Editing Duckworth–Lewis–Stern method (section)

==Theory==
{{Further|topic=the cricket calculation tool|WASP (cricket calculation tool)|label1=WASP}}

===Calculation summary===
The essence of the D/L method is 'resources'. Each team is taken to have two 'resources' to use to score as many runs as possible: the number of [[over (cricket)|over]]s they have to receive; and the number of [[Dismissal (cricket)|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 they have left. 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.<ref>{{cite web|url=http://www.daa.com.au/analytical-ideas/cricket/|title=The Duckworth-Lewis Method|publisher=Data Analysis Australia|date=September 2006|access-date=13 June 2008|archive-date=13 July 2011|archive-url=https://web.archive.org/web/20110713052219/http://www.daa.com.au/analytical-ideas/cricket/|url-status=dead}}</ref>

[[File:(Duckworth Lewis) method of adjusting target scores.PNG|thumb|300px|A published table of resource remaining percentages, for all combinations of wickets lost and whole overs left]]

The D/L method converts all possible combinations of overs (or, more accurately, balls) and wickets left into a combined resources remaining [[percentage]] figure (with 50 overs and 10 wickets = 100%), and these are all stored in a published table or computer. The target score for the team batting second ('Team 2') can be adjusted up or down from the total the team batting first ('Team 1') achieved using these resource percentages, to reflect the loss of resources to one or both teams when a match is shortened one or more times.

In the version of D/L most commonly in use in international and [[First-class cricket|first-class]] matches (the 'Professional Edition'), the target for Team 2 is adjusted simply in [[Proportionality (mathematics)|proportion]] to the two teams' resources, i.e.

: <math>\text{Team 2's par score }=\text{ Team 1's score} \times \frac{\text{Team 2's resources}}{\text{Team 1's resources}}.</math>

If, as usually occurs, this 'par score' is a non-[[integer]] number of runs, then Team 2's target to win is this number rounded up to the next integer, and the score to [[The result in cricket#Tie|tie]] (also called the par score), is this number rounded down to the preceding integer. If Team 2 reaches or passes the target score, then they have won the match. If the match ends when Team 2 has exactly met (but not passed) the par score then the match is a tie. If Team 2 fail to reach the par score then they have lost.

For example, if a rain delay means that Team 2 only has 90% of resources available, and Team 1 scored 254 with 100% of resources available, then 254&nbsp;×&nbsp;90%&nbsp;/&nbsp;100%&nbsp;=&nbsp;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,<ref name=FAQ>{{cite web|url=http://static.espncricinfo.com/db/ABOUT_CRICKET/RAIN_RULES/DL_FAQ.html|title=D/L method: answers to frequently asked questions|author1=Frank Duckworth|author2=Tony Lewis|work=Cricinfo|publisher=ESPN Sports Media|date=December 2008}}</ref> so a computer which has this software loaded must be used.

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

: <math>\text{Team 2's par score }=\text{ Team 1's score} \times \text{Team 2's resources}. </math>

===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 makes no change to the target score, because both sides are aware of the total number of overs and wickets throughout their innings, thus they will have the same resources 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 the numbers of overs in the two innings are reduced (but still the same as each other), then 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. If Team 1 had known that their innings was going to be shorter, they would have batted less conservatively, and scored more runs (at the expense of more wickets). They saved some wicket resource to use up in the overs that ended up being cancelled, which Team 2 does not need to do, therefore Team 2 ''does'' have more resource to use in the same number of overs. Therefore, increasing Team 2's target score compensates Team 1 for the denial of some of the overs they thought they would get to bat. The increased target is what D/L thinks Team 1 would have scored in the overs it ended up having, if it had known throughout that the innings would be only as long as it was.
:For example, if Team 1 batted for 20 overs before rain came, thinking they would have 50 overs in total, but at the re-start there was only time for Team 2 to bat for 20 overs, it would clearly be unfair to give Team 2 the target that Team 1 achieved, as Team 1 would have batted less conservatively and scored more runs, if they had known they would only have the 20 overs.
*If there are interruption(s) to Team 2's innings, either before it starts, during, or it 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 whether Team 2's resource loss is large enough.

===Mathematical theory===
The original D/L model started by assuming that the number of runs that can still be scored (called <math>Z</math>), for a given number of overs remaining (called <math>u</math>) and wickets lost (called <math>w</math>), takes the following [[exponential decay]] relationship:<ref name=OrigPaper>{{Cite journal |last1=Duckworth |first1=FC |last2=Lewis|first2= AJ |year=1998 |title=A fair method for resetting the target in interrupted one-day cricket matches |journal=Journal of the Operational Research Society |volume=49 |issue=3 |pages=220–227 |doi=10.1057/palgrave.jors.2600524 |citeseerx=10.1.1.180.3272 |s2cid=2421934 }}</ref>

:<math>Z(u,w) = Z_0(w)\left({1 - e^{-b(w)u} } \right),</math>

where the constant <math>Z_0</math> is the [[Asymptotic analysis|asymptotic]] average total score in unlimited overs (under one-day rules), and <math>b</math> is the exponential decay constant. Both vary with <math>w</math> (only). The values of these two parameters for each <math>w</math> from 0 to 9 were [[Estimation|estimated]] from scores from 'hundreds of one-day internationals' and 'extensive research and experimentation', though were not disclosed due to 'commercial confidentiality'.<ref name="OrigPaper"/>

[[Image:DuckworthLewisEng.png|right|Scoring potential as a function of [[wicket]]s and [[over (cricket)|overs]].]]
Finding the value of <math>Z</math> for a particular combination of <math>u</math> and <math>w</math> (by putting in <math>u</math> and the values of these constants for the particular <math>w</math>), and dividing this by the score achievable at the start of the innings, i.e. finding
:<math>P(u,w) = \frac{Z(u,w)}{Z(u=50,w=0)},\,</math>
gives the proportion of the combined run scoring resources of the innings remaining when <math>u</math> overs are left and <math>w</math> wickets are down.<ref name="OrigPaper"/> These proportions can be plotted in a graph, as shown right, or shown in a single table, as shown below.

This became the Standard Edition. When it was introduced, it was necessary that D/L could be implemented with a single table of
resource percentages, as it could not be guaranteed that computers would be present. Therefore, this single formula was used giving average resources. This method relies on the assumption that average performance is proportional to the mean, irrespective of the actual score. This was good enough in 95 per cent of matches, but in the 5 per cent of matches with very high scores, the simple approach started to break down.<ref>{{Cite journal |last=Duckworth |first=Frank |year=2008 |title=The Duckworth/Lewis method: an exercise in Maths, Stats, OR and communications |journal=MSOR Connections |volume=8 |issue=3 |pages=11–14 |url=https://www.heacademy.ac.uk/system/files/msor.8.3d.pdf |publisher=HE Academy|doi=10.11120/msor.2008.08030011 |doi-broken-date=1 November 2024 }}</ref> To overcome the problem, an upgraded formula was proposed with an additional parameter whose value depends on the Team 1 innings.<ref name=SecondPaper>{{Cite journal |last1=Duckworth |first1=FC |last2=Lewis|first2= AJ |year=2004 |title=A successful Operational Research intervention in one-day cricket |journal=Journal of the Operational Research Society |volume=55 |issue=7 |pages=749–759 |doi=10.1057/palgrave.jors.2601717 |s2cid=28422411 |doi-access=free }}</ref> This became the Professional Edition.