Duckworth–Lewis method

In the sport of cricket, the Duckworth–Lewis method (D/L method) is a controversial mathematical way to calculate the target score for the team batting second in a one-day cricket or Twenty20 cricket match interrupted by weather or other circumstance. It is generally accepted to be a fair and accurate method of setting a target score, but as it attempts to predict what would have happened had the game come to its natural conclusion, it generates some controversy. The D/L method was devised by two English statisticians, Frank Duckworth and Tony Lewis.

Examples

Stoppage in first innings

In the 4th India – England ODI in the 2008 series, the first innings was interrupted by rain on two occasions, resulting in the match being reduced to 22 overs a side. India (batting first) made 166/4. England's target was therefore set by the D/L method at 198 from 22 overs.

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 and South Africa scored 250-9. The D/L method was applied which adjusted the target to 268. As the number of overs was reduced in between South Africa's innings, this method takes into account what South Africa would have scored before the first interruption. Despite Yusuf Pathan's lower order century, India were bowled out for 234 resulting in a 33 run victory for South Africa who clinched the series 3-2.

Both examples illustrate how the D/L method is applied. In the case of the first match, 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)". This is not a perfect method in some conditions.

Stoppage in second innings

A simple example of the D/L method being applied was the first One Day International (ODI) between India and Pakistan in their 2006 ODI series. India batted first, and were all out in the 49th over for 328. Pakistan, batting second, were 7 wickets down for 311 when bad light stopped play after the 47th over.

In this example, Pakistan's target, had the match continued, was 18 runs in as many 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 target score of 304 at the end of the 47th over, with the officially listed result as "Pakistan won by 7 runs (D/L Method)".

Examples in T20 matches

During the 2010 ICC World T20, the D/L method was used in the group stage match between Sri Lanka and Zimbabwe. Sri Lanka scored 173/7 in 20 overs batting first and Zimbabwe were 29/1 in 5 overs when rain interrupted play. Sri Lanka won the match by 14 runs according to the D/L method.

On the same day, another group match between England and West Indies was also decided by the D/L method. England scored 191/5 in 20 overs, and rain interrupted play after 2.2 overs of the chase when West Indies had scored 30/0. According to the D/L method, West Indies were set a target of 60 runs in 6 overs, which they achieved with a ball to spare. The then English captain Paul Collingwood heavily criticized the usage and appropriateness of the D/L method in T20 matches.

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.

Using a published table 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. This percentage is then used to calculate a target (sometimes called a 'par score') that is usually a fractional number of runs. If the second team passes the target then the second team is taken to have won the match; if the match ends when the second team has exactly met (but not passed) the target (rounded down to the next integer) then the match is taken to be a tie.

Application

The Duckworth/Lewis method is relatively simple to apply, but requires a published reference table and some simple mathematical calculations. As with most non-trivial statistical derivations, however, the D/L method can produce results that are somewhat counterintuitive, 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.

Applied to 50 over matches, each team has to face at least 20 overs before D/L can decide the game. In Twenty20 games, each side has to face at least 5 overs.

History and creation

The D/L method was created by two British statisticians, Frank Duckworth and Tony Lewis. It was first used in international cricket in the second game of the 1996/7 Zimbabwe versus England One Day International series, which Zimbabwe won by 7 runs,and was formally adopted by the International Cricket Council in 2001 as the standard method of calculating target scores in rain shortened one-day matches.

Various different methods had been previously used to achieve the same task, including run-rate ratios, the score that the first team had achieved at the same point in their innings, and targets derived by totaling the best scoring overs in the initial innings. All of these methods have flaws that are easily exploitable. For example, run-rate ratios do not account for how many wickets the team batting second have lost, but simply reflect how quickly they were scoring at the point the match was interrupted; thus, 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. Notoriously, the "best-scoring overs" method, used in the 1992 Cricket World Cup, left the South African cricket team requiring 21 runs from one ball (when the maximum score from any one ball is generally six runs). Prior to a brief rain interruption, South Africa was chasing a target of 22 runs from 13 balls but, following the stoppage, the team's amended target became 21 (less only one run) to be scored off just one ball (reduced by 12 balls) due to the best-scoring overs method . The D/L method removes – or at least normalises – this flaw: in this match, the revised D/L target would have been four runs to tie or five to win from the final ball.

Updates

The published table that underpins the D/L method is regularly updated, most recently in 2004, as it became clear 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. 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.

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."

Criticism

The D/L method has been criticized based on the fact that wickets are (necessarily) a much more heavily weighted resource than overs, leading to the observation 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 asking rate was 6.1, it could be enough to score at 4.75 an over for the first 20–25 overs).

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

More common informal criticism from cricket fans and journalists of the D/L method is that it is overly complex and can be misunderstood. 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 realising 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.