Basis of the method
The D/L method works using the notion that teams have two resources with which to make as many runs as they can - these are the number of overs they have to receive and the number of wickets they have in hand. From any stage in their innings, their further run-scoring capability depends on both these two resources in combination. The table gives the percentage of these combined resources that remain for any number of overs left and wickets lost. An extract from the table is given in Table 1. Information is provided later on how the full table can be obtained, including the ball-by-ball version which is used when stoppages occur mid-over.
When a match is shortened after it has begun, the resources of one or both teams are depleted and the two teams usually have different amounts of resource for their innings. In this case a revised target must be set. The D/L method does this in accordance with the relative resources available to the two teams. If stoppages cause the team batting second (referred to here as Team 2) to have less resources available, as is more often than not the case, then their target will be revised downwards. If, on the other hand, as often happens when Team 1's innings has been interrupted, the stoppages usually result in Team 2 having more resources available then their target is revised upwards to correct for the extra resources they have at their disposal.
Table 1: Extract from the table of resource percentages remaining
Wickets lost
Overs left 0 2 5 7 9
50 100.0 83.8 49.5 26.5 7.6
40 90.3 77.6 48.3 26.4 7.6
30 77.1 68.2 45.7 26.2 7.6
25 68.7 61.8 43.4 25.9 7.6
20 58.9 54.0 40.0 25.2 7.6
10 34.1 32.5 27.5 20.6 7.5
5 18.4 17.9 16.4 14.0 7.0
Reading the table
The single table applies to all lengths of one-day matches from 50 overs-per-side downwards. Because this length of match is by far the most common, the resources listed in the table are expressed as percentages of those available at the start of a 50-over innings. Thus when there are 50 overs still to be received and no wickets have been lost, the resource percentage available is 100%. A 40-over innings starts with a resource percentage of 90.3% relative to a 50 over innings. An innings shortened to 25-overs before it starts commences with a resource percentage of 68.7% relative to 50-over innings. (Although such innings have only half the overs of a 50-over innings they have all 10 wickets and so have much more than half the resources.)
In order to determine the correct resource percentage the batting side has remaining at any stage of its innings, the number of overs left must be identified. This number of overs left, in conjunction with the number of wickets lost, is then used to read the resource percentage remaining from the table.
For example, suppose that after 20 out of 50 overs a team have lost 2 wickets. They have 30 overs left. From the table you will see that the resource percentage remaining is 68.2%.
Suppose now that there is an interruption in play and 10 overs are lost from the innings. When play can resume there are only 20 overs left but there are still, of course, 2 wickets down, and the table now tells us that the resource percentage remaining is 54.0%. Thus the shortening of the innings has caused the team to lose a resource percentage of 68.2 - 54.0 = 14.2%.
Having started with a resource percentage of 100% and lost 14.2%, then if they complete their innings with no further loss of overs, they will have had a resource percentage available for their innings of 100 - 14.2 = 85.8%.
Applying the D/L method
The procedure for setting a revised target, which is the same for any number of stoppages at any stage of the match, is as follows.
For each team's innings
(a) from the table note the resource percentage the team had available at the start of their innings;
(b) using the table, calculate the resource percentage lost by each interruption;
(c) hence calculate the resource percentage available.
If Team 2 have less resources available than Team 1, then calculate the ratio of the resources available to the two teams. Team 2's revised target is obtained by scaling down Team 1's score by this ratio. The figure so obtained is rounded down to the next whole number to give the score needed for a tie. The target is one more run than this. The procedure by which a tie is always possible is a consequence of a change in playing conditions introduced internationally from April 1999.
If Team 2 have more resources available than Team 1, then calculate the amount by which Team 2's resource percentage exceeds Team 1's. Work out this excess as a percentage of 225 [the average 50-over score in 'first class' matches and one-day internationals (ODIs)]. Rounding this down to the next whole number gives the extra runs to add on to Team 1's score to give the score to tie. Adding one run gives Team 2's target.
When a match is shortened after it has begun, the resources of one or both teams are depleted and the two teams usually have different amounts of resource for their innings. In this case a revised target must be set. The D/L method does this in accordance with the relative resources available to the two teams. If stoppages cause the team batting second (referred to here as Team 2) to have less resources available, as is more often than not the case, then their target will be revised downwards. If, on the other hand, as often happens when Team 1's innings has been interrupted, the stoppages usually result in Team 2 having more resources available then their target is revised upwards to correct for the extra resources they have at their disposal.
