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The dummy's guide to Duckworth-Lewis

During one of the discussions in the course of the Aiwa Cup telecast, Tony Greig asked Robin Jackman what he thought of the Duckworth/Lewis (D/L) method to reset targets in one-day matches affected by rain.

Jackman attempted a serious enough explanation: "It tries to take into account both the overs remaining and the wickets remaining ..." and then trailed off. But he did say: "I have talked to Steve Waugh about it and he thinks it's the best we have".

It doesn't surprise me that Steve Waugh, arguably the most intelligent cricket captain in the business today, at least tried to understand what D/L is all about. Most other captains (e.g. Sanath Jayasuriya) are quite content to say: "If we play good cricket we'll win. We don't worry too much about D/L .."

Given the high stakes in the game, and the great passion that cricket arouses, this attitude simply won't do. Cricketers and cricket commentators should, like Steve Waugh, take half a day off to understand the D/L method. This could result in better winning strategies for the team and more enlightened discussions on TV (while I usually like everything that Geoff Boycott says on TV, I can't forgive him for calling the D/L method "rubbish").

Old rain rules

What then is D/L all about? Before we begin to explain the method we'll consider a few examples and review some of the rain rules used before D/L came along.

Let's start with an extreme (fictitious) example. Imagine that in a key match in 1990 India bat first and score 300/2 against Pakistan in 50 overs. Returning to the field the Indians bowl brilliantly and reduce Pakistan to 151/9 in 25 overs. Just a ball separates India from a famous victory, but that ball can't be bowled! It begins to rain cats and dogs and the match can't resume. Having scored 151 in 25 overs, Pakistan are a whisker ahead of the required run rate of 6.0. They are declared winners ..

This rule, in which the team having a higher run rate won, was used till 1992 or so. Realising that the rule tended to favour the side batting second, and attempting to correct this situation, the Australian Cricket Board went to the other extreme with its 'most productive overs' rule.

This rule went as follows: suppose Team 1 scores 250 in 50 overs, and a rain curtailment leaves only 40 overs for Team 2. Then the rule first required one to arrange the runs scored by Team 1 in every over in a descending order (e.g. 12,9,8,8,7,7,5,5,5,4,4 ... 3,2,2,2,1,1,0,0,0,0) and then 'chop off' the runs scored in the last 10 overs. In our example, Team 2 would be required to score 250-11+1=240 in 40 overs to win.

Another way to understand this rule is to think of the 'Manhattan' and imagine that it is 're-arranged' so that the tallest building is to the extreme left, followed by the next tallest building and so on .. till you reach all those one-storey and 'zero storey' buildings, corresponding to the most economical overs, at the extreme right. If eight overs are lost, chop off the eight tiniest buildings from the right and revise the target to be the sum of the runs depicted by the height of the remaining buildings plus one. If two more overs are lost, chop the next two from the right ..

This was blatantly unfair to Team 2. We all recall how South Africa, requiring 22 off 13 balls to reach the World Cup finals, were suddenly asked to score 21 runs in one ball when a short shower took two overs away! (the rule merely chopped off two 'tiny' overs, which yielded 1,0 runs, from the right extreme of the re-arranged 'Manhattan'). Some of us will also recall how India's target, while batting second in an early World Cup match against Australia, kept getting stiffer with every brief shower. Despite very good knocks by Azharuddin and Sanjay Manjrekar, India lost narrowly because the overs which kept getting chopped off were relatively low run overs (the irony was galling: if only Kapil Dev hadn't been so economical in his bowling, our victory target could've been lower!).

Although the Australian Board made some effort to salvage this 'most productive overs' rule, it eventually died as it deserved to. We then went through a series of other rules such as the parabola method and Clark curves which, although not obviously unfair to either team, had their blemishes and glitches. It was a terrible situation: everyone agreed that all existing rules were inadequate, but no one was coming up with a winner. And then came Duckworth/Lewis .. (actually preliminary studies on the method began in 1992, and the D/L proposal was officially submitted to the TCCB, now ECB, in October 1995).

