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Duckworth–Lewis method

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Duckworth–Lewis method

The Duckworth–Lewis method (often written as D/L method) is a mathematical formulation designed to calculate the target score for the team batting second in a limited overs cricket match interrupted by weather or other circumstances. It is generally accepted to be the most accurate method of setting a target score. The D/L method was devised by two English statisticians, Frank Duckworth and Tony Lewis.[1]

The basic principle is that each team in a limited-overs match has two resources available with which to score runs: wickets remaining, and overs to play. Where overs are lost, setting an adjusted target for the team batting second is not as simple as reducing the run target proportionally to the loss in overs, because a team with ten wickets in hand and 25 overs to bat can be expected to play more aggressively than if they had ten wickets and a full 50 overs, for example, and can consequently achieve a higher run rate. The Duckworth–Lewis method is an attempt to set a statistically fair target for the second team's innings, based on the score achieved by the first team, taking their wickets lost and overs played into account.

In November 2014, the Duckworth–Lewis method was renamed the Duckworth-Lewis-Stern (or D/L/S) method.[2]

Contents

• Calculation summary 1
• Summary of impact on Team 2's target 2
• Examples 3
• Stoppage in first innings 3.1
• Stoppage in second innings 3.2
• Stoppages in both innings 3.3
• Theory 4
• Application 5
• Standard Edition and Professional Edition 5.1
• Calculations 6
• Standard Edition 6.1
• G50 6.1.1
• Professional Edition 6.2
• Example Standard Edition calculations 7
• Reduced target: Team 1's innings completed; Team 2's innings delayed 7.1
• Reduced target: Team 1's innings completed; Team 2's innings cut short 7.2
• Reduced target: Team 1's innings completed; Team 2's innings interrupted 7.3
• Increased target: Team 1's innings cut short; Team 2's innings completed 7.4
• Increased target: Multiple interruptions in Team 1's innings; Team 2's innings completed 7.5
• Calculation of an innings' resource percentage 8
• Other uses 9
• Ball-by-ball par score 9.1
• Net Run Rate calculation 9.2
• History and creation 10
• Previous methods 10.1
• 2002 11.1
• 2004 11.2
• 2009 11.3
• Criticism 12
• Cultural influence 13
• References 14

Calculation summary

In the version of D/L most commonly in use in international and first class matches (the 'Professional Edition'), the target for the team batting second ('Team 2') is adjusted up or down from the total the team batting first ('Team 1') scored, in proportion to the two teams' resources (combination of overs and wickets available), i.e.

\text{Team 2's par score }=\text{ Team 1's score} \times \frac{\text{Team 2's resources}}{\text{Team 1's resources}}.

This 'par score' is usually a non-integer number of runs. Team 2's target to win is this number rounded up to the next integer, and the score to tie (also called the par score), is this number rounded down to the preceding integer. For example, if a rain delay means that Team 2 only has 90% of the resources that were available to Team 1, and Team 1 scored 254, then 254 x 90% = 228.6, so Team 2's target is 229, and the score to tie is 228. The actual resource values used in the Professional Edition are not publicly available,[3] so a computer must be used which has the software loaded.

If it's a 50-over match and Team 1 completed its innings uninterrupted, then they had 100% resource available to them, so the formula simplifies to:

\text{Team 2's par score }=\text{ Team 1's score} \times \text{Team 2's resources}.

Summary of impact on Team 2's target

• If there is a delay before the first innings starts, so that the numbers of overs in the two innings are reduced (but still the same as each other), then D/L will make no change to the target score. This is because both sides will be in the same position of having the same number of overs and 10 wickets available, and knowing this throughout their innings, so having the same amount of resource available.
• Team 2's target score is first calculated once Team 1's innings has finished. If there were interruption(s) during Team 1's innings, or Team 1's innings was cut short, so that the numbers of overs in the two innings are reduced (but still the same as each other), then (in the Professional Edition) D/L will adjust Team 2's target score as described above. The adjustment to Team 2's target after interruptions in Team 1's innings is often an increase, implying that Team 2 has more resource available than Team 1 had. Although both teams have 10 wickets and the same (reduced) number of overs available, an increase is fair as, for some of their innings, Team 1 thought they would have more overs available than they actually ended up having. They would have batted less conservatively, and scored more runs at the expense of more wickets, if they had known that their innings was going to be shorter than initially thought. They saved some wicket resource to use up in the overs that ended up being cancelled, which Team 2 doesn't need to do, therefore Team 2 has more resource to use in the same number of overs. Therefore, increasing Team 2's target score neutralizes the injustice done to Team 1 when they were denied some of the overs to bat they thought they would get.
• If there are interruption(s) before or during the Team 2's innings, or Team 2's innings is cut short, then D/L will reduce Team 2's target score from the initial target set at the end of Team 1's innings, in proportion to the reduction in Team 2's resources. If there are multiple interruptions in the second innings, the target will be adjusted downwards each time.
• If there are interruptions which both increase and decrease the target score, then the net effect on the target could be either an increase or decrease, depending on which interruptions were bigger.