Table 1: Extract from the table of resource percentages remaining
Wickets lost
Overs left 0 2 5 7 9
50 100.0 83.8 49.5 26.5 7.6
40 90.3 77.6 48.3 26.4 7.6
30 77.1 68.2 45.7 26.2 7.6
25 68.7 61.8 43.4 25.9 7.6
20 58.9 54.0 40.0 25.2 7.6
10 34.1 32.5 27.5 20.6 7.5
5 18.4 17.9 16.4 14.0 7.0
Reading the table
The single table applies to all lengths of one-day matches from 50 overs-per-side downwards. Because this length of match is by far the most common, the resources listed in the table are expressed as percentages of those available at the start of a 50-over innings. Thus when there are 50 overs still to be received and no wickets have been lost, the resource percentage available is 100%. A 40-over innings starts with a resource percentage of 90.3% relative to a 50 over innings. An innings shortened to 25-overs before it starts commences with a resource percentage of 68.7% relative to 50-over innings. (Although such innings have only half the overs of a 50-over innings they have all 10 wickets and so have much more than half the resources.)
In order to determine the correct resource percentage the batting side has remaining at any stage of its innings, the number of overs left must be identified. This number of overs left, in conjunction with the number of wickets lost, is then used to read the resource percentage remaining from the table.
For example, suppose that after 20 out of 50 overs a team have lost 2 wickets. They have 30 overs left. From the table you will see that the resource percentage remaining is 68.2%.
Suppose now that there is an interruption in play and 10 overs are lost from the innings. When play can resume there are only 20 overs left but there are still, of course, 2 wickets down, and the table now tells us that the resource percentage remaining is 54.0%. Thus the shortening of the innings has caused the team to lose a resource percentage of 68.2 - 54.0 = 14.2%.
Having started with a resource percentage of 100% and lost 14.2%, then if they complete their innings with no further loss of overs, they will have had a resource percentage available for their innings of 100 - 14.2 = 85.8%.
Applying the D/L method
The procedure for setting a revised target, which is the same for any number of stoppages at any stage of the match, is as follows.
For each team's innings
(a) from the table note the resource percentage the team had available at the start of their innings;
(b) using the table, calculate the resource percentage lost by each interruption;
(c) hence calculate the resource percentage available.
If Team 2 have less resources available than Team 1, then calculate the ratio of the resources available to the two teams. Team 2's revised target is obtained by scaling down Team 1's score by this ratio. The figure so obtained is rounded down to the next whole number to give the score needed for a tie. The target is one more run than this. The procedure by which a tie is always possible is a consequence of a change in playing conditions introduced internationally from April 1999.
If Team 2 have more resources available than Team 1, then calculate the amount by which Team 2's resource percentage exceeds Team 1's. Work out this excess as a percentage of 225 [the average 50-over score in 'first class' matches and one-day internationals (ODIs)]. Rounding this down to the next whole number gives the extra runs to add on to Team 1's score to give the score to tie. Adding one run gives Team 2's target.
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Introduction
The Duckworth/Lewis method (D/L) is a system used to set victory targets for the side batting second in rain-affected one-day cricket matches.
The system was first used in 1997, and is now used in all official International Cricket Council (ICC) competitions, and some domestic competitions.
The Duckworth/Lewis method derives its name from its co-creators, Frank Duckworth and Tony Lewis. Duckworth is a consultant statistician, and Lewis is a lecturer in quantitative business.
How D/L works
The aim of the Duckworth/Lewis method is to ensure that if a one-day match has to be shortened (ie the number of overs either side will bowl is reduced), the target score for the side batting second will be fair.
In order to calculate this, D/L considers the resources available to both sides. At the start of a match, it is assumed that both sides will bat for 50 overs, and will have 10 wickets. As the game progresses, these resources are consumed (overs are bowled, leaving fewer overs to bat for, and wickets are lost). When a rain interruption results in fewer overs being bowled, D/L is used to calculate what percentage of resources have been lost to both sides because of the weather.
This might sound counusing, so let's take a quick example. The side batting first (Team 1) has scored 200 runs without losing a wicket after 40 overs when the heavens open. The umpires take the players off the field. So much time is lost that the umpires reduce the game to 40 overs per side.
Fair enough, you might think, the side batting second (Team 2) needs to score 201 runs to win.
Before D/L this is the kind of analysis that would have been done, the target would not have been revised, and team 2 would have been set 201 in 40 overs to win the match. But this is clearly unfair on Team 1. All the time they are batting, they are thinking in terms of the 50 overs they will bat for. The batsmen will have paced the innings, and now they are all set for an onslaught in the last five or ten overs, during which they will expect to score a heap of runs (100 or so even) much quicker than they already have, given that they still have ten wickets in hand. Had they known at the start of their innings that they would only receive 40 overs, they would have started increasing their run-rate earlier, and would certainly have ended up with more than 200 after the end of 40 overs.
