# The anomalous contraction of the Duckworth-Lewis method

On May 3, 2010, there were two curious applications of the Duckworth-Lewis method in the ICC World Twenty20. West Indies defeated England even though they only scored 60 for 2 in 6 six overs in reply to England's 191 for 5, and Sri Lanka came perilously close to being eliminated because Zimbabwe were only required to score 44 in five overs to defeat Sri Lanka's 173 for 7.

There was outrage at the D/L targets and also surprise. The D/L method is now proven in 50-over matches, so why was it giving wonky targets in Twenty20 matches? Clearly it was because the ICC was trying to fit a model designed for 50-over matches to 20-over games. The fit wasn't working. The trousers were too big.

What can a young lad do if he's forced to wear his dad's trousers? Essentially one of two things: cut off the long legs and walk around pretending the trousers now fit, or put the trousers in a washing machine and hope they shrink sufficiently.

The ICC has so far chosen the first method. It's surprising we didn't have situations like the ones on May 3 earlier. Our little calculation, which we will explain as we go along, suggests that the shrunken trousers might have done a better job that day.

**The Duckworth-Lewis rationale**

Till D/L came along, ODI targets for the team chasing were only based on overs remaining: if the team batting first scored 250 in 50 overs and then rain washed out 25 overs, the team chasing only had to score 125+1 = 126 to win, with all 10 wickets in hand, which was obviously unfair.

New variants were therefore tried, including the much-maligned "most-productive overs" (MPO) idea. MPO isn't such a bad idea if the interruption occurs between the two innings, but it wasn't capable of handling one or more during-the-innings interruptions. The horror of a South Africa target of 22 runs in 13 balls turning into the impossible 21 runs off one ball was an extreme manifestation of this inability.

The D/L method cleverly combined overs remaining and wickets in hand into a single combined resource. It wasn't easy to model this complex interplay between the two resources, but Duckworth and Lewis did it quite brilliantly. Better still, their method solved the vexing problem of during-the-innings interruptions.

Over the years the D/L method has become even better, especially after the Professional Edition, which required the use of a computer, was introduced. It would be fair to say that the method now resolves interruptions in 50-over matches almost perfectly.

The D/L method is well explained in the first graph to the right. When the team chasing begins its innings, it is like an ant sitting at the top left corner of the graph. There are 50 overs remaining and the "combined resource" available is the full 100%.

There are 10 curves in the graph, corresponding, to the number of wickets remaining. At the start of the innings our ant is at the topmost point of the top curve, corresponding to "10 wickets remaining". After every ball is bowled, the ant moves a step to the right along the curve. If a wicket falls, the ant drops vertically to the curve below.

Now, suppose 20 overs have been bowled and two wickets have been lost. The ant will be in line with "30 overs remaining", and on the green curve, corresponding to "8 wickets remaining". A visual guess suggests that about 65% of the combined resource is still available. If, instead, five wickets are down at this stage, the ant would be on the orange curve and the combined resource would only be 45%.

Finally, imagine that 20 overs are lost at this stage due to rain. What would our ant do? Well, it would simply trot down the green (or orange) curve and stop at the point corresponding to 10 overs remaining. The combined resource now available would be about 30% for the green line and about 25% for the orange line.

So at every stage of the match we know exactly how much of the combined resource percentage is still available. Let's call this R2. And let R1 be the combined resource that was available to the team batting first (would be 100% if the 50 overs are completed or all 10 wickets are lost). Let S be the score of the team batting first. The D/L target and the par score are then calculated by playing around with R1, R2 and S.

If we stare a little more at these curves, we'll find that D/L has been rather clever. Look at the higher curves (corresponding to 10, 9 or 8 wickets remaining). They aren't coming down too quickly to start with, because wickets in hand tend to be more valuable in the early part of the chase, but towards the end all the curves seem to fuse into one single, fat curve, because as the match nears its end, overs and balls remaining are far more valuable than wickets in hand.

The complete evolution of a limited-overs match can therefore be gleaned by looking at the entire span of the curves. That's how dad's trousers look.

**D/L for Twenty20 by cutting the trousers**

Let's now see what we are doing when we try to employ D/L for Twenty20. We're cutting off our full curves and only retaining the part to the right of "20 overs remaining", i.e. pretending that a Twenty20 is simply an ODI reduced to 20 overs a side. This is a convenient assumption, *but it may not be entirely valid*. How many ODIs can we recall in which not a single wicket has fallen after 30 overs? And what do we do about the field restrictions in the first six overs of a Twenty20?

To get an idea of how things change when we look at D/L curves only to the right of "20 overs remaining", compare the second graph with the first.

These curves are a lot flatter, and when only five or six overs remain, the top six or seven curves collapse to become almost a diagonal. It is as if D/L is saying, "I don't care about wickets in hand at this stage, my targets only depend on overs remaining." In other words, we're almost back to the pre-D/L days of run rate-based targets, even if we go through the motions of using D/L.

In the England-West Indies match. England scored at 9.55 runs per over. Using a simple run rate-based target, West Indies only needed to score 9.55*6 = 58. The D/L target, if the match had been reduced to six overs at the start of the innings, would have be 66, i.e. a run rate of 11 per over. An increase of 66 - 58 = 8 runs just can't offset the very considerable advantage that West Indies would have enjoyed: of having all 10 batsmen, favourable field restrictions for two of the six overs, and a hard ball.

However, the rain interruption didn't occur at the innings break. West Indies came out to chase believing they would play their full 20 overs. After a 14-ball blitzkrieg, in which they scored 30 runs without losing a single wicket, there was a spell of rain and the match was curtailed to six overs. This very severe "during-the-innings" interruption further hurt D/L and gave West Indies a massive advantage: they now had to get six runs fewer, i.e. 60 in six overs!

West Indies further helped their cause by not losing a single wicket before the interruption. If they had been, say, 30 for 2 instead, the D/L target would have been 71. All the good fairies had apparently come together to bless West Indies and damn D/L.

If the Sri Lanka-Zimbabwe match had been reduced to five overs at the start of the innings, the D/L target for Zimbabwe would have been 52, i.e. a run rate of 10.4 to counter Sri Lanka's run rate of 8.65. This again seems excessively generous, especially with two overs of field restrictions.

That match, too, went awry. It rained after Zimbabwe had scored 4 without loss after the first over. They returned to bat imagining they had about 100 more to get in 10 more overs. Sadly, they failed to stay ahead of the par score; if Zimbabwe had scored 40 in the next four overs with just the loss of that one wicket, they would have won a thoroughly undeserved victory.

These examples provide further proof that D/L is in deep trouble if the overs come down to just five or six in a Twenty20, though it is noteworthy that even with a minimum of 10 overs, the D/L model regains reasonable control.

