
This problem has now been overcome by use of the Professional Edition and this has been in general use for most matches at the top level of the game, including ODIs, since early in 2004. It can only be operated by using a computer program.
12. How can copies of the full tables be obtained?
The over-by-over tables for use with the Standard Edition may be found on the ICC website (see http://icc-cricket.yahoo.com/; go to ‘more regulations’ then ‘Duckworth Lewis’. The full tables are available to all cricket authorities. The general public may obtain these by purchase of the booklet ‘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
13. How do the results of the Professional Edition differ from those of the previous (Standard) Edition?
For innings when the side batting first (Team 1) score at or below the average for top level cricket (which would be about 235 for an uninterrupted 50-over innings), the results of applying the Professional Edition are generally similar to those from the Standard Edition. For higher scoring matches, the results start to diverge and the difference increases the higher the first innings total. In effect there is now a different table of resource percentages for every total score in the Team 1 innings, and so a computer is essential to operate the system.
14. How do we know whether to use the Professional Edition or the Standard Edition?
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 CODA.
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 (see Q1).
15. How does one obtain the computer software for operating the Professional Edition? And how does one obtain the resource percentage tables for the Professional Edition?
This software is not yet available for sale to the general public; when it is details will be available on the ICC website (http://icc-cricket.yahoo.com/). The tables of resource percentages, which would enable the calculations to be carried out manually, can only be produced when Team 1’s innings has been completed and then only by using the computer software.
16. Shouldn't the revised target take account of the quality of the players at the crease when play is stopped and of those who still have to bat? And should not account also be taken of the number of overs the top line bowlers will still have to bowl when play is resumed?
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 can be computed. As well as leading to contention, such a procedure would be quite impractical to implement.
17. Does the D/L method take account of the various rules on fielding restrictions, eg PowerPlay overs?
If any allowance were made for the different scoring abilities for overs with fielding restrictions, then the identities of the different types of overs would have to be input into the target calculation, and this would be a considerable and unwelcome complication for the scorers and would prevent targets and par scores being known instantly they are required. But a thorough analysis of several thousand match scorecards covering the different rules in place over the years has shown that the effects of these rules on scoring patterns are not statistically significant. So no allowance for the effect of rules on fielding restrictions has been considered necessary.
18. Does the D/L method take account of ‘substitute’ rules?
It would be extremely difficult, and highly inconvenient to the scorers, to make an allowance for the recently trialled ‘SuperSub’ rule (whereby the 11 players 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. However, ICC decided in March 2006 that the rule would be discontinued.
Games between teams of 12 (and in some cases of 13) are still 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 and the D/L computer program CODA can be customised for such competitions so that due allowance for this effect is made.
19. Can the method be used for Twenty20 matches?
The method is used in Twenty20 matches in exactly the same way as in 50 overs/innings games. It is no different from a standard ODI reduced to 20 overs/side by rain before the start.
The Professional Edition was introduced in 2003 to allow for the increasing prevalence of higher scoring games. In essence this distributes the run-scoring expectations more evenly throughout the innings compared with the original (Standard) Edition which is more geared to average performances. Twenty20 games also have higher run scoring rates and experience has shown that the Professional Edition works particularly well in these games.
20. How can I get hold of the formula of the D/L method?
There are two academic papers that explain how the D/L tables are created. Whereas they provide the necessary explanation of the mathematics involved, due to commercial sensitivity, some fine details are omitted. But the papers will provide the information required for those cricket fans interested in the mathematics behind D/L.
Students of mathematics, and others, may be able to obtain the papers from their campus library, which may well stock the Journal of the Operational Research Society , or they may be able to obtain the papers from other sources. Otherwise the papers can be obtained from the journal’s publishers Palgrave Macmillan http://www.palgrave-journals.com/pal/index.html.
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.
Comments to the authors, via email, are welcome.