Examples

Stoppage in first innings

Increased target

In the 4th IndiaEngland ODI in the 2008 series, the first innings was interrupted by rain on two occasions, resulting in the match being reduced to 22 overs each. India (batting first) made 166/4. England's target was set by the D/L method at 198 from 22 overs. As England knew they had only 22 overs the expectation is that they will be able to score more runs from those overs than India had from their (interrupted) innings. England made 178/8 from 22 overs, and so the match was listed as "India won by 19 runs (D/L method)".[4]

During the fifth ODI between India and South Africa in January 2011, rain halted play twice during the first innings. The match was reduced to 46 overs each and South Africa scored 250/9. The D/L method was applied which adjusted India's target to 268. As the number of overs was reduced during South Africa's innings, this method takes into account what South Africa are likely to have scored if they'd known throughout their innings that it would only be 46 overs long.

Decreased target

On 3 December 2014, Sri Lanka played England and batted first, but play was interrupted when Sri Lanka had scored 6/1 from 2 overs. At the restart both innings were reduced to 35 overs, and Sri Lanka finished on 242/8. England's target was set by D/L at 236 from 35 overs.[5] Although Sri Lanka had less resource remaining to them after the interruption than England would have for their whole innings (about 7% less), they'd used up so much resource before the interruption (2 overs and 1 wicket, about 8%), that the total resource used by Sri Lanka was still slightly more than England would have available, hence the slightly decreased target for England.

Stoppage in second innings

A simple example of the D/L method being applied was the first ODI between India and Pakistan in their 2006 ODI series.[6] India batted first, and were all out for 328. Pakistan, batting second, were 311/7 when bad light stopped play after the 47th over. Pakistan's target, had the match continued, was 18 runs in 18 balls, with three wickets in hand. Considering the overall scoring rate throughout the match, this is a target most teams would be favoured to achieve. And indeed, application of the D/L method resulted in a retrospective target score of 305 (or par score of 304) at the end of the 47th over, with the result therefore officially listed as "Pakistan won by 7 runs (D/L Method)". In another match between Pakistan and Sirilanka in 2007, Pakistan fielders forfeited the ground claiming bad light, and using D/L method Pakistan won by 17 runs.

The D/L method was used in the group stage match between Sri Lanka and Zimbabwe at the 20/20 World Cup in 2010. Sri Lanka scored 173/7 in 20 overs batting first, and in reply Zimbabwe were 4/0 from 1 over when rain interrupted play. At the restart Zimbabwe's target was reduced to 108 from 12 overs, but rain stopped the match when they had scored 29/1 from 5 overs. The retrospective D/L target from 5 overs was a further reduction to 44, or a par score of 43, and hence Sri Lanka won the match by 14 runs.[7][8]

Stoppages in both innings

During the 2012/13 KFC Big Bash League, D/L was used in the 2nd semi-final played between the Melbourne Stars and the Perth Scorchers. After rain delayed the start of the match, it interrupted Melbourne's innings when they had scored 159/1 off 15.2 overs, and both innings were reduced by 2 overs to 18, and Melbourne finished on 183/2. After a further rain delay reduced Perth's innings to 17 overs, Perth returned to the field to face 13 overs, with a revised target of 139. Perth won the game by 8 wickets with a boundary off the final ball.[9][10]

Theory

The essence of the D/L method is 'resources'. Each team is taken to have two 'resources' to use to make as many runs as possible: the number of overs they have to receive; and the number of wickets they have in hand. At any point in any innings, a team's ability to score more runs depends on the combination of these two resources. Looking at historical scores, there is a very close correspondence between the availability of these resources and a team's final score, a correspondence which D/L exploits.[11]

Using a published table or computer which gives the percentage of these combined resources remaining for any number of overs (or, more accurately, balls) left and wickets lost, the target score can be adjusted up or down to reflect the loss of resources to one or both teams when a match is shortened one or more times. The two teams' resource percentages are found, and used to calculate a 'par score' for the second team that is usually a fractional number of runs. The target score is this number rounded up to the next integer, and the score to tie (also called the par score), is this number rounded down to the preceding integer. If the second team reaches or passes the target score, then they have won the match. If the match ends when the second team has exactly met (but not passed) the par score then the match is a tie. If the second team fail to reach the par score then they have lost.

An example of such a tie was found in the one day international between England and India on 11 September 2011. This match was frequently interrupted by rain in the final overs, and a ball-by-ball calculation of the Duckworth–Lewis 'par' score played a key role in the tactical decisions made during those overs. At one point, India were ahead according to this calculation, during one rain delay (and would have won if play was unable to be resumed). At a second rain interval, England, who had scored some quick runs (precisely because they were aware of the need to get ahead in D/L terms) would correspondingly have won if play had not resumed. Play was finally called off with just 7 balls of the match remaining and England's score equal to the Duckworth–Lewis 'par' score, therefore resulting in a tied match.