**D/L for Twenty20 by shrinking the trousers**

Imagine for a moment that we had to describe Twenty20 cricket to someone who woke up after a ten-year slumber. Would we say that Twenty20 is "like the last 20 overs of an ODI", or would we say that it is "like a complete ODI in which everything happens much, much faster"?

We rather fancy the latter. Why not, then, assume that the complete D/L curves, designed for 50-over matches, also adequately depict the evolution of a Twenty20 match?

We fiddled around with the over-by-over D/L standard tables (available in the public domain) to do exactly that. Here's (third graph) how the curves appear now, with the recalibrated combined resources.

How would the England-West Indies match have panned out using this "shrunken trousers" model for D/L? Our calculations indicate that West Indies would have needed to score 69 in 6 overs if the match had evolved in exactly the same fashion with that interruption at 30 for no loss after 14 balls.

A target of 69 certainly appears more reasonable, but what if the interruption had occurred between the innings and West Indies knew from the start that they only had six overs to bat? The target then would have been 87, or 14.4 runs per over, with only two overs of field restrictions. This appears steep, but we mustn't forget that 191 too is a lot of runs.

If Zimbabwe knew from the start that they could only bat for five overs in reply to Sri Lanka's 173, their target to win would have been 68, at 13.5 runs per over, with two overs of field restrictions. But given the way the match actually panned out, with an interruption at 4 without loss after the first over, Zimbabwe's target using this shrinking model would have been 60 in five. This must appear much more reassuring than a mere 44.

**Send for the tailor**

The shrunken-trousers model certainly appears to give more satisfactory results than the cut-trousers one for the two matches on May 3, but we would need to look at many more examples before we can recommend the former with any degree of authority; there is a lurking fear that it may set very high targets, especially if the interruption occurs between innings.

There is also the problem of field restrictions in the first six overs. D/L has never accommodated this batting phase into its mix. The D/L explanation is that the more adventurous batting during the field-restriction phase is compensated by the greater propensity to lose wickets. This explanation seems valid in 50-over matches, where the loss of a wicket significantly reduces the combined resource percentage. But in Twenty20, especially with the cut-trousers model practiced currently, losing a wicket brings down the combined resource percentage by much less, if at all, and it is much less likely to inhibit adventurous strokeplay. Given the nature of Twenty20, and the sort of audience they attract, it may be worthwhile to retain field restrictions during the first six overs at all times, whatever the state of the match!

All these fixes are only for the short term. There is a compelling and urgent need to redraw D/L-like curves for Twenty20 based on actual Twenty20 match data. The IPL repository itself contains about 175 matches, and there must be at least 100 match-data sets from international games. We feel certain that D/L can come up with these new curves, and if they don't feel so inclined, we would be happy to participate in a parallel initiative.

Rajeeva Karandikar is a professor at Chennai Mathematical Institute. Srinivas Bhogle heads the Bangalore centre of TEOCO Software Pvt Ltd. The authors would like to thank K Vijay for his suggestions via Twitter

Comments have now been closed for this article

It does seem unfair I would also say that for a ODI match to count there should be a least 8 overs bowled as a 5 over match is a bit unfair. In 50 over cricket there has to be 20 overs a side to count for a match which is 40% of the total number of overs.

Going back to the original point of this article, there seems to be a misapprehension that the exisitng tables were designed for 50 over cricket only. The original 1997 rules, which I still have, gave separate tables for different lengths of innings, but these were based on scaled versions of the same curves, just changing the definition of 100% resource. On the first revision, these were combined into a single table of 50 overs (the maximum now needed), but this was always intended for any match length up to 50 overs. This is why the values are defined in terms of "overs left". D/L have repeatedly checked the effect of powerplays and found no need to alter the tables for them. Therefore, there is no need to "pretend" that a 20 over innings is like the last part of a 50 over innings: simple logic will tell you that the scoring possibilities are the same. I simply do not understand the logic behind the "shrunken trousers" model, and in any case the data do not fit that model.

@Pelham_Barton: I like the uniform idea as an analogy. I still disagree that Rule 2 is unfair: I think that in your example Rule 1 is unfair on the fielding side. I think that the target increasing with runs scored sounds counter-intuitive, but that it's the right way to go. Look at it this way: what's actually the only relevant statistic in a run chase? It's the runs required, not the runs scored. You can be 200-2 after 30 overs but if you're chasing 600 you're still not doing very well. So any argument based on the runs scored is going to lead to some unhappy conclusions. In your example, the game has been reduced to 80 to win off 20 overs. Any other numbers are irrelevant: 80 off 20 is now "the game" in its entirety. So this should be reduced after the rain to 40 to win off 10 overs. (Agreed, a team on 320 chasing 400 is more likely to win than a team on 120 chasing 200, but to exactly the same extent that they're more likely to get to 360 than the team on 120 is to get to 160).

@hattima: Yep, that's the right formula - it's the same as the one in my 20:36 post, because "Required runs after resumption" = New Target - Runs Scored. Under the current D/L method WI at 2-0 go from needing 190 off 106 to 59 off 22; at 60-0 they go from 132 off 106 to 1 off 22. With mine, 190 off 106 becomes 48 off 22 and 132 off 106 becomes 33 off 22. It's worth pointing out that this is what the resources formula says is the "equivalent" run chase, and there's plenty of statistical analysis to back it up. The issue of *exploitability* that you raise is a separate, and valid one. You certainly can't build "whether the batting captain thinks it's going to rain" into the model, just as you can't factor in team strength, pitch conditions etc. It's true that D/L, in assuming teams don't know it's going to rain, becomes more exploitable in twenty20, but this is mostly due to the problems we've already outlined. Preserve probabilities, and it becomes much less of an issue.

For anyone who still has an open mind, the following may help. Suppose (never mind how) we could redesign cricket so that teams could reasonably be expected to score at a uniform rate through an innings and that Team1 has scored 250 off 50 overs. If Team2's innings is reduced to 40 overs before the start, we would all agree they should be set 200 to tie, 201 to win. If the match was ended by its first interruption after 40 overs of Team2's innings, we would also agree to apply the par score of 200 to tie. If Team2 had reached 170 off 30 overs and then 10 overs are lost, we could have (Rule1) maintain the expected scoring rate across the whole of Team2's innings: still 200 to tie, or (Rule2) maintain the rate now required by Team2 at 4rpo: 210 to tie. Rule1 is unfair because Team2's asking rate has come down from 4rpo to 3rpo: Rule2 is unfair because Team2's target has gone up by 10 runs as a result of their own success. You have to choose between them. D/L is analogous to Rule1.