Enter Duckworth/Lewis

The first thing to understand about D/L (and Robin Jackman started saying it on TV) is that it recognises that there are two resources available to the batting side: overs and wickets, and argues that a combination of these two resources must be used to reset the victory target. Think of our example: if Pakistan had been 151 for no loss in 25 overs we wouldn't have grudged them their victory in the rain-curtailed match; it was the fact that they had already lost 9 wickets which bothered us. Put another way, it was the fact that the rule considered only the runs per over rate, and not also wickets remaining, which was the essential weakness.

The second merit of D/L, which I consider to be the real breakthrough, is that it takes into account the timing of rain interruption/s and allows for multiple interruptions; the earlier rules (parabola method and Clark curves) worked reasonably well for interruptions between innings but went into a difficult tizzy when required to reset targets after interruptions occurred in the middle of an innings.

One aspect of D/L, which was severely 'rubbished' by Geoff Boycott, Ian Chappell and others, was the fact that a victory target could be revised upwards. But consider the following scenario: having chosen to bat on a good batting wicket, Team 1 are aiming to get about 275 in their 50 overs. After 25 overs, Team 1 are cruising comfortably at 100/2. Then it rains horribly and the match referee determines that only 25 more overs can be played. So Team 1's innings is terminated and Team 2 are asked to bat for 25 overs. What should Team 2's target be? Surely not 101! After all, Team 1 were 'pacing' their innings imagining that they had a full 50 overs, and didn't take undue early risks. Team 2 know from the start that they only have 25 overs and won't mind losing 5, 6 or even 9 wickets as long as they get to their target! What options do we then have to ensure that it's a fair contest? Only two in this case: we could ask Team 2 to reach their target with the same number of wickets lost or we could ask Team 2 to score more than 100 (about 170?) to win. Reducing the number of wickets still appears highly impractical in one-day cricket .. so we must perforce choose the second option (in many cases there can be other options too, but each is riddled with rather serious problems. We will discuss these towards the end of this narrative).

D/L table and the concept of resource percentage

So how do we start our D/L orientation programme? It's all about looking up one table. The table has 'rows' and 'columns' (see the illustration).

	Wickets Lost
Overs left	0	2	5	7	9
50	100.0	83.8	49.5	26.5	7.6
40	90.3	77.6	48.3	26.4	7.6
30	77.1	68.2	45.7	26.2	7.6
25	68.7	61.8	43.4	25.9	7.6
20	58.9	54.0	40.0	25.2	7.6
10	34.1	32.5	27.5	20.6	7.6
5	18.4	17.9	16.4	14.0	7.0

The rows correspond to the number of overs left and the columns to the number of wickets lost. Let's take a first look at this table and try to read off the value when the overs left are 50 and the wickets lost are 0. The tabulated value is 100.0. Let's try again (and get slightly more ambitious): suppose the overs left are 20 and wickets lost are 7. First run your finger down the 'overs left' column till you reach 20; then move your finger along this row till you reach the entry corresponding to 7 wickets lost. The tabulated value is 25.2. Got it? Good!

How do we interpret this tabulated value? It is defined to be the resource percentage remaining. In the first instance the value is 100% -- this make sense, you have all your wickets and overs intact and have so far 'lost' no resource! In the second instance it is 25.2 -- which too makes sense; after batting for 30 overs and having lost 7 wickets your resource remaining is only about 25%; or, in other words, you have already used about 75% of your resources.

A few clarifications before we proceed: the table accompanying this write-up is only an extract; the original table lists the resource percentage (RP) available corresponding to every wicket lost (0,1,2,3,..up to 9) and every ball left in every over (there are values corresponding to 50,49.5,49.4,49.3 ..etc. right up to 1,0.5,0.4,..0.1 balls left). So a rain interruption at every stage of the game can be taken into account.

How did Duckworth and Lewis come up with these RP values? The numbers arose out of a detailed analysis of match data spanning hundreds of matches and appear to 'capture' most of the variation in one-day cricket matches. Occasionally the numbers can appear a shade unrealistic. For example, if India are 10/1 after two overs the D/L table indicates that the RP remaining is still 91.1%; but, if Sachin Tendulkar is the batsman who is out, most of us might feel that the RP remaining value should be 75%! But no rule can really account for the phenomenal ability of a Tendulkar. For most 'normal' batsmen, the RP value appears very reasonable.

