Editing Duckworth–Lewis–Stern method (section)

===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.