@David, I see your point regarding the formula, what I had in mind was to decrease the required runs according to the resources, so it should have been: Required runs after resumption = (Required runs before resumption)*(resources in hand after resumption)/ (resources in hand before resumption). It was a mistake not to divide by the resources as it would not have stayed 100%. But, I still think preserving the" probability" is not feasible: the formula you gave would help teams like ZIM in the SL match, or even worse, teams like Afganisthan in the AUS match. Teams always know the weather forecasts, and play accordingly. With D/L there is still the need to score at a good rate to achieve what WI did, and that involves risk. But with your formula, they'd preserve wickets, score, say, 5-6 runs in 4-5 overs; and would still have a chance to go broke for the 17-18 runs in 1 over after the break. Work out the target score of WI in 6 assuming they were 2-0 in 2.2 and you'll see my point.

@hattima: That formula can't work, because the target gets bigger as the break progresses (as "unused resources after break" gets smaller). Also you need it to be true that for a degenerate 0-over break, you get New Target = Old Target. So you could have New Target = Runs Scored + (1-X+Y) * Runs Required, where X is unused resources at break, Y is unused resources after break, but this is equivalent to my formula (I divided by total resources because by convention, total resources = 100% only for a 50-over match), which doesn't work because as Y tends to 0, the runs required tends to (1-X)*(Runs required at break) and it needs to tend to 0. The system you propose is pretty much what we have at the moment, but I'm arguing it's flawed: how's it fair for a team with a 60% chance of winning at the break to have a 99% chance of winning after it? The fact that the match might have been terminated is irrelevant once it resumes. You want to give them "about as hard a chase as they had before".

@David, what I had in my mind is: New Target=Runs Scored+(unused resources at break - unused resources after break)* (Run required at the time of interruption). The total resources should be 100% and hence no division needed in the formula. So it is closer to your original formula than what you have written above. However, I do not fully like the "prob preserving idea" (assuming it's achievable). Suppose a team would win at the break if no further play is possible. So their winning chance is 1. Now if just 1 ball can be bowled after the resumption, by D/L system in most cases they'd still win. But according to your system the winning prob will have to be reduced to, say, 0.6. So, D/L would reward heavily the team who anticipates the break and you'd penalize them. I'd want a more continuous system, which should change their prob of win to, say, 0.99, and gradually reduce 0.6 as more and more balls are left. It seems fairer to me to give some credit to a team playing the situation.

@Anupam Mathur (and lucyferr): I agree with you that part of this article contains as clear an explanation of how D/L works as any that I have ever seen. However, as has been pointed out by several people (including me) in earlier comments, D/L have already done the analysis that the article's authors are asking for. Far from reaching weird conclusions, their analysis indicates that T20 does behave exactly like the last 20 overs of a 50 over match amd so there is no need for a separate set of tables.

@hattima - As for using the difference instead of the ratio, the only formula that makes sense to do that would have to be New Target = Old Target - {[(unused resources before break - unused resources after break)/(total resources)] * Runs required}, since if you have a degenerate 0-over break then the target has to stay the same. I don't think this preserves probabilities, though: in the case where Zim are 5-1 after 5 overs, it sets a new target of 57. The problem is that if you have x% of your total resources unused before the break, and there's not long left after the break, then you basically just times the runs required by about x%: this suffers from a similar problem to the current method. A mathematician would say that as "unused resources" tends to 0, you want "runs required" to tend to 0, but it doesn't, it tends to "runs required before the break" * x%.

It does seem unfair I would also say that for a ODI match to count there should be a least 8 overs bowled as a 5 over match is a bit unfair. In 50 over cricket there has to be 20 overs a side to count for a match which is 40% of the total number of overs.

Going back to the original point of this article, there seems to be a misapprehension that the exisitng tables were designed for 50 over cricket only. The original 1997 rules, which I still have, gave separate tables for different lengths of innings, but these were based on scaled versions of the same curves, just changing the definition of 100% resource. On the first revision, these were combined into a single table of 50 overs (the maximum now needed), but this was always intended for any match length up to 50 overs. This is why the values are defined in terms of "overs left". D/L have repeatedly checked the effect of powerplays and found no need to alter the tables for them. Therefore, there is no need to "pretend" that a 20 over innings is like the last part of a 50 over innings: simple logic will tell you that the scoring possibilities are the same. I simply do not understand the logic behind the "shrunken trousers" model, and in any case the data do not fit that model.

@Pelham_Barton: I like the uniform idea as an analogy. I still disagree that Rule 2 is unfair: I think that in your example Rule 1 is unfair on the fielding side. I think that the target increasing with runs scored sounds counter-intuitive, but that it's the right way to go. Look at it this way: what's actually the only relevant statistic in a run chase? It's the runs required, not the runs scored. You can be 200-2 after 30 overs but if you're chasing 600 you're still not doing very well. So any argument based on the runs scored is going to lead to some unhappy conclusions. In your example, the game has been reduced to 80 to win off 20 overs. Any other numbers are irrelevant: 80 off 20 is now "the game" in its entirety. So this should be reduced after the rain to 40 to win off 10 overs. (Agreed, a team on 320 chasing 400 is more likely to win than a team on 120 chasing 200, but to exactly the same extent that they're more likely to get to 360 than the team on 120 is to get to 160).

@hattima: Yep, that's the right formula - it's the same as the one in my 20:36 post, because "Required runs after resumption" = New Target - Runs Scored. Under the current D/L method WI at 2-0 go from needing 190 off 106 to 59 off 22; at 60-0 they go from 132 off 106 to 1 off 22. With mine, 190 off 106 becomes 48 off 22 and 132 off 106 becomes 33 off 22. It's worth pointing out that this is what the resources formula says is the "equivalent" run chase, and there's plenty of statistical analysis to back it up. The issue of *exploitability* that you raise is a separate, and valid one. You certainly can't build "whether the batting captain thinks it's going to rain" into the model, just as you can't factor in team strength, pitch conditions etc. It's true that D/L, in assuming teams don't know it's going to rain, becomes more exploitable in twenty20, but this is mostly due to the problems we've already outlined. Preserve probabilities, and it becomes much less of an issue.

For anyone who still has an open mind, the following may help. Suppose (never mind how) we could redesign cricket so that teams could reasonably be expected to score at a uniform rate through an innings and that Team1 has scored 250 off 50 overs. If Team2's innings is reduced to 40 overs before the start, we would all agree they should be set 200 to tie, 201 to win. If the match was ended by its first interruption after 40 overs of Team2's innings, we would also agree to apply the par score of 200 to tie. If Team2 had reached 170 off 30 overs and then 10 overs are lost, we could have (Rule1) maintain the expected scoring rate across the whole of Team2's innings: still 200 to tie, or (Rule2) maintain the rate now required by Team2 at 4rpo: 210 to tie. Rule1 is unfair because Team2's asking rate has come down from 4rpo to 3rpo: Rule2 is unfair because Team2's target has gone up by 10 runs as a result of their own success. You have to choose between them. D/L is analogous to Rule1.