Calculating resource percentages lost and available

We are now ready for our first exercise involving D/L: In an India-New Zealand one-dayer held at Taupo in January 1999, India scored 257/5 in their full 50 overs. New Zealand (NZ) were 168/3 in 30.4 overs when the floodlights failed. Who had the upper hand in the match at that point?

Since India's innings ended in the normal course they used up their entire (100%) RP. NZ had 19.2 overs left and 3 wickets lost. At that point, NZ's RP remaining was 49.5% (I am reading off this value from my 'full' D/L table; it can't be read off from our mini D/L table), so they had, at that point, used only about half (100-49.5=50.5%) of their total RP. This figure suggests that NZ were probably winning at that stage; being only 90 runs away from their target, and having enough overs and wickets left, this conclusion seems reasonable.

It took 50 minutes for the match to resume and, consequently, an unforeseen loss of 11 overs in the NZ innings. Would the interruption affect NZ's chances? What would be NZ's new winning target? We shall soon find the answers, but what's important (and reassuring) is that in practically every such case D/L provides a very satisfactory solution.

Let's take a second example. In a 50-over match, Team 1 make 200/2 in 40 overs when rain stops play. Team 1's innings is closed at that point and, when play resumes, Team 2 start their 40-over innings. What are the RP remaining values of Teams 1 and 2?

If we look up our mini D/L table (which we can in this case) for 10 overs left and 2 wickets lost we get a tabulated value of 32.5. So when the rain came down, Team 1 still had 32.5% of their RP remaining, i.e. they used only 100.0-32.5=67.5% of their resources. What about Team 2? It has 40 overs and 10 wickets left and our tabulated value of RP remaining is 90.3%.

We therefore have a situation where Team 1 used only 67.5% of their available resources while Team 2 have the opportunity to use 90.3% of their available resources. Clearly Team 2 have the advantage! So how do we neutralise this advantage? Well, the best course is to ask Team 2 to score more runs than Team 1, in their 40 overs, if they are to win. How much more? We'll explain that a little later.

We are now almost ready for our first target resetting tutorial using the D/L method. But, before we do so, let us make sure that we understand what exactly is meant by RP remaining and RP available. 'RP remaining' is easy: it is simply the value that you read off from the D/L table. 'RP available' tells you how much of the RP was used or can be used by each team. The RP available for Team 1 and Team 2 is respectively designated as R1 and R2.

Setting revised targets

Let us return to the Taupo match where a floodlights failure has reduced NZ's innings to 39 overs. What was NZ's revised target?

To determine revised targets using the D/L method one only needs to know the values of R1 and R2. In our example, R1 (RP available for India) was 100.0% since India used all their available batting resources. When NZ came off following the floodlight interruption, their RP remaining value (read off from the D/L table, corresponding to 19.2 overs left and 3 wickets lost) was 49.5%. Then 11 overs were lost. So, when play resumed, NZ had 19.2-11=8.2 overs left and, of course, still 3 wickets lost. The corresponding tabulated value in the D/L table is 27.2% RP remaining.

Let's quickly understand what's going on. The disruption meant that NZ had 'lost' a RP equal to 49.5-27.2=22.3%. So while India have a RP available (R1) of 100%, NZ's RP available has been depleted by 22.3% due to the floodlight failure; in other words, NZ's RP available (R2) has become less than 100%; it is now 100% minus 22.3%=77.7%. Being lower than India's 100%, NZ's winning target must be reduced. By how much? Now it's down to school arithmetic. We argue as follows: with a RP available (R1) of 100%, India scored 257. So with a RP available of 77.7%, NZ must score 257 x 77.7/100 = 199.69 to level. Since we can't accept run fractions we knock off the 0.69 and add 1 to get a target of 200 to be scored in 39 overs. (Postscript: NZ scored 200/5 in 38 overs to win the match).

So we've done it! We have just completed our first exercise to reset a target using the D/L method. And it wasn't a trivial exercise because the freak interruption occurred in the course of the NZ innings; not between the India and NZ innings.

Let us now complete the discussion of the match in which Team 1's innings was curtailed at 200/2 in 40 overs and Team 2 had to be set a winning target in its 40 overs. We have already seen that the RP available (R1) to Team 1 was 67.5% while R2 was 90.3%. So since Team 2 has more resources, their target must be raised upwards.