This example does show how crucial (and difficult) the decisions of the umpires can be, in assessing at exactly what point the rain is heavy enough to justify ceasing play. If the umpires of that match had halted play one ball earlier, England would have been ahead on D/L, and so would have won the match. Equally, if play had stopped one ball later, without England scoring off that ball, India would have won the match – indicating how finely-tuned D/L calculations can be in such situations.

Application

As with most non-trivial statistical derivations, the D/L method can produce results that are somewhat counter intuitive, and the announcement of the derived target score can provoke a good deal of second-guessing and discussion amongst the crowd at the cricket ground. This can also be seen as one of the method's successes, adding interest to a "slow" rain-affected day of play.

For 50-over matches, each team must face at least 20 overs before D/L can decide the game, unless one or both sides have been bowled out in less than 20 overs and/or the team batting second has reached its target in less than 20 overs. For Twenty20 games, each side must face at least five overs before D/L can decide the game, unless one or both sides have been bowled out in less than five overs and/or the team batting second has reached its target in less than five overs. If these prerequisites are not met, the match is declared a no result.

Standard Edition and Professional Edition

Until 2003, a single version of the D/L method was in use. This used a single published reference table of total resource percentages remaining for all possible combinations of overs and wickets,[12] and some simple mathematical calculations, and was relatively transparent and straightforward to implement. However, it had a known flaw in how it handled very high first innings scores (350+). Following the 2003 World Cup, a second version was introduced which overcomes this flaw by using substantially more sophisticated statistical modelling. However, this version does not use a single table of resource percentages, instead the percentages also vary with score, so the percentages must be obtained from a computer. Therefore, it loses some of the previous advantages of transparency and simplicity. The original version was named the Standard Edition, and the new version was named the Professional Edition.

Tony Lewis said, 'We were then [at the time of the 2003 Cricket World Cup Final] using what is now known as the Standard Edition. We knew that under normal circumstances it worked well, but we had also known for a while that in case of very big totals the approach was not really that good. We did have in place a computerised version, but it meant that the transparency was lost. You couldn’t do it manually by looking up the tables. Up until that point the ICC were very happy with the manual version and the transparency that came with it. But Australia got 359 and that showed up the flaws and straightaway the next edition was introduced which handled high scores much better. The par score for India is likely to be much higher now.'[13]

Duckworth and Lewis wrote, 'When the side batting first score at or below the average for top level cricket..., the results of applying the Professional Edition are generally similar to those from the Standard Edition. For higher scoring matches, the results start to diverge and the difference increases the higher the first innings total. In effect there is now a different table of resource percentages for every total score in the Team 1 innings.'[14]

The decision on which edition should be used is for the cricket authority which runs the particular competition.[15] The ICC Playing Handbook,[16] the official handbook for international matches, says 'The Professional Edition of the Duckworth–Lewis method shall be used in all [international] matches... Where possible, arrangements shall be made for the provision of back-up capability, in case of computer malfunction, for the operation or continued operation of the Professional Edition. In the event of computer non-availability or malfunction where no such provision has been made, the Standard Edition (the method in use prior to October 2003) shall be used.'[17] This also applies to most countries’ national competitions.[15] At lower levels of the game, where use of a computer cannot always be guaranteed, the Standard Edition is used.[18]

Calculations

Using the notation of the ICC Playing Handbook,[16] the team that bats first is called Team 1, their final score is called S, the total resources available to Team 1 for their innings is called R1, the team that bats second is called Team 2, and the total resources available to Team 2 for their innings is called R2.

Percentage total resources remaining reference table (D/L Standard Edition)
Overs remaining Wickets in hand
10 8 6 4 2
50 100.0 85.1 62.7 34.9 11.9
40 89.3 77.8 59.5 34.6 11.9
30 75.1 67.3 54.1 33.6 11.9
20 56.6 52.4 44.6 30.8 11.9
10 32.1 30.8 28.3 22.8 11.4
5 17.2 16.8 16.1 14.3 9.4

Standard Edition

For each reduction in overs, the loss in total resources available to the batting team is found using a published reference table,[12] then Team 2's target score is changed as follows:

• If R2 < R1, reduce Team 2's target score in proportion to the reduction in total resources, i.e. S × R2/R1.
• If R2 = R1, no adjustment to Team 2's target score is needed.
• If R2 > R1, increase Team 2's target score by the extra runs that could be expected to be scored on average with the extra total resource, i.e. S + G50 × (R2 – R1)/100, where G50 is the average 50-over total. Team 2's target score is not simply increased in proportion to the increase in total resources, i.e. S × R2/R1, as this 'could lead to some unrealistically high targets if Team 1 had achieved an early high rate of scoring [in the powerplay overs] and rain caused a drastic reduction in the overs for the match.'[19] Instead, D/L Standard Edition requires average performance for Team 2's additional resource over Team 1.