@David, I see your point regarding the formula, what I had in mind was to decrease the required runs according to the resources, so it should have been: Required runs after resumption = (Required runs before resumption)*(resources in hand after resumption)/ (resources in hand before resumption). It was a mistake not to divide by the resources as it would not have stayed 100%. But, I still think preserving the" probability" is not feasible: the formula you gave would help teams like ZIM in the SL match, or even worse, teams like Afganisthan in the AUS match. Teams always know the weather forecasts, and play accordingly. With D/L there is still the need to score at a good rate to achieve what WI did, and that involves risk. But with your formula, they'd preserve wickets, score, say, 5-6 runs in 4-5 overs; and would still have a chance to go broke for the 17-18 runs in 1 over after the break. Work out the target score of WI in 6 assuming they were 2-0 in 2.2 and you'll see my point.

@hattima: That formula can't work, because the target gets bigger as the break progresses (as "unused resources after break" gets smaller). Also you need it to be true that for a degenerate 0-over break, you get New Target = Old Target. So you could have New Target = Runs Scored + (1-X+Y) * Runs Required, where X is unused resources at break, Y is unused resources after break, but this is equivalent to my formula (I divided by total resources because by convention, total resources = 100% only for a 50-over match), which doesn't work because as Y tends to 0, the runs required tends to (1-X)*(Runs required at break) and it needs to tend to 0. The system you propose is pretty much what we have at the moment, but I'm arguing it's flawed: how's it fair for a team with a 60% chance of winning at the break to have a 99% chance of winning after it? The fact that the match might have been terminated is irrelevant once it resumes. You want to give them "about as hard a chase as they had before".

@David, what I had in my mind is: New Target=Runs Scored+(unused resources at break - unused resources after break)* (Run required at the time of interruption). The total resources should be 100% and hence no division needed in the formula. So it is closer to your original formula than what you have written above. However, I do not fully like the "prob preserving idea" (assuming it's achievable). Suppose a team would win at the break if no further play is possible. So their winning chance is 1. Now if just 1 ball can be bowled after the resumption, by D/L system in most cases they'd still win. But according to your system the winning prob will have to be reduced to, say, 0.6. So, D/L would reward heavily the team who anticipates the break and you'd penalize them. I'd want a more continuous system, which should change their prob of win to, say, 0.99, and gradually reduce 0.6 as more and more balls are left. It seems fairer to me to give some credit to a team playing the situation.

@Anupam Mathur (and lucyferr): I agree with you that part of this article contains as clear an explanation of how D/L works as any that I have ever seen. However, as has been pointed out by several people (including me) in earlier comments, D/L have already done the analysis that the article's authors are asking for. Far from reaching weird conclusions, their analysis indicates that T20 does behave exactly like the last 20 overs of a 50 over match amd so there is no need for a separate set of tables.

@hattima - As for using the difference instead of the ratio, the only formula that makes sense to do that would have to be New Target = Old Target - {[(unused resources before break - unused resources after break)/(total resources)] * Runs required}, since if you have a degenerate 0-over break then the target has to stay the same. I don't think this preserves probabilities, though: in the case where Zim are 5-1 after 5 overs, it sets a new target of 57. The problem is that if you have x% of your total resources unused before the break, and there's not long left after the break, then you basically just times the runs required by about x%: this suffers from a similar problem to the current method. A mathematician would say that as "unused resources" tends to 0, you want "runs required" to tend to 0, but it doesn't, it tends to "runs required before the break" * x%.

@Pelham_Barton: My argument against that is that if a team's task is made impossible by the rain break (as in the extreme example I gave when they come back with 1 over left) then neither the difficulty of the task "remaining" nor the difficulty of the task over the whole innings can be preserved, since the "whole innings" includes the fact that the team didn't know there was going to be an interruption.

@Hattima: Yep, that's right, it's wrong for the 50-over game too. It's most likely to be shown up in twenty20, though, because in that format you're much more likely to resume batting with not many overs left. Where Zim are 5-1 after 5 overs (needing 99 from 36), my method gives a target of 23 (18 needed from 1 over). I agree, 18 from 6 is easier than 99 from 36. This may be down to an imperfection of the formula at extremes. The version on their website is the Standard Edition, not the Professional one, so that might be the problem. In any case it's a better target than 50 from 6.

Great article. Firstly, simply great analysis. It has been a big education for me. I for one, did not know that in a T20 game, they assume that it to be the last 20 overs of an ODI. For some reason, I had always assumed that they would do a 2.5 overs of T20 is the same resource as an over of T20 - i.e. the model u have proposed. Given how the graphs work out, "last 20 overs of an ODI" doesn't seem logical to be applied to T20 as such.

Completely agree, 175 sample size is probably good enough to draw the curves for T20. D/L are smart guys - I am sure they would have tried it themselves and probably reached very weird conclusions at this stage and that's why they haven't come up with a T20 version.

Finally, I think, getting to "the right cuves" for T20 is a tough ask. The reason being that most teams now seem to wait for the last 5-7 overs to go for the big bang, looking to have wickets in hand till then. And that is something difficult to scale up by 2.5 times for an ODI.

the current form of d/l method assume that 20 over match is actually a 50 over match in which both teams have scored equal runs in their first 30 overs with zero wicket loss. In england vs wi match, if we assume both team scored 140 runs in first 30 overs, so total score of england would be 140+191= 331 in 50 overs. In return wi would hv made 170 in 194 balls n dat too without a loss of wicket when it started to rain. Thus the target of 30 runs from next 16 balls would look better.

This is great! Thank you so much for the explanation - I understand the idea behind D/L much better now. And yes, your suggestions for recalibrating D/L make sense. I hope the ICC pay the boffins to do something like this, with options for future adjustments as more stats become available as more T20 matches are played.

I'd like to quit by pointing out that although it does seem that David has an excellent point, his ideas are valid (or not valid) for any limited overs game. So, it doesn't prove that D/L is any worse for T20. Whereas there may be many scopes of improving D/L in general, I still feel it is as good for T20 as it is for 50 over matches, and it is certainly more logical than the "shrunken trouser" model proposed above. The statisticians love to point out that no model is correct or wrong, they are only better or worse for a particular purpose. So the only way we could know which method is better is by performing some cross-validation. It is clearly impossible in this case as we can never conclusively check what is a fairer target for a truncated game with any degree of accuracy. So we have to leave it to speculation, and there would always be dissatisfied parties no matter which method is used.

@David, are you sure you are not undermining the performance of the second team till the break a bit too much? For example, what would have been Zim's target score been in the sixth over according to your calculations if they were, say, 5-1 in 5 overs? I have done some rough calculations and it seems to me that would be around 16-17, and I dare say that it would have then increased the chance of Zim winning considerably from a much tighter situation. However, it does seem you have a very good point; how about looking at the difference of resources instead of the ratio?

@david_franklin: We are agreed that the current D/L tables are the best available measure of difficulty of task. We differ in how to apply them when an innings is interrupted and resumed. I believe that the fair thing is to preserve difficulty across the whole of the innings, while you believe that it is to preserve difficulty of the remaining task.