How do we do it? In this case we proceed as follows: Calculate the difference between the RP's available, i.e. 90.3-67.5=22.8%. We then ask how many runs is this advantage worth? Since, on the average, a typical one-day score in a 50-over match is 225, we estimate the 'run equivalent' of 22.8% to be 22.8/100 x 225=51.30. So Team 2 must score an additional 51.30 runs to attain parity with Team 1's RP -- and their victory target is therefore 200+51.30+1=252 runs to win.

This procedure must raise a few eyebrows. When R2 is less than R1 (as in our India-NZ example), we simply multiply Team 1's score by R2/R1 and add one run to obtain Team 2's target. But when R2 is greater than R1, we don't adopt the same procedure; we do it slightly differently as discussed above. Why? In their guide, D/L only explain that "there are very good reasons" for not doing so. (In a remarkable analysis, which appeared in The Hindu in March 1999, Chetan Shah discusses these 'reasons' in greater detail). As far as we are concerned, it is enough to satisfy ourselves that in both cases, D/L returns fair verdicts (which it does, in fact in practically every case).

And why use 225 as the average score in a fifty over match? Why not 250, which may be closer to the real average, at least in the Indian sub-continent? It turns out that the difference has a very minimal effect on the reset target (no more than 1-4 runs in most cases). But, as D/L wrote in response to my e-mail, nothing prevents the Indian or Pakistani Board from changing 225 to 250 "as long as all the parties involved are aware of the change from the beginning".

Indeed, one of the joys of attempting this 'popular' introduction to the D/L method has been the wonderful interaction with Frank Duckworth and Tony Lewis, and their patience in responding to every question (including some really dumb questions: when I asked D/L what would be a good run scoring strategy if there are frequent rain interruptions, D/L's answer made me feel silly: "just keep playing your normal game", they wrote, "and D/L will ensure that there are no rude surprises").

Par scores

A key question which bothers the cricket enthusiast with dark clouds looming across the horizon (or when the TV coverage shows the first few drops of rain on the glass panel) is: "who will win the match if it is abandoned right now?". A recent example is India's World Cup match against England. India scored 232/8 in their 50 overs and when England were 73/3 after 20.3 overs a heavy downpour caused the match to be called off for the day. It was a tense night for the Indian supporter. His horror scenario was that Thorpe and Fairbrother come out the next morning, slam a few quick boundaries to reach 100/3 at the 25-over mark, and then no further play is possible! (In such a situation India's RP (R1) was 100%, England's RP (R2) with 25 overs left and three wickets lost was 100-57.1=42.9%. With R2 less than R1, England's winning target was 232 x 42.9/100= 99.52+1, i.e., exactly 100. So England would've won!). That's why the real turning point was when Thorpe was adjudged lbw by Javed Akhtar. Not only had England's most dangerous batsman gone, but England's RP available (R2) had gone up! (If England had been 100/4 at the 25-over mark, they would have lost because D/L had pushed up the winning target to 232 x 48.8/100=113.21+1=114).

Now it can be no great pleasure to keep calculating who will win after every over is completed, or every wicket falls, in a rain-threatened match. That's why D/L has come up with the concept of the par score. The par score is defined as the score that would tie the match, under the D/L method, if the match were abandoned at that point. In other words, it is simply the 'break-even score' under the D/L method. If your score at the point of interruption is greater than the par score, you win; if your score equals the par score it is a tie; if your score is less than the par score you lose!

How do we calculate the par score? Exactly the same way that we calculate the D/L winning target! Let's take the example of the match (discussed above) where Team 2 have been set a winning target of 252 in 40 overs in reply to Team 1's 200/2 in 40 overs. Suppose Team 2 are 190/5 after 30 overs when conditions get murky. Who's winning at that point? We have already calculated that, when Team 2 started their innings, R1=67.5% and R2=90.3%. Now the RP remaining at this point, corresponding to 10 overs left and 5 wickets lost (look at your mini D/L table), is: 27.5. So out of a total RP of 90.3% which Team 2 can use, they still have an unused RP of 27.5%, i.e. they have so far used a RP equal to 90.3-27.5=62.8%. If the match is stopped at this stage, then we have the situation where Team 2 could only use a RP of 62.8%. In these circumstances R1=67.5%, R2=62.8%. Since R2 is less than R1, the par score, using the D/L method, would be 200 x 62.8/67.5=186.07. So Team 2's winning target at that point was 187. Since they are 190, they would win by a whisker!