G50

G50 is the average score expected from the team batting first in an uninterrupted 50 overs-per-innings match. This will vary with the level of competition and over time. The annual ICC Playing Handbook[16] gives the values of G50 to be used each year when the D/L Standard Edition is applied:

 Period Matches involving ICC full member nations Matches between teams that play first class cricket Under-19 internationals Under-15 internationals Matches between ICC associate member nations Women’s ODIs 1999 − 31 Aug 2002 [20] 225 ? 1 Sep 2002 − 2006 [21] 235 2006/07[22] 235 200 190 175 2007/08 2008/09[16] 2009/10[16] 245 200 2010/11[16] 2011/12[16] 2012/13[16] 2013/14[16]

Duckworth and Lewis write, 'We accept that the value of G50, perhaps, should be different for each country, or even for each ground, and there is no reason why any cricket authority may not choose the value it believes to be the most appropriate. In fact it would be possible for the two captains to agree a value of G50 before the start of each match, taking account of all relevant factors. However, we do not believe that something that is only invoked if rain interferes with the game should impose itself on every game in this way. In any case, it should be realised that the value of G50 usually has very little effect on the revised target. If 250 were used, for instance, instead of 235, it is unlikely that the target would be more than two or three runs different.'[23]

Professional Edition

For each reduction in overs, the loss in total resources available to the batting team is found using a computer, then Team 2's target score is changed as follows:

• If R2 < R1, reduce Team 2's target score in proportion to the reduction in total resources, i.e. S × R2/R1.
• If R2 = R1, no adjustment to Team 2's target score is needed.
• If R2 > R1, increase Team 2's target score in proportion to the increase in total resources, i.e. S × R2/R1. The problem of early high scoring rates potentially producing anomalously high targets has been overcome in the Professional Edition, which is essentially 'a different table of resource percentages for every total score in the Team 1 innings.'[14] Therefore, Team 2's target score can be simply increased in proportion to the increase in total resources when R2 > R1,[19] and there is no G50.

However, the resource percentages used in the Professional Edition are not publicly available,[3] so a computer must be used which has the software loaded.

Example Standard Edition calculations

As the resource percentages used in the Professional Edition are not publicly available, it is difficult to give examples of the D/L calculation for the Professional Edition. Therefore, examples are given from when the Standard Edition was widely used, which was up to early 2004.

Reduced target: Team 1's innings completed; Team 2's innings delayed

Percentage total resources remaining reference table (D/L Standard Edition)[12]
Overs remaining Wickets in hand
10 8 6 4 2
31 76.7 68.6 54.8 33.7 11.9
30 75.1 67.3 54.1 33.6 11.9
29 73.5 66.1 53.4 33.4 11.9
28 71.8 64.8 52.6 33.2 11.9
27 70.1 63.4 51.8 33.0 11.9

On 18 May 2003, Lancashire played Hampshire in the National League.[24][25][26] Rain before play reduced the match to 30 overs each. Lancashire batted first and scored 231–4 from their 30 overs. Before Hampshire began their innings, it was further reduced to 28 overs.

 Total resources available to Lancashire (R1) 30 overs and 10 wickets 75.1% Total resources available to Hampshire (R2) 28 overs and 10 wickets 71.8% Hampshire's par score 231 x R2/R1 = 231 x 71.8/75.1 220.850 runs

Hampshire's target was therefore 221 to win (in 28 overs), or 220 to tie. They were all out for 150, giving Lancashire victory by 220 − 150 = 70 runs.

If Hampshire's target had been reduced simply in proportion to the reduction in overs, their par score would have been 231 x 28/30 = 215.6. This would have kept the required run rate the same as Lancashire achieved (7.7 runs per over), but this would have given an unfair advantage to Hampshire, as it's easier to achieve a run rate for a shorter period. Increasing Hampshire's target from 215 overcomes this.

Reduced target: Team 1's innings completed; Team 2's innings cut short

Percentage total resources remaining reference table (D/L Standard Edition)[12]
Overs remaining Wickets in hand
10 8 6 4 2
50 100.0 85.1 62.7 34.9 11.9
40 89.3 77.8 59.5 34.6 11.9
30 75.1 67.3 54.1 33.6 11.9
20 56.6 52.4 44.6 30.8 11.9
10 32.1 30.8 28.3 22.8 11.4
5 17.2 16.8 16.1 14.3 9.4

On 3 March 2003, Sri Lanka played South Africa in the 2003 Cricket World Cup Pool B.[27][28] Sri Lanka batted first and scored 268–9 from their 50 overs. Chasing a target of 269, South Africa had reached 229–6 from 45 overs when play was abandoned.

 Total resources available to Sri Lanka (R1) 50 overs and 10 wickets 100.0% Total resources available to South Africa at the start of their innings 50 overs and 10 wickets 100.0% Total resources remaining to South Africa when play abandoned 5 overs and 4 wickets 14.3% Total resources available to South Africa (R2) 100.0% − 14.3% 85.7% South Africa's par score 268 x R2/R1 = 268 x 85.7/100.0 229.676 runs

Therefore, South Africa's retrospective target from their 45 overs was 230 runs to win, or 229 to tie. In the event, as they had scored exactly 229, the match was declared a tie.