@Hattima - That's not really what I'm saying: resources and probability aren't the same thing, even though they're both a percentage. Resources is a measure of how many wickets and overs you have left and has nothing to do with how many runs you need or have scored. Win probability is definitely dependent on the number of runs you require.

@Pelham_Barton: I guess we'll have to agree to disagree then, but I can't see how anything other than probability-preserving can be fair, pretty much by definition! A team resumes their innings with a different chance of winning than they had before the rain, which to me has to be unfair. Like I said, a resumption is a completely different situation to when a match is terminated, where you want to declare whoever has the highest win probability the winner.

@Pelham_Barton - Yep, I agree that D/L worked fine in SL-Zim, because the match was terminated by the rain so the problems it has didn't apply. Suppose, though, that the rain had relented with time for one over left. They would've needed 26 to win from that over, which is unfair: 26 from 6 is a bit harder than 75 from 36. If they'd been 54-1, needing 60 from 36, they'd have needed 1 from 6, which is MUCH easier. If they'd been 55-1, needing 59 from 36, they'd be declared the winners already! D/L helps the team ahead at an interruption: the problem is that the target needs to vary with the number of runs already scored in order to be fair. The current formula is Target = Original Target * Total Resources Available. It SHOULD be Target = Runs Scored + (Unused Resources after break / Unused Resources before break) * (Runs Required at break). Using this, Zim on 29-1 would've been set 14 from the last over (target 43) whereas if they'd been 55-1, say, they'd have been set 66.

Dear David, I now understand what you mean; but basically you are just describing the D/L idea! What you and I are calling probability, they just refer to it as "resource". As you can check out at Duckworth-Lewis.com (page 40-41), they calculate the "resource lost" when the game resumes and computes the target accordingly to compensate for that. They are not calling it "probability" because it is not probability according to its true definition, but it is nevertheless a percentage and hence is just what you are saying. So, the example you have given is not an actual one, it wouldn't happen under D/L! Just shows again that it is quite a fair method in terms of the ideas. The only possible problem is whether we should trust their tables of resoures (or your probabilities); but as many of us have pointed out, they have mentioned that they have cross-validated it. So I'd say that at the end of the day, their methods are quite fair; but the ICC rule to finish the match after 5 overs is not.

I must agree with Thomas et al; rather than reworking D/L curves to fit a 5-over innings, the Laws must be changed for T20 such that a result can only be had by each side playing 20 overs. (For the record, I'm not a fan of the 1-over "Eliminator".) If we continue to allow shortened Twenty20 matches, I'm afraid we'll eventually move to packing in a tournament's worth of Five5 matches in a weekend. T20 was bad enough for us Test purists, but please don't bastardize this game anymore than it already has been. Long live Test cricket!

@david_franklin: If you prefer to apply different principles to decide a match when there is time for one more over and when there is not, I can accept that as a valid preference, but it is not one that I share. I have no quarrel with anyone who accepts that this is a matter of preference, and whose preferences differ from my own. I do not accept that probability preserving is the only definition of fairness.

As I write this, no-one has commented on the target in the SL-Zim match, so let us look at that here. When the second lot of rain came, Zim were 14 runs short of the par score after 5 overs. To get to a winning score on D/L, they would have had to score (on average) 3 more runs in each over without losing more than the one wicket. Suppose they had managed that, and were 45-1 after 5 overs, and the second lot of rain had not come. They would then only have needed another 59 off the remaining 6 overs, an increase in scoring rate of less than one run per over, with Mendis only able to bowl one of the remaining overs. In other words, if they had been ahead of the par score at 5 overs, they would have been in a good position to win the match. As it was, they came nowhere near to the par score. Given all that, I cannot accept that the par score in that match was unreasonable, although I agree with those who say that the minimum overs for a result should be a lot more than 5.

@hattima and Pelham_Barton: There's a *big* difference between a match which is resumed following an interruption (at which point you want to give the teams back the same win probabilities as they had before) and a match which is terminated by an interruption (where you declare the team with the higher win probability "the winner"). In the first you want to set a reasonable target, in the second you want to decide what the par score was and find whether they were above or below it. As OzzieLondon and DVC have said, they've done detailed testing of the formula, and shown there's nothing wrong with it, even in twenty20. It's perfect for terminated matches. But for resumed matches they don't apply it correctly: they need to use it to scale down the *runs required*, not the whole target. Preserving the win probabilities after an interruption has to be the aim: no other aim makes sense.

Dear David, if it is not possible to continue the game anymore, how would you preserve the "60% chance of winning"? You'd have to declare one team as winner, thereby increasing their "chance of winning" to 100%. So, this "probability" that you want to preserve just can not stay fixed.

@david_franklin: Probability preserving rules - if they can be correctly specified - may make sense if an innings is interrupted and resumed, but cannot work if the innings is interrupted and the match is abandoned. The best you can do is award the match to the team with the higher probability of winning at the time of the interruption. This means that you are essentially applying different principles in the two different cases. Difference-preserving rules do not have this problem.

I think the key point of the article and most comments on the article, is that for 20-20 cricket the match should be played until both sides have had their 20 overs!

D/L solves a real problem with 50 over games, which cannot be easily rescheduled in a tournment context, but surely it is quite reasonable to just wait until you get your 20 overs per side.

Here are some arguments for this: - in 50 over cricket, D/L does not attempt to set par scores until each side has 20 overs (!) - the 'chris gayle' problem: sides will have their best batsmen up first, and so are likely to score at a higher rate if they only have to bat a shorter amount of time (so the 'wickets in hand' factor is not just about getting bowled out - lower order batsmen are less effective than higher order batsman, on average) - Paul Collingwood's point that a side would always prefer to chase 60 off 6 overs rather than 190+ off 20. I think this is pretty unarguable.

Most people know whether a target is fair/unfair. Unfortunately none of the rules for truncated matches are that exact. We need to understand what dictates our common sense. 'Is the chasing team on course?' , 'Have the conditions changed? If so, in whose favor, and by how much?', 'Historically how do these matches go?' etc. We need to quantify these factors as well. The math needs to get better.

Like OzzieLondon was saying, D & L have actually checked the model we are currently using against recent T20 data, whereas you guys are just guessing and going off what seems right. It may be that additional data for rain affected games will induce changes, but I'd prefer that we rely on data to make our decisions rather than anecdotes. It comes down to whether we want justice or perceived justice.

Yawnnnn ... Too geeky for my liking.

contd The bottomline is just the fact that there are ample wickets in hand doesn't mean a batsman can score at thrice the normal run rate and get out in 1/3rd the time. This is merely a statistical fact and cannot account for the fact that cricket is played by humans. In layman's terms, the shrunken D/L assumes that a 50-ball 50 is equivalent to a 5-ball 20, or something of this sort. Anybody can tell you that its much easier to score the former than the latter.

