Duckworth-Lewis method
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In the sport of cricket, the Duckworth-Lewis method (D/L method) is a 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 assessing a target score, however as it attempts to predict what "would have happened" had the game come to its natural conclusion, it will likely always generate some controversy.
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History
The D/L method was devised by two statisticians, Frank Duckworth and Tony Lewis in 1998, and has been adopted by the International Cricket Council as the standard method of calculating target scores in shortened one-day matches.
Previous methods used to achieve the same task included run-rate ratios, selecting the score that the first team had achieved at the equivalent point in their innings, and targets derived by totalling the best scoring overs in the initial innings.
All of these previous methods, under analysis, have easily exploitable flaws. A run rate ratio, for example, takes no account of how many wickets the team batting second have lost, only how fast they were scoring at the point the match was interrupted. Thus if a team feels a rain stoppage is 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. This flaw is not present in the D/L method.
Application
The D/L method is relatively simple to apply, but requires a published reference table and some simple mathematical calculation. 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.
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 "par score" that is usually a fractional number of runs. If the second team passes the par score set by the first team 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 par score (rounded down to the next integer) then the match is taken to be a tie.
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.
Example
A simple example of the D/L method being applied was the first one-day International between India and Pakistan in their 2006 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 favoured to easily 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 par score was 304, so the result of the match is officially "Pakistan won by 7 runs (D/L Method)"[1].
Reference
- 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