South Africa scored no runs off the very last ball. If play had been abandoned without that ball having been bowled, the resource available to South Africa at the abandonment would have been 14.7%, giving them a par score of 228.6, and hence victory.

Reduced target: Team 1's innings completed; Team 2's innings interrupted

On 16 February 2003, New South Wales played South Australia in the ING Cup.[29][30] New South Wales batted first and scored 273 all out (from 49.4 overs). Chasing a target of 274, rain interrupted play when South Australia had reached 70–2 from 19 overs, and at the restart their innings was reduced to 36 overs (i.e. 17 remaining).

 Total resources available to New South Wales (R1) 50 overs and 10 wickets 100.0% Total resources available to South Australia at the start of their innings 50 overs and 10 wickets 100.0% Total resources remaining to South Australia at the interruption 31 overs and 8 wickets 68.6% Total resources remaining to South Australia at the restart 17 overs and 8 wickets 46.7% Total resources lost to South Australia by the interruption 68.6% − 46.7% 21.9% Total resources available to South Australia (R2) 100.0% − 21.9% 78.1% South Australia's par score 273 x R2/R1 = 273 x 78.1/100.0 213.213 runs

South Australia's new target was therefore 214 to win (in 36 overs), or 213 to tie. In the event, they were all out for 174, so New South Wales won by 213 − 174 = 39 runs.

Increased target: Team 1's innings cut short; Team 2's innings completed

On 25 January 2001, West Indies played Zimbabwe.[31][32] West Indies batted first and had reached 235–6 from 47 overs (of a scheduled 50) when rain halted play for two hours. At the restart, both innings were reduced to 47 overs, i.e. West Indies' innings was closed immediately, and Zimbabwe began their innings.

 Total resources available to West Indies at the start of their innings 50 overs and 10 wickets 100.0% Total resources remaining to West Indies when innings was closed 3 overs and 4 wickets 10.2% Total resources available to West Indies (R1) 100.0% − 10.2% 89.8% Total resources available to Zimbabwe (R2) 47 overs and 10 wickets 97.4% Zimbabwe's par score 235 + G50 x (R2−R1)/100 = 235 + 225 x (97.4−89.8)/100 252.100 runs

Zimbabwe's target was therefore 253 to win (in 47 overs), or 252 to tie. It is fair that their target was increased, even though they had the same number of overs to bat as West Indies, as West Indies would have batted more aggressively in their last few overs, and scored more runs, if they had known that their innings would be cut short at 47 overs. Zimbabwe were all out for 175, giving West Indies victory by 252 − 175 = 77 runs.

These resource percentages are the ones which were in use back in 2001, before the 2002 revision, and so do not match the currently used percentages for the Standard Edition, which are slightly different. Also, the formula for Zimbabwe's par score comes from the Standard Edition of D/L, which was used at the time. Currently the Professional Edition is used, which has a different formula when R2>R1. The formula required Zimbabwe to match West Indies' performance with their overlapping 89.8% of resource (i.e. score 235 runs), and achieve average performance with their extra 97.4% − 89.8% = 7.6% of resource (i.e. score 7.6% of G50 (225 at the time) = 17.1 runs).

Increased target: Multiple interruptions in Team 1's innings; Team 2's innings completed

On 20 February 2003, Australia played Netherlands in the 2003 Cricket World Cup Pool A.[33][34][35][36] Rain before play reduced the match to 47 overs each, and Australia batted first.

• Rain stopped play when they had reached 109–2 from 25 overs (i.e. 22 remaining). At the restart both innings were reduced to 44 overs (i.e. 19 remaining for Australia).
• Rain stopped play again when Australia had reached 123–2 from 28 overs (i.e. 16 remaining), and at the restart both innings were reduced further to 36 overs (i.e. 8 remaining for Australia).

Australia finished on 170–2 from their 36 overs.

 Total resources available to Australia at the start of their innings 47 overs and 10 wickets 97.1% Total resources remaining to Australia at first interruption 22 overs and 8 wickets 55.8% Total resources remaining to Australia at restart 19 overs and 8 wickets 50.5% Total resources lost by first interruption 55.8% − 50.5% 5.3% Total resources remaining to Australia at second interruption 16 overs and 8 wickets 44.7% Total resources remaining to Australia at restart 8 overs and 8 wickets 25.5% Total resources lost by second interruption 44.7% − 25.5% 19.2% Total resources available to Australia (R1) 97.1% − 5.3% − 19.2% 72.6% Total resources available to Netherlands (R2) 36 overs and 10 wickets 84.1% Netherlands' par score 170 + G50 x (R2−R1)/100 = 170 + 235 x (84.1−72.6)/100 197.025 runs

The Netherlands' target was therefore 198 to win (in 36 overs), or 197 to tie. It is fair that their target was increased, even though they had the same number of overs to bat as Australia, as Australia would have batted less conservatively in their first 28 overs, and scored more runs at the expense of more wickets, if they had known that their innings would only be 36 overs long. Increasing the Netherlands' target score neutralizes the injustice done to Australia when they were denied some of the overs to bat they thought they would get. The Netherlands were all out for 122, giving Australia victory by 197 − 122 = 75 runs.