Hence although the cutting version is rubbish, the shrunken one has its own flaws, and a golden mean needs to be introduced. Extensive T20 match data needs to be analysed and used, and since very little international T20 has been played thus far, statistics from domestic T20 leagues also needs to be considered.

There is still one doubt that I have - how is the vertical spacing betwween the lines done? how is the available resource lost when a wicket falls calculated? Why is it less for the 1st, 2nd wickets and more for 5/6th?

The major reason why the shrunken version of D/L will produce massive targets of the order of 13 runs per over if the chase is reduced to 5/6 overs is because the value of 10 wickets as a resource is overestimated. Agreed, you have all 10 wickets at your disposal for just 6 overs instead of conserving them overt a 20-over period, but does it really makes sense that the knowledge of having 10 wickets in hand will give batsmen reassurance to go berserk? 6 overs is a mere 36 balls; so if you ask any player, or even a person remotely understanding cricket if he needs 10 wickets to chase absolutely any target from 0 to 100 in 36 balls, he'll tell you he doesn't. 3/4 are enough, for that means you have 3/4 balls less to score those runs. In any case, its really difficult to lose 10 wickets in 36 balls even if you wanted to.

Alongside with the changes in D/L calculations, Normally in 50-50 match atleast 20 overs is mandatory for any sort of result, i.e. 40% of an innings. Similarly in the Twenty20 it should be a minimum of 8 overs (same 40%). Why did they choose 5 overs to be minimum???

Terrific article - please send a copy to Sky Sports commentators. Then we might have less of the"'Duckworth Lewis makes my brain hurt' nonsense from the very people who are paid a fortune to explain what is going on.

Thank you for a lucid explanation of D/L and its shortcomings for T20 cricket. One really has to have one's head in the sand to think that D/L for T20 is just fine as it is, as D and L believe. IMO, the fairest solution for T20 is going to be to both increase the asking rate, and reduce the number of wickets available. In ODIs you can get away with only increasing the asking rate for shortened matches, but in T20 it will be impractical, as your target of 87 in six overs shows. More appropriate would be something like a target of 70-75 with 5 wickets in hand, for a 6 overs innings.

Suppose that at a team has a 60% chance of winning when the rain comes down. The aim should be that whenever they resume, that's still at 60%. But actually, the more overs are lost, the higher their win probability becomes. They even acknowledge this in Q8 of their FAQ.

It's not a T20 problem, it's a problem with targets set *when innings are resumed with not long left* (which happens most often in T20).

Suppose a team are chasing 250 to win in an ODI, and the rain comes after 30 overs. The D/L par score is 120. Let's see what happens to teams on 90-2 and 130-2 if the innings is resumed with 1 over left. The win target becomes 129. So a team on 130-2 is declared to have won already: they go from having ~60% chance of winning to 100%. A team on 90-2 (needing 160 in 20 overs with 8 wickets in hand, so ~30% chance of winning) needs 39 from the last over, so they now have a 0% chance of winning.

They need to *preserve win probabilities across an interruption*. They don't.

The authors contradict themselves somewhat in this article. First they praise D/L for recognising that the value of wickets-in-hand diminishes the less overs there are remaining, and then they remodel their revised D/L method to place more weighting on wickets remaining. Which is it? Personally I believe the current D/L method applies as far as about 10-10 over matches, but beyond that it becomes a bit of a lottery. To use the West Indies example, chasing 60 off 6 overs, with one bad over (say 3 runs) means 57 off 5 which is up over 11 per over. England would have had a few such overs in their first innings I'm sure, but the Windies were not afforded that degree of freedom. I don't think you can blame the D/L system for any of this, it is simply not ideal to have a game that is only 6-overs in length. All that said, T20 is about crowd entertainment first and foremost (anyone who says different is a bit naive) - so from their perspective 6 overs is better than none..

Dear Margarath, Prof. Karandikar is one of the most respected statisticians in India. However, I agree that D/L seems much more sensible than Prof. Karandikar's 'shrunken trouser' model. I think looking at the example they have provided is ample to describe the problem with their model: to score 191 in 20 overs, with 36 powerplay deliveries, if you hit just 2/3 ofl the powerplay deliveries for, say, fours, you need to score 95 in the other 84 balls; to score 87 in 6 overs, with only 12 powerplay deliveries, even if you hit all fours, you still need 39 of 24 balls in the second case! You can clearly see which one is steeper. I feel that contrary to the claim of the authors, the D/L as it is now is much more sensible. The authors' model puts too much emphasis on the difference in no. of wickets in hand and underplay the value of dot balls, which does not seem that sensible. As for the WI-ENG game, just like WI knew it was going to rain, so did ENG, but they did not plan for that.

@kiwimike: The parts of the curves with 7 and 8 wickets down and 19 or more overs remaining can never actually be used in a 20 over match, but can be if a team loses 7 wickets in the first 30 overs of a 50 over match. In that case, there is a good chance that the team will not last its full allocation of overs. Losing an 8th wicket makes it even more unlikely that the team will bat its full overs, unless the batsmen make very little attempt to score runs, in which case they are wasting the overs anyway. That is why the loss of the 8th wicket with 20 overs to go is worth so much.

In theory, using the last 20 overs of the D/L method for a 20/20 game should work fine. The problem is, the D/L method hasn't been adjusted to reflect modern trends in ODIs in general. If a team plays conservatively, keeps wickets in hand, and gets to 90-0 after 30 overs, it says they will only score another 117 in the final 20 overs to end on 207. That is a reasonable figure for 20 years ago, and optimisticly high for the early days of ODIs, but nowadays, that's way too low. 20/20 is just focusing attention on this problem with D/L in general.

The analysis presented here is good, but I am also concerned about the "granularity" of the different "cricketing terms" in T20, e.g. we may keep the current D/L score, but redefine a 4 as a 3 and a 6 as a 4.5, so that the team batting second may compete even if they are not capable of hitting three sixes in each over (which is exceptional stuff) or the first team may have some advantage, if one of the overs goes bad. In ODI's the possible score per bowl averages out quite well, but in T20s, specially in the reduced ones, I don't see it balancing out well. Secondly, we may redefine the ball per over (the problem is that we cannot make it fractional, like we can do with the score) or we may give the option of a bowling change at certain points within an over, for reduced matches. I don't think that reducing the number of wickets is a good solution, that would make the team batting second much more cautious and will reduce the charm and passion of the game.

I was watching an interview with Mr D & L and they had recently put a whole lot of T20 data into the models and found the current D&L works for T20. Personally I think the problem here is WI. DL assumes a pretty even distribution of batting talent in your side. WI have 1 batsman who would eat 60 of 6 for breakfast, and another 10 that would stuggle against my daughters under 14 side. One must also remember that to get 8 an over is easy (hence a <160 score usually gets mowed down) because you only need one boundry an over. (1x4 + 4x1 + dot ball). To score at just 3 runs an over more you need to take twice the risk (2x4 + 3x1 + dot ball) and therefore loose wickets at twice the rate. When Gayle came back out WI needed 160 more with 17.4 overs at a rr of 8.9 with 10 wickets. Now like I said, unless Gayle bats most of the 20, against good bowling I would not expect the WI to get the runs but I would back Australia or England to do it from there more often that not.