A moment's reflection will tell you that it is an easy matter to have a full set of pre-calculated par score values for every team batting second (Team 1's score and R1 can't change; only R2 keeps continually changing, but the D/L table tells us exactly how it will change). In fact D/L has a computer program (called CODA) which can give par score printouts at every stage for the captains and scorers to keep track (during the recent World Cup in England we saw some scoreboards showing a figure like +12, meaning that the batting side is currently 12 runs ahead of its par score as calculated by the D/L method).

A winning strategy?

Indeed the par score concept offers the side batting second a most elementary winning strategy: always stay ahead of the par score. Chetan Shah's illuminating analysis in The Hindu also suggests a potentially very rewarding strategy for Team 2: when you are chasing a large total, and there's a possibility of a long disruption (15-25 overs?) due to rain, work hard to keep your wickets intact, even if it means batting at a relatively lower run rate. I'll mention one example from Shah: Team 1 bat for 50 overs to score 300. In reply Team 2 are 115/2 in 25 overs when rain stops further play. Who wins? Surprisingly, D/L decides that Team 2 wins! (R1=100%, R2=38.2%. Par score=300 x 38.2/100=114. So 115 wins). D/L agree that this is a "weakness" and even have a better mathematical model to correct this. But they can't, because a superior mathematical model will require a computer!

D/L for the TV viewer

I find this surprising. To have to compromise mathematical sophistication because there might not be a computer available at the ground appears ridiculous. If we are, in particular, looking at 'big' matches for which TV channels use equipment worth millions of dollars, surely a modest Pentium computer is a peanut. The stump vision camera (which, I'll confess, shows me nothing interesting in most cases) costs thousands of dollars!

I don't really think that's the way D/L should go. In fact there's a strong case for TV channels, and interactive Web sites, to embrace D/L in a big way. Consider, for example, the par score. This is a concept full of exciting possibilities for TV. I suggest that just as the telecast shows 'Manhattans' and 'worms', it should show charts making even ball-by-ball comparisons with the par score. A good commentator (who, today, is bothered by D/L, and is trying to hide this confusion with a brave grin) can actually heighten the match drama by using par score charts. Examples: "the loss of Thorpe's wicket isn't going to help England ... there you are! England now need 14 more runs to get at the 25-over mark if they are to stay in contention for that Super Six berth"; or "after that blistering onslaught by Klusener, Cronje's actually smiling. Two overs ago, South Africa were praying that it wouldn't rain .. now they'd like to see the biggest cloudburst of the decade!".

To be sure, showing par score charts will be a little more difficult than showing the crawling worm. One complication is that a new par score curve has to be generated every time a wicket falls. But that's not a difficult problem. Indeed, once TV producers participate in the effort to make D/L more 'friendly', and include D/L software to their inventory of gadgets, this truly remarkable initiative will finally receive the acclaim that it deserves. It will also permit Duckworth and Lewis to use their more sophisticated model to give even sharper results. I really believe that animated par score charts on TV will add to the live cricket drama (imagine how dull it would be if we didn't have TV run out replays and third umpires pressing the green light for Sanath Jayasuriya).