This formula for Netherlands' par score comes from the Standard Edition of D/L, which was used at the time. Currently the Professional Edition is used, which has a different formula when R2>R1. The formula required Netherlands to match Australia's performance with their overlapping 72.6% of resource (i.e. score 170 runs), and achieve average performance with their extra 84.1% − 72.6% = 11.5% of resource (i.e. score 11.5% of G50 (235 at the time) = 27.025 runs).

After the match there were reports in the media[34] that Australia had batted conservatively in their final 8 overs after the second stop, to avoid losing wickets rather than maximising their numbers of runs, in belief that this would further increase the Netherlands' par score. However, if this is true, this belief was mistaken, in the same way that conserving wickets rather than maximising runs in the final 8 overs of a full 50-over innings would be a mistake. At that point the amount of resource available to each team was fixed (as long as there were no further rain interruptions), so the only undetermined number in the formula for Netherlands' par score was Australia's final score, so they should have tried to maximise this.

Calculation of an innings' resource percentage

Although the examples above may make it look like finding the total resource percentage requires a different calculation for each different scenario, the formula is actually the same each time − it's just that different scenarios, with more or less interruptions and restarts, need to use more or less of the same formula. The calculations above just show and use the parts of the formula that are relevant for that scenario.

The total resources available to a team are given by:[12]

\begin{matrix} \text{Total} \\ \text{resources} \\ \text{available} \end{matrix} \ \ \ = \ \ \ \begin{matrix} \text{Resources} \\ \text{at start} \\ \text{of innings} \end{matrix} \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{matrix} \text{Resources lost by} \\ \text{first interruption} \end{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \begin{matrix} \text{Resources lost by} \\ \text{second interruption} \end{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \begin{matrix} \text{Resources lost by} \\ \text{third interruption} \end{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \text{etc}...

         = \ \ \
\begin{matrix}
\text{Resources} \\
\text{at start} \\
\text{of innings}
\end{matrix}
\ \ \ - \ \ \
\left(
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at first} \\
\text{interruption}
\end{matrix}
\ - \
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at restart}
\end{matrix}
\right)
\ \ \ - \ \ \
\left(
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at second} \\
\text{interruption}
\end{matrix}
\ - \
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at restart}
\end{matrix}
\right)
\ \ \ - \ \ \
\left(
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at third} \\
\text{interruption}
\end{matrix}
\ - \
\begin{matrix}
\text{Resources} \\
\text{remaining} \\
\text{at restart}
\end{matrix}
\right)
\ \ \ - \ \ \ ...



Each time there's an interruption or a restart after an interruption, the resource remaining percentages at those times (obtained from a reference table for the Standard Edition, or from a computer for the Professional Edition) can be entered into the formula, with the rest left blank. In the examples above:

 South Africa innings v Sri Lanka (3 Mar 2003) South Australia innings v New South Wales (16 Feb 2003) West Indies innings v Zimbabwe (25 Jan 2001) Australia innings v Netherlands (20 Feb 2003) Resources at start of innings 100.0% 100.0% 100.0% 97.1% Resources remaining at first interruption 14.3% 68.6% 10.2% 55.8% Resources remaining at restart 46.7% 50.5% Resources remaining at second interruption 44.7% Resources remaining at restart 25.5% Total resources available 100.0% − (14.3%) = 85.7% 100.0% − (68.6% − 46.7%) = 78.1% 100.0% − (10.2%) = 89.8% 97.1% − (55.8% − 50.5%) − (44.7% − 25.5%) = 72.6%

Other uses

There are uses of the D/L method other than finding the current official final target score for the team batting second in a match that has already been reduced by the weather.

Ball-by-ball par score

During the second team's innings, the number of runs a chasing side would expect to have scored on average with this number of overs used and wickets lost, if they were going to successfully match the first team's score, called the D/L par score, may be shown on a computer printout, the scoreboard and/or TV alongside the actual score, and updated after every ball. This can happen in matches which look like they're about to be shortened by the weather, and so D/L is about to be brought into play, or even in matches completely unaffected by the weather. This is:

• To help spectators and players understand whether the chasing side are doing better or worse than they would need to do on average to reach the target score.
• The score the batting team's score would be compared to determine which side had won, if the match had to be abandoned right then. It is the par score which is displayed, i.e. the score to tie. The target, to win, score is one run more than this. South Africa exited the 2003 World Cup after a resulting a draw with Sri Lanka by mistakenly believing the par score on the printout was the target score.[37][38]

Net Run Rate calculation

It has been suggested that when a side batting second successfully completes the run chase, the D/L method could be used to predict how many runs they would have scored with a full innings (i.e. 50 overs in a One Day International), and use this prediction in the net run rate calculation.[39]

This suggestion is in response to the criticisms of NRR that it doesn't take into account wickets lost, and that it unfairly penalizes teams which bat second and win, as those innings are shorter and therefore have less weight in the NRR calculation than other innings which go the full distance.