(continuing) Also, scoring quickly against the hard ball may not be as easy as it sounds above; ask the Asian teams! It is a skill, and WI won because they had it; Zimbabwe lost because they did not.

Middle of the innings breaks can help the fielding sides as well. Take the England-WI match for example. England could have started with their best bowlers, and could have tried to keep them bowling until the rain break, and could have nullified the disadvantage of reduced bowling quotas if the overs were reduced post-break. Their game plan could have involved opening the bowling and continuing with slow bowlers until rain came in (just like WI kept hitting with the same fact in mind), but they went on with Sidebottom spraying the ball around.

Also, there is very little margin of error for the batsmen as they have to start hitting without getting set, and one dot ball counts a lot more compared to a full match. So, in the balance of things, I'd side with D/L.

One thing I do have to disagree with in the article, however, is your comment that, in the England vs Windies match, the Windies came out to bat believing they would play their full 20 overs. This is rubbish. Everyone - players, commentators, spectators - knew that the rain was coming and that D/L would come into play. The Windies knew that the more runs they scored before the break, the better the D/L calculation would be for them as D/L assumes that a rain break is unexpected and that any play before a rain break is as would be in a normal innings. I realize that D/L can't really compensate for this, but let's not kid ourselves that those 1st 14 balls the Windies faced were, to all intents and purposes, were played by them on a D/L reduced over basis, but under normal 20 over rules

I guess we are forgetting a few points here. First of all, the point about the powepoint overs and 'hard white ball' applies to one day matches as well; so it is not that 20-20 is all that different from 50-50 matches in that respect.Secondly, as we have repeatedly seen in 20-20 matches, wickets are not as crucial as they are in 50-50 matches; in the sense that many teams do not lose beyond 6 or 7 wickets. So, we should not use the arguments like that if a team is chasing 160 in 20 overs, they should have to score 80 in 5 because they have 10 wickets. In the context of 20-20, the significance of losing a wicket or two is much less, (that is what Duckworth tried to suggest in his interview.) There is a limit to which a batsman can succeed trying to hit every ball, scoring 8-an over for 20 overs with 5 bowlers is much easier than scoring 80 in 5 overs with the same 5 bowlers (look at the powerplay scores if you do not believe me.) (cont.)

I firmly believe that DL is not a good solution for T-20 cricket. Most of the time an easy target given to the team who bat second. I think the best solution is. RE MATCH ha ha ha ha

Thanks for a good article. Totally agree that D/L is flawed for T20 in it's current format and something needs to be done. Your suggestions sound intriguing and hopefully a specific T20 D/L method will be created soon. As well as the IPL and international results to work with, we could also use the other domestic T20 matches to use in the database - at least until we have sufficient international matches to use.

England could have just bowled a few dot balls instead of a bunch of wides. That might have helped.

Fantastic article! I hope is implemented as soon as possible in all T20s (if it was up to me I would start immediately with the semi-finals, but I know that is not going to happen). I am an England supporter, so perhaps feel the need to replace D/L more strongly than others, but actually it did not cost England. If the system is not changed, a semi-final or final is going to end in absurdity. God knows what the reaction would be if India lost a final after scoring 191 runs while the opposition coasted to 60 off 6 overs.

The reason they haven't changed this is because they *did* do a study on just 20-20 matches, and the curves are almost identical to your 'sawn off trousers' model, i.e. the one they are using, according to Mr. Duckworth. This is why they haven't changed it yet. Maybe you should do the statistics before writing an article. I have to say I agree with ABail on the reduction of T20 matches. I think they do need to be reduced, but should be reduced to the same % as 50 over matches can be - i.e. 8 overs. We wouldn't think it fair to reduce a 50 over match to 12 overs would we? Because that is what we are doing with T20 at the moment. Possibly this is why D/L seems a little odd, if we calculate some 'targets' for a reduced to 12 overs 50 over match, they might seem equally odd.

There is also the starnge result where the value of the wickets is not the right way around. Just looking at the 20-20 graph it appears that the value of the 8th wicket is more than the first 5 combined. Your proposed graph makes that closer but it still ranks bowlers wickets above the batmen, which can't be right. The lines should be changed so the top order is worth more, or at least no less than, the tail.

This is a wonderful article. A friend and I had had a similar discussion, but didn't get as far in our analysis. The issue in the revised 2020 scenario here is the penalty for the first wicket seems too high, and perhaps it will need a model that takes that into account.

I agree that Duckworth Lewis is an unparalleled method for setting targets for odi's, and I have used it to build a score predictor, with quite considerable success. It seems to estimate the score to within 5% about 80% of the time after 40 overs. That is really quite impressive. It does not have nearly the accuracy with 2020, part of which is the smaller sample size, and also the nature of 2020 with the inherent twists and turns.

Overall congratulations on a great article, and a well thought out first look at this vexing problem.

Nice article...finally we get a good simple explanation of how D-L has been used for 20-20 matches! However, the targets for Zimbabwe and West Indies seem similar (run rate required are roughly the same) even though England scored 191 and Sri Lanka only 173.

We need to see one more aspect here : For how many overs England had maintained the same or higher run rate along with the considering fall of wickets in those overs as compared to previous overs i.e. if ENG was at 90/1 in 10 overs and then they maintained 9.5 RR till the next 5 overs without losing wkt OR let say they maintained NRR of 9.5 but lost 2 wkts at the end of 15 overs ..... In the second case ENG had done a better job becuase maintain the NRR despite of losing wkts is far hard task as compared to the first case !! While considering the revised target for the chasing team such overs and their NRR should be taken into consideration rather than simply considering total runs made in 20 overs because the team who is chasing is required to play only a small portion of the 20 overs which should be compansated with such high RR overs by the team batted first.

" West Indies came out to chase believing they would play their full 20 overs"

Not true. They knew of the weather reports and clearly had local knowledge of the conditions. The 14-ball blitzkrieg was a deliberate tactic as part of a D/L exploitation strategy.

Having said that, great article guys!

This method won't work, I'm afraid. It will introduce more serious flaws. For example, in the England - West Indies match, while chasing 191 from 20 overs, say rain interrupt after 10 overs have been played and the match is called off. WI"s target from 10 overs would be just 64/0 or 69/1 or 76/2. How can that be fair? WI would win the match despite being well behind the eight ball.

1. The data used for the original Duckworth/Lewis tables included some matches which, although scheduled for 40 or more overs, had been reduced to 20 or fewer overs before the start of the match. There may not have been much data to fill those part of the curves, but they did not need 50 over matches with no wickets in the first 30 overs to fill them.

