Resources, Not Runs: How the Duckworth-Lewis-Stern Method Sets a Target

Few numbers in cricket are argued over as much as a rain-revised target. A side that expected to chase 250 is suddenly told the figure is 213 from 38 overs, and the supporter at home reaches for the nearest suspicion. The method behind that figure is Duckworth-Lewis-Stern, usually shortened to DLS. It is not a mystery, and its central idea can be explained without a single equation.

The problem it was built to solve

A limited-overs match is a fair contest only if both teams get the same opportunity. Rain breaks that. If the second innings loses overs, the team batting second has less time, but it has also lost nothing in the way of wickets. Simply scaling the target by the number of overs left, the old run-rate approach, ignores something every batter knows: a side with ten wickets in hand can play recklessly for the last few overs, while a side with three wickets in hand cannot.

The 1992 World Cup semi-final between England and South Africa at Sydney is the example that is always cited. According to Wikipedia, the tournament used a Most Productive Overs rule, which replaced the earlier Average Run Rate method. When rain interrupted South Africa's chase, their requirement was revised to something close to impossible from a single ball. The two Wikipedia pages I read give slightly different figures for the requirement (22 or 21 runs from one ball), so I have not quoted one. What both agree on is that the rule had exposed its flaw in front of a world audience.

The central idea: two resources

The method, devised by the English statisticians Frank Duckworth and Tony Lewis, treats a batting side as holding two resources at any moment: overs still to be bowled and wickets still in hand. According to Wikipedia, a full innings of 50 overs with ten wickets in hand is counted as 100 per cent. Every other combination, say 20 overs left and six wickets in hand, maps to a percentage of that full resource.

That is the whole concept. If the team batting first used 100 per cent of its resources and scored 250, the question after a stoppage becomes: given the resources the chasing team has left, how many runs is a fair share of that 250? The answer is the par score, the total the chasing side would be expected to have reached at that point to be level. Per Wikipedia, the target to win is the par score rounded up and the score that would tie is the par rounded down.

An illustration, using made-up round numbers rather than a real match: if a chasing team, after a delay, has only 85 per cent of the resources of the side that batted first, the method scales the first team’s total accordingly. It does not simply take 85 per cent of the runs, because the real formula is a curve rather than a straight line, but the logic of proportion is the same. The attraction is that wickets lost earlier carry weight. A team that loses early wickets before a delay arrives with fewer resources, and the target moves in its disfavour.

A short history

According to Wikipedia, the method was first used in international cricket on 1 January 1997, in the second one-day international between Zimbabwe and England. The ICC then adopted it in 1999 as the standard method for rain-shortened one-day matches; that date is confirmed by the ICC’s own release on the death of Tony Lewis and by a profile of Steven Stern published by Bond University.

The third name in the title arrived later. Stern, an American-born statistician who moved to Australia in 1994 and later became a professor of data science at Bond University, first sent Duckworth and Lewis a paper on improving their method in the late 1990s, according to Bond. Duckworth read it in full but judged the suggestions too complicated for use in live games. By 2006 Stern was the method’s official programmer, and the ICC says that after Duckworth and Lewis retired he became its custodian. His name was added in 2014; Wikipedia gives the date as November of that year, and the method was in use in that form at the 2015 World Cup. Bond reports that Stern developed refinements for Twenty20 using newer data and that the ICC adopted them ahead of that tournament.

Lewis, who studied mathematics and statistics at Sheffield and retired as a lecturer at Oxford Brookes University, died aged 78; the ICC’s release is dated 2 April 2020. It quoted the ICC’s general manager of cricket, Geoff Allardice, saying that the contribution was huge.

Two versions, one idea

Wikipedia describes two editions. The Standard Edition uses a single published table of resource percentages, so a scorer with the table can work out a target by hand. The Professional Edition uses a different resource table depending on the first team’s total, and it needs a computer; its figures are not public. The page says the Professional Edition has been used in international one-day matches since early 2004. The idea behind this split is, in my reading, that a high first-innings total changes how much a team can profitably accelerate, so a single table cannot fit every match.

Why it still draws complaints

There are three reasons people stay unconvinced, and they are worth separating from the method’s arithmetic.

• It is an estimate, not a measurement. No formula can know how a particular chase would have unfolded. It offers a statistically fair target, not the actual outcome.
• The inputs are invisible in the moment. Supporters see a new number and not the resource percentages behind it, so the figure can feel arbitrary. Showing the par score on the screen, as broadcasters now do, helps.
• Conditions change. A stoppage that hands the pitch to a different bowling attack, or turns day into floodlit evening, alters the contest in ways no table can include.

These are interpretations rather than facts from the sources, but they match why the method keeps being refined rather than replaced.

How to read the next target

When you next see a revised number, ask two things. How many overs did the chasing side lose, and how many wickets did it have in hand when the rain came? A side that is two wickets down with ten overs lost is treated very differently from one that is seven down with the same loss. The par score is the live figure to watch: it moves with every ball, and if the chasing side is ahead of it, the rain has favoured them. That single comparison turns DLS from a source of grumbling into something you can follow.