History and creation

The D/L method was devised by two British statisticians, Frank Duckworth and Tony Lewis, as a result of the outcome to the semi-final in the 1992 Cricket World Cup between England and South Africa, where the most productive overs method was used. Rain stopped play for 12 minutes with South Africa needing 22 runs from 13 balls chasing England's 252/6 off 45 overs. The revised target left South Africa needing 21 runs from one ball, which was a reduction of only one run compared to a reduction of two overs, and a preposterous target given that the maximum score from one ball is generally six runs.[40] The D/L method avoids this flaw: in this match, the revised D/L target would have left South Africa four to tie or five to win from the final ball.[41] Duckworth said, "I recall hearing Christopher Martin-Jenkins on radio saying 'surely someone, somewhere could come up with something better' and I soon realised that it was a mathematical problem that required a mathematical solution."[42][43]

It was first used in international cricket in the second game of the 1996–97 Zimbabwe versus England One Day International series, which Zimbabwe won by seven runs,[44] and was formally adopted by the International Cricket Council in 1999 as the standard method of calculating target scores in rain shortened one-day matches.

Previous methods

Various different methods had been used previously, with the most common being the average run-rate method, and the most productive overs method.

All of these methods have flaws that are easily exploitable:

• The average run-rate method takes no account of how many wickets the team batting second have lost, but simply reflects how quickly they were scoring when the match was interrupted, so if a team felt a rain stoppage was likely they could attempt to force the scoring rate without regard for the corresponding highly likely loss of wickets, skewing the comparison with the first team.
• The most productive overs method also takes no account of how many wickets the team batting second have lost, and also has the further effect of penalizing the team batting second for good bowling, as their best overs are ignored in setting the revised target.

An example of this is in the 1988/89 Benson and Hedges World Series Cup, where the average run rate method was used: in the third final between Australia and the West Indies, rain stopped play for one hour and 25 minutes with the West Indies needing 180 off 31.2 overs chasing Australia's 226/4 off 38 overs. The revised target left the West Indies needing 61 off the 11.2 overs that remained, and the West Indies won the match and the competition with 4.4 overs remaining and eight wickets in hand after Desmond Haynes hit a Steve Waugh full toss for six. Australian fans loudly booed this unsatisfactory conclusion, and criticism from the media and both captains led to the average run rate method being replaced by the most productive overs method for setting revised targets in interrupted matches.[45] In this match, the D/L method would have increased the West Indies target to 232 to take into account a two-hour rain delay during Australia's innings, and then revised the target to 139 after the second interruption.

The published table that underpins the D/L method is regularly updated, using source data from more recent matches.

2002

The resource percentages were revised, and G50 for ODI's was changed to 235, following an extensive analysis of limited overs matches in recent years. These changes came into effect on 1 September 2002.[21] As of 2014, these resource percentages are the ones still in use in the Standard Edition, though G50 has subsequently changed.

The tables show how the percentages were in 1999 and 2001, and what they were changed to in 2002. Mostly they were reduced.

Percentage total resources remaining: 1999[46] and 2001[47]
Overs remaining Wickets in hand
10 8 5 3 1
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
20 58.9 54.0 40.0 25.2 7.6
10 34.1 32.5 27.5 20.6 7.5
5 18.4 17.9 16.4 14.0 7.0
Percentage total resources remaining: 2002[21]
Overs remaining Wickets in hand
10 8 5 3 1
50 100.0 85.1 49.0 22.0 4.7
40 89.3 77.8 47.6 22.0 4.7
30 75.1 67.3 44.7 21.8 4.7
20 56.6 52.4 38.6 21.2 4.7
10 32.1 30.8 26.1 17.9 4.7
5 17.2 16.8 15.4 12.5 4.6

2004

From the 1999 Cricket World Cup match in Bristol between India and Kenya, Tony Lewis noticed that there was an inherent weakness in the formula used at the time that would give a noticeable advantage to the side chasing a total in excess of 350. A correction was very soon built into the formula and the software to correct this, by including a 'match' factor. However, this minor correction was not fully adopted by users until the 2004 update. Updating the source data in its own right would reflect the overall trend that one-day matches were achieving significantly higher scores than in previous decades, affecting the historical relationship between resources and runs.

At the same time as this update, the D/L method was also split into a Professional Edition and a Standard Edition.[48] The main difference is that while the Standard Edition preserves the use of a single table and simple calculation – suitable for use in any one-day cricket match at any level – the Professional Edition uses substantially more sophisticated statistical modelling, and requires the use of a computer. The Professional Edition has been in use in all international one-day cricket matches since early 2004.

