Duckworth-Lewis method

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In the sport of cricket, the Duckworth-Lewis method (D/L method) is a mathematical way to calculate the target score for the team batting second in a one-day 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. It is not used in first-class cricket or Test matches.

Contents 1 History 2 Application 3 Theory 4 Updates 5 Example 6 Criticisms 7 References 8 Further reading 9 External links

[edit] History

The D/L method was devised by two English 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.^[1]

In 2001, it was formally adopted by the International Cricket Council as the standard method of calculating target scores in rain shortened one-day matches.

Previous methods used to achieve the same task included the use of run-rate ratios, the use of the score that the first team had achieved at the same point in their innings, and the use of targets derived by totalling the best scoring overs in the initial innings.

All of these older methods have flaws that are easily exploitable. 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. The best-scoring overs method, used in the 1992 Cricket World Cup, left the South African cricket team requiring 21 runs from one ball – where the maximum score from any one ball is generally six runs – where prior to the brief rain interruption, they had a target of 22 from 13 balls, a target that although difficult, was at least attainable, leaving an exciting conclusion to the game in prospect, a finish destroyed when the team was given the same target off 12 fewer balls.^[2] These flaws are not present, or at least effectively normalised, by the D/L method.

[edit] Application

The D/L method is relatively simple to apply, but requires a published reference table and some simple mathematical calculation (or use of a computer). 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.

[edit] 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.

[edit] 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.^[3] 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 modeling, and requires the use of a computer. The Professional Edition has been in use in all international one-day cricket matches since early 2004.

[edit] Example

A simple example of the D/L method being applied was the first One-Day International 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.

This is a relatively uncontroversial example, because with three full overs left to play (18 balls) and three wickets in hand, most cricket fans would agree that Pakistan would be almost certain to close the 17-run gap and take the match. In fact, application of the D/L method showed that at the end of the 47th over, the target was 304, so the result of the match is officially listed as "Pakistan won by 7 runs (D/L Method)".^[4]

[edit] Criticisms

Some observers believe that the D/L method does not take into account certain noticeable trends in the scoring rates during an innings, such as the sharp rise in difficulty as higher numbers of runs need to be scored per over, or the trend to increased scoring rates towards the end of an innings. However, any such trend that is statistically real over the historical record of one-day matches is automatically taken into account by the fact that the D/L tables use historical match data to determine the relationship between resources and run rates.

More detailed criticism is 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, it is more sensible not to lose wickets and score at just an above average rate (if the asking rate, for example, is 6.1, it's enough to score at 4.75 or 5 an over for the first 20-25 overs).^[5]

Another criticism is that the D/L method does not account for changes in the number of overs during which field restrictions are in place.^[6]

[edit] References

• Duckworth, F "A Role for Statistics in International Cricket" [1] Teaching Statistics, (June 2001) Volume 23, No. 2 pp 38-44
• Duckworth, FC & Lewis, AJ "A fair method of resetting the target in interrupted one-day cricket matches" Journal of the Operational Research Society, (Mar 1998) Volume 49, No. 3 pp 220-227

[edit] Further reading

• Duckworth, FC & Lewis, AJ "Your Comprehensive Guide to The Duckworth Lewis Method for Resetting Targets in One-day Cricket", Acumen Books, 2004. ISBN 0-9548718-0-4

[edit] External links

• Cricinfo's introduction to the D/L method
• Cricinfo's D/L method FAQ
• Duckworth/Lewis online calculator
• Duckworth/Lewis standard edition table of resource percentages
• ICC's Duckworth/Lewis Method of re-calculating the target score in an interrupted international match
• BBC Sport's explanation of the D/L method
• Acumen Books