So, the question becomes, how do we revise the target for the chasing side? Helpfully, a D/L resource table is available for calculating the percentage of a team's resource is left according to the number of overs and wickets remaining (see the links below for extracts from the table). According to the tables, with 10 overs of 50 left, and no wickets lost, the side batting first still has 32.1% of its resources available. In other words, the rain delay has deprived them of just over one third of the available resource they could have used to score runs. They have only had 67.9% of resources available for their innings.
Resource available to Team 1 so far = (Resource available at start) - (Resource lost because of rain)
= 100% - 32.1%
= 67.9%
Team 2 are about to start their innings, which will last for 40 overs. They haven't suffered as much from the loss of overs, because they know from the start of their innings that it will only last 40, and not 50 overs. In other words, they have lost less resource. According to the D/L tables, they have 89.3% of their resources available. This means that they currently have 21.4% more resource available to them. With greater resource, you would expect them to score more runs, so D/L needs to set Team 2 a revised target that is higher than Team 1's total.
Using a figure of 235 as the average score in 50 overs across all One Day Internationals (originally 225 was used, but was revised) Team 2 will need to score an extra 21.4% of 235 runs in order to win the game. That's 50.29 runs, which gives Team 2 a target of 251 (rounding up) to win, or 250 (rounding down) to tie the match.
Resource available to Team 2 so far = 89.3%
Extra resource available to Team 2 = 89.3% - 67.9%
= 21.4%
Revised target = Team 1 score + (extra resource * 235)
= 200 + (21.4% * 235)
= 250.29
By D/L standards, that was a relatively simple calculation. Fortunately, software is available to speed the process of calculating victory targets, although all you really need is a calculator and the full resource tables.
D/L can adjust victory targets regardless of when and for how long a rain interruption occurs, whether it is during the first team's innings, between innings, or during the second team's innings (including multiple interruptions).
All is fair in love and rain-affected ODIs.
With its current revisions, D/L usually seems to set reasonable targets for the side batting second. Occasionally the target might look a little generous, or harsh, but it's certainly an improvement on previous rain rules, as any South African cricket fan will tell you.
On 22 March 1992, South Africa played England for a place in the final of the World Cup. England batted first in a match reduced to 45 overs, and made 252, built on a knock of 83 from Graeme Hick, and a late flurry of runs from Dermot Reeve. South Africa had reached 231 for the loss of 6 wickets, with 13 balls remaining. The rain came down, and England captain Graham Gooch accepted the umpires' offer to take the players off the field. When play resumed, two overs had been lost, leaving South Africa just one ball to score the winning runs. Unfortunately, according to the rain rules in use at the time, the target was not adjusted, so Brian MacMillan and Dave Richardson returned to the crease, with the scoreboard unforgettably showing a victory target of 22 runs from 1 ball.
Not that D/L has been much kinder to South Africa. In the 2003 World Cup, needing to beat Sri Lanka to progress to the super six stage of the competition, the game was abandoned with South Africa, chasing 269 to win, on 229 for the loss of six wickets with 5 overs remaining. Cruelly for the South African side and supporters, D/L calculations gave the result of the match as a tie, and South Africa were knocked out of the competition.
As well as D/L, other systems are used in various domestic competitions around the world. The Australian Rain Rule reduces targets for teams batting second according to the highest scoring overs of the team batting first, while in South African domestic cricket and ODIs played in South Africa, the Clark Curves are used to set a 'par' total. If Team 2 is ahead of the par score when a game is abandoned for rain they are declared winners. I'm no expert on the Clark Curves, but my calculations lead me to the conclusion that they, too, give the result of the 2003 World Cup Sri Lanka v South Africa game as a tie, although I am not certain to what extent South Africa's wickets in hand would be taken into account under that system.
Sources
Duckworth and Lewis have published a guide to making the calculations: "Your comprehensive guide to the Duckworth/Lewis method for target resetting in one-day cricket" (ISBN 1-86043-189-5)
Tables containing some figures for calculation can be found on a number of web pages, a good place to start is
http://www.cricket.org/link_to_database/ABOUT_CRICKET/RAIN_RULES/
Also, the following cricket sites contain articles on D/L:
* www.cricinfo.com
* www.cricket365.com/WC_Rules/index.shtml
* www.rediff.com/cricket
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