2009

In June 2009, it was reported that the D/L method would be reviewed for the Twenty20 format after its appropriateness was questioned in the quickest version of the game. Lewis was quoted admitting that "Certainly, people have suggested that we need to look very carefully and see whether in fact the numbers in our formula are totally appropriate for the Twenty20 game."[49]

Criticism

The D/L method has been criticized on the grounds that wickets are a much more heavily weighted resource than overs, leading to the suggestion that if teams are chasing big targets, and there is the prospect of rain, a winning strategy could be to not lose wickets and score at what would seem to be a "losing" rate (e.g. if the required rate was 6.1, it could be enough to score at 4.75 an over for the first 20–25 overs).[50]

Another criticism is that the D/L method does not account for changes in proportion of the innings for which field restrictions are in place compared to a completed match.[51]

More common informal criticism from cricket fans and journalists of the D/L method is that it is unduly complex and can be misunderstood.[52][53] For example, in a one-day match against England on 20 March 2009, the West Indies coach (John Dyson) called his players in for bad light, believing that his team would win by one run under the D/L method, but not realizing that the loss of a wicket with the last ball had altered the Duckworth–Lewis score. In fact Javagal Srinath, the match referee, confirmed that the West Indies were two runs short of their target, giving the victory to England.

More recently, concerns have been raised as to its suitability for Twenty20 matches, where a high scoring over can drastically alter the situation of the game and variability of the run-rate is higher over matches with a shorter number of overs.[54]

Cultural influence

"The Duckworth Lewis Method" is the name of a band formed by Neil Hannon of The Divine Comedy and Thomas Walsh of Pugwash, which recorded a self-titled concept album of cricket songs.[55][56]

References

1. ^ "A Decade of Duckworth–Lewis".
2. ^ "Introducing Duckworth-Lewis-Stern method". Cricbuzz. 12 February 2015. Retrieved 2015-03-30.
3. ^ a b espncricinfo D/L FAQ's Q15
4. ^ Scorecard for the rain-affected 4th ODI between India and England on 23 November 2008, from Cricinfo.
5. ^ Sri Lanka v England Scorecard
6. ^ Pakistan v India Scorecard, 6 Feb 2006
7. ^ Scorecard
8. ^ Report
9. ^ Perth v Melbourne scorecard
10. ^ Perth v Melbourne report
11. ^ Data Analysis Australia's detailed mathematical analysis of the Duckworth–Lewis Method daa.com.au.
12. ^ a b c d e Duckworth/Lewis Method of Re-calculating the Target Score in an Interrupted Match
13. ^ Tony Lewis, of Duckworth-Lewis, Interview: Journalists denigrate system by publishing 'rubbish' without understanding 27 August 2013
14. ^ a b espncricinfo D/L FAQ's Q13
15. ^ a b espncricinfo D/L FAQ's Q14
16. ^ a b c d e f g h i ICC Playing Handbook
17. ^ ICC Playing Handbook 2013/14 Section 6
18. ^ espncricinfo D/L FAQ's Q1
19. ^ a b espncricinfo D/L FAQ's Q4
20. ^ The dummy's guide to Duckworth-Lewis
21. ^ a b c The Duckworth-Lewis Method (2002) from Cricinfo.
22. ^ ICC Playing Handbook 2006-07
23. ^ espncricinfo D/L FAQ's Q6
24. ^ Scorecard
25. ^ Scorecard
26. ^ Article
27. ^ Scorecard
28. ^ Report
29. ^ Scorecard
30. ^ Scorecard
31. ^ Scorecard
32. ^ Scorecard
33. ^ Scorecard
34. ^ a b Report
35. ^ Report
36. ^ Over by over
37. ^ South Africa left to lick wounds (BBC)
38. ^ Being Duckworth-Lewis: cricket's weather-break mathematicians (The Guardian)
39. ^ SportTaco.com
40. ^ "22 off one ball – A farcical rain rule leaves everyone bewildered", from Cricinfo.
41. ^ "Stump the Bearded Wonder", Bill Frindall explains how D/L would apply to 1992 WC semi-final
42. ^ espncricinfo
43. ^ BBC
44. ^ Scorecard of the 2nd ODI between England and Zimbabwe, 1 January 1997, from Cricinfo.
45. ^ 3rd Final, 1988/89 Benson and Hedges World Series Cup
46. ^ The Duckworth-Lewis Method (1999) from Cricinfo.
47. ^ The Duckworth-Lewis Method (2001) from Cricinfo.
48. ^ Rain affected rules from Cricinfo.
49. ^ Duckworth–Lewis to review their formula for T20 matches
50. ^ Bhogle, Srinivas, The Duck worth/Lewis Factor, Rediff.com.
51. ^ Booth, Shane, quoted in For a Fair Formula, The Hindu.
52. ^ Varma, Amit, Simple and subjective? Or complex and objective?, ESPNcricinfo
53. ^ Charlie Brooker, AV campaigners have created a stupidity whirlpool that engulfs any loose molecules of logic, dismisses the claim of the simplicity by citing the method's formula, The Guardian, 25 April 2011. Retrieved 2011-04-28
54. ^ The anomalous contraction of the Duckworth–Lewis method
55. ^ BBC news interview with The Duckworth Lewis Method
56. ^ Interview with band