2. The authors call for D/L to recalculate the curves using actual T20 data. If you read the last two paragraphs of the link "Duckworth defends rain-rules formula", you will see that they have recently done exactly that, and concluded that T20 scoring patterns fit to their curves.

It should be a 'floating target'. Runs required should increase when a wicket falls. Too many teams send tailenders early in truncated games to reduce a 50 off 30 to a 10 off 10 despite losing 3-4 wickets in doing so. It's only fair to tell a team,' your target is 60 off 6 overs, lose a wicket and it's 65, lose 2 and it's 71, lose 3 and it's 78' and so on. More games will go down to the wire. Some may say it's more confusing to teams and viewers, but i say it's fairer, and no less confusing than a D-L chart!

Good article. The facts contained within it emphasise how disappointing Duckworth's initial response was to the concerns raised about DL in 20/20 matches following the England-West Indies game. He basically argued that there wasn't a problem, and sought to question the motives of those saying that there was. I thought he acted pretty unprofessionally in seeming to not even acknowledge that there may possibly be some kind of difficulty with the application of his method to the shortest form of the game.

All those who came on this site and dismissed Collingwood's remarks after the West Indies game as a just an Englishman having whine after losing need to learn what the difference is between a whine and the articulation of legitimate concerns.

Excellent, well-considered article that offers an opportunity for sensible discussion to remove what is an obvious unfairness for teams batting second in t20s. I hope this is investigated further.

The group matches in the World T20 referred to in this article came very close to undermining the credibility of the tournament. It must be lunacy to have a situation where one side bats 20 overs and gets a big score and another side can overtake the run rate needed in the first 5 overs when the fielding restrictions are in place and then win if it rains. How can that be a credible contest. D/L needs to take account of that urgently. WI should have been set about 75 in the first 5 overs against England for a more realistic comparison.

This is a really nice article clearly explaining all the permutations and combinations of D/L . Another possible suggestion would be like the one that was made by Anil Kumble in a newspaper article reg: reducing the no: of wickets when a revised target is set, just as it is there for the "super over" .

This is a really nice article clearly explaining all the permutations and combinations of D/L . Another possible suggestion would be like the one that was made by Anil Kumble in a newspaper article reg: reducing the no: of wickets when a revised target is set, just as it is there for the "super over" .

yeah true...T20 provides too less sample space for the D/L method to work out any suitable playing conditions that is acceptable for both the teams...

This is a wonderful article. Though I was not a fan of Collingwood's whining, I found it unfortunate that Duckworth did not see his concerns as valid. WI ultimately had to score at 10 runs per over in only 6 overs, this after England scored at almost 9.6 in 20. It obviously was not fair. And while your projections may seem steep, we must remember that scoring at almost 10 per over in 20 overs is no easy feat, and as such the target bar must be set high over small sets of overs. I must say that even I, as a West Indian, felt for England, and also found it almost laughable that Zimbabwe got such an undeserved opportunity to go through. I hope you gentlemen follow through on your proposal and that the necessary revisions come as a result.

I will say, though, that I really don't believe in revised targets for T20s. As an avid cricket fan and lover, 5 overs doesn't sit well with me as constituting an innings. It's almost a mockery of the game. Imagine a 100M race being cut to 20M!

So you've taken the existing D/L ODI curves and transplanted then onto Twenty/20 variables, i.e. 20 overs.... why couldn't D/L understand this in the first place??

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So you've taken the existing D/L ODI curves and transplanted then onto Twenty/20 variables, i.e. 20 overs.... why couldn't D/L understand this in the first place??

This is a wonderful article. Though I was not a fan of Collingwood's whining, I found it unfortunate that Duckworth did not see his concerns as valid. WI ultimately had to score at 10 runs per over in only 6 overs, this after England scored at almost 9.6 in 20. It obviously was not fair. And while your projections may seem steep, we must remember that scoring at almost 10 per over in 20 overs is no easy feat, and as such the target bar must be set high over small sets of overs. I must say that even I, as a West Indian, felt for England, and also found it almost laughable that Zimbabwe got such an undeserved opportunity to go through. I hope you gentlemen follow through on your proposal and that the necessary revisions come as a result.

I will say, though, that I really don't believe in revised targets for T20s. As an avid cricket fan and lover, 5 overs doesn't sit well with me as constituting an innings. It's almost a mockery of the game. Imagine a 100M race being cut to 20M!

yeah true...T20 provides too less sample space for the D/L method to work out any suitable playing conditions that is acceptable for both the teams...

This is a really nice article clearly explaining all the permutations and combinations of D/L . Another possible suggestion would be like the one that was made by Anil Kumble in a newspaper article reg: reducing the no: of wickets when a revised target is set, just as it is there for the "super over" .

The group matches in the World T20 referred to in this article came very close to undermining the credibility of the tournament. It must be lunacy to have a situation where one side bats 20 overs and gets a big score and another side can overtake the run rate needed in the first 5 overs when the fielding restrictions are in place and then win if it rains. How can that be a credible contest. D/L needs to take account of that urgently. WI should have been set about 75 in the first 5 overs against England for a more realistic comparison.

Excellent, well-considered article that offers an opportunity for sensible discussion to remove what is an obvious unfairness for teams batting second in t20s. I hope this is investigated further.

Good article. The facts contained within it emphasise how disappointing Duckworth's initial response was to the concerns raised about DL in 20/20 matches following the England-West Indies game. He basically argued that there wasn't a problem, and sought to question the motives of those saying that there was. I thought he acted pretty unprofessionally in seeming to not even acknowledge that there may possibly be some kind of difficulty with the application of his method to the shortest form of the game.

All those who came on this site and dismissed Collingwood's remarks after the West Indies game as a just an Englishman having whine after losing need to learn what the difference is between a whine and the articulation of legitimate concerns.

It should be a 'floating target'. Runs required should increase when a wicket falls. Too many teams send tailenders early in truncated games to reduce a 50 off 30 to a 10 off 10 despite losing 3-4 wickets in doing so. It's only fair to tell a team,' your target is 60 off 6 overs, lose a wicket and it's 65, lose 2 and it's 71, lose 3 and it's 78' and so on. More games will go down to the wire. Some may say it's more confusing to teams and viewers, but i say it's fairer, and no less confusing than a D-L chart!

1. The data used for the original Duckworth/Lewis tables included some matches which, although scheduled for 40 or more overs, had been reduced to 20 or fewer overs before the start of the match. There may not have been much data to fill those part of the curves, but they did not need 50 over matches with no wickets in the first 30 overs to fill them.

2. The authors call for D/L to recalculate the curves using actual T20 data. If you read the last two paragraphs of the link "Duckworth defends rain-rules formula", you will see that they have recently done exactly that, and concluded that T20 scoring patterns fit to their curves.