The unnecessary confusion between D/L and net run rates

There is apparently an ICC ruling that if the D/L method is used to reset the target in a rain-affected match, then the scores in that game will not be used in the calculation of net run rates (NRR). The NRR calculation is simple: suppose India meets Australia and Sri Lanka twice in the Aiwa Cup league. Then India's NRR is calculated as follows: first add the runs scores by India in their four matches and divide it by the sum of overs they faced in each of the four matches, next add the runs scored against India in the four matches and divide by the sum of overs bowled by India in the four matches. The difference between the first and second terms gives India's NRR. In the normal course a win fetches the winning side a positive NRR component while the losing side gets an equal negative NRR component. Now, in case D/L is invoked, it is possible that the losing side actually scores at a higher run rate than the winning side. This messes up the NRR calculation, and, worse still, questions the very rationale of using the NRR to resolve a tie in league points. Clearly, therefore, D/L and NRR cannot easily co-exist. The ICC ruling, if true, is a clumsy effort to shrug off this incompatibility and can in some cases be very unfair (although, in the Aiwa Cup, Sri Lanka were fair victors). The best course is to replace the NRR criterion with a D/L-based scheme. Such a scheme is implicit in the par score definition. I quote from the D/L guide (page 45): "The D/L methodology allows a match result to be expressed in runs whichever side wins. If Team 1 wins, there is no problem as the margin is in runs anyway. But if Team 2 win, then all we need to do is to note how how much their score at the point of victory exceeds the par score". The suggestion therefore is to use some scheme based on the margin of runs, as determined by D/L, instead of the NRR.

Other D/L odds and ends

We will finally return to the question of raising Team 2's target upwards. There are many cricket enthusiasts who still feel uncomfortable with this option (I know Rediff's Prem Panicker doesn't like it. In his match report on the abandoned India-West Indies final at Singapore he calls it a "travesty of justice"). The question is essentially related to a cricket regulation which states that if a certain number of overs are lost due to interruptions in a one-day match then the lost overs should be divided equally among the two teams. So if 30 overs are lost out of the total of 100, both Teams 1 and 2 should lose 15 each. Now if the disruption occurs after Team 1 has played 30 overs then the rules require Team 1 to get only five more overs while Team 2 gets 35. It's clear that Team 2 would enjoy a much higher RP available which must somehow be compensated. There is of course the option to 'tinker' with the overs allotted to Teams 1 and 2 so that the number of overs is different but the number of runs to be scored is nearly the same. This option, which might involve some juggling using computers, is interesting but violates the regulation of equal distribution of lost overs between teams. Any other options? Well, we could, for example, pretend that there has been no disruption and ask Team 1 to bat on for their full 50 overs when the match resumes and somehow hope that Team 2 also get their full quota of 50 overs -- this might be possible if the ground has lights, good super soppers etc. or if, like in the recent World Cup, we have an additional day available, but, in a one day match, that's always a dangerous option because there may no result if Team 2 can't get 25 overs! It is in view of all this that the target raising option still appears to be the most feasible (it is also good viewing especially for those who like cricket of the "bang, bang" variety).

A question often posed to D/L is how their models take into account pinch hitting, especially in the first 15 overs. D/L say that they are not particularly worried about distortions caused by a rampaging Jayasuriya or Afridi because with the faster scoring there is also the tendency to lose more wickets. So while run rates are higher, the RP values are not very different. "We still don't have enough evidence to justify a change in our tables", they write, "but we will continue to monitor the trend".

Finally there is an added complication if the match is affected by rain and the match referee has reduced the number of overs which Team 2 can bat. How does D/L handle this? Let's consider an imaginary example where India are playing Zimbabwe. Zimbabwe bat first and score 250 in their 50 overs. But since India take a little longer than their allocated time (and Cammie Smith is the match referee), it is decided that India can bat only 46 overs. What should the values of R1 and R2 be in this case?

Let's do our D/L criterion one last time. In the normal course one would have R1=100% and R2 at the beginning of India's innings to be 96.5% (based on the RP remaining for 46 overs). But since the D/L target in that case would be 250 x R2/100, the revised target will reduce to India's advantage if there is an interruption (remember, it is India which is being penalised!). That's why D/L requires that R1 must be reduced from 100% to a resource equivalent of the run penalty (in this case to 96.5%) to ensure that no advantage is inadvertently available to India.

Acknowledgements

It is a great pleasure to thank Frank Duckworth and Tony Lewis for their wonderful co-operation and support; we have exchanged over half a dozen e-mails and the D/L response in every case has been remarkably prompt. I have also lifted several examples directly from the D/L guide; but I am sure they won't mind. I must acknowledge that Chetan Shah's articles in The Hindu were very educative; he appears to be a modeller of real mettle. I also thank my colleague and friend, P S Swathi, another modeller of mettle, who brought the D/L guide for me from England and is still refusing my offer to pay the rupee equivalent of about six pounds.