Understanding the Duckworth-Lewis system in cricket
HISTORY
Over the years if a one day cricket match
was shortened by poor weather calculations
were made about how many runs the team chasing the total
needed. This total was based on the average runs scored by
the other side over all their overs. this wasn't always fair
so another way was needed. One that was fair and could work
ball by ball as the target total, one that could be known all
the way through the game. The solution was the famous and sometime
infamous Duckworth-lewis system, a system not everyone understands.
The D/L method was devised by two British statisticians, Frank Duckworth and Tony Lewis.
the revised D/L target would have been four runs to tie or five to win from the final ball.
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 still 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 single table gives the percentage of these combined resources that remain for any number of overs left and wickets lost. An extract of the over-by-over table is given in Table 1. (A ball-by-ball version of the table has also been produced to enable scorers to deal with instances when play is interrupted 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 run-scoring 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 result in Team 2 having more resources available, then their target is revised upwards to compensate for the extra resources they have at their disposal.
HOW DOES IT WORK?
For example: a team have lost five wickets after receiving 25 of their 50 overs
when rain stops play.
At this point, using the table produced by the Duckworth-Lewis method, the team's
remaining resources are valued at 42.2%.
If 15 overs are then lost because of the weather, the innings will be completed
after only 10 more overs.
The D/L method says that, with 10 overs left and five wickets lost, the team has
26.1% of their resources left.
To compensate for the lost overs, we must calculate the resource
% lost.
This works out to 42.2 - 26.1 = 16.1.
If the team had been chasing a total of 250 runs, their new target is calculated in
the following way.
Resources available at the start = 100%
Resources lost = 16.1
Resources available after rain interruption = 83.9%
Then reduce team one's score in the following way. Multiply team one's runs scored
by the recalculated resources divided by the resources available at the start.
That is: 250 x 83.9/100 = 209.75.
The target is then rounded to the nearest whole number, so the team batting second
would be set a target of 210 to win.
Simple!
STILL YOU HAVE SOME DOUBTS..?
See this examples
Example 1:
Premature curtailment of Team 2's innings
Team 1 have scored 250 runs from their 50 available overs and Team 2 lose 5 wickets in scoring 199 runs in 40 overs. Play is then stopped by the weather, the rain refuses to relent and the match is abandoned. A decision on the winner is required.
Team 1's innings: this was uninterrupted, so the resource percentage available is 100%.
Team 2's innings: resource % available at start of innings = 100%
After 40 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at suspension of play = 27.5%
As play is abandoned all this remaining resource is lost.
Hence resource % available for Team 2's innings = 100 - 27.5 = 72.5%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 72.5/100
Team 1 scored 250, so Team 2's 'target' is 250 x 72.5/100 = 181.25
As there is to be no further play, the winner is decided according to whether or not this target has been exceeded. With 199 runs on the board, they have exceeded their required target by 17.75 and so are declared the winners by 18 runs.
Note : The above result is quite fair as Team 2 were clearly in a strong position when play was stopped and would very likely have gone on to win the match if it hadn't rained. Most other methods of target revision in use would, unfairly, make Team 1 the winners. The average run rate method gives 201 to win, the Current ICC method gives 227 and the parabola method gives 226. [Setting the target by the method of Discounted Total Runs - the Australian rain-rule - requires knowledge of the runs made by Team 1 from their most productive overs but the target would almost certainly be no lower than that required under average run rate and would probably be much higher so that Team 2 would very probably lose by this method as well.
Example 2:
Interruption to Team 2's innings
In an ECB Axa Life (Sunday) League match Team 1 have scored 200 runs from their 40 available overs and Team 2 lose 5 wickets in scoring 140 runs in 30 overs. Play is then suspended and 5 overs are lost. What is Team 2's revised target?
Team 1's innings: At the start of 40 over innings resource percentage available = 90.3%
Team 2's innings: resource % available at start of 40 over innings = 90.3%
After 30 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at start of suspension = 27.5%
5 overs are lost, so when play is resumed 5 overs are left.
From table, resource % left at resumption of play = 16.4%
Hence resource % lost = 27.5 - 16.4 = 11.1%
so resource % available for Team 2's innings = 90.3 - 11.1 = 79.2%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 79.2/90.3
Team 1 scored 200, so Team 2's target is 200 x 79.2/90.3 =175.42, or 176 to win, and they require a further 36 runs from 5 overs with 5 wickets in hand.
Example 3:
Interruption to Team 1's innings
In an ODI, Team 1 have lost 2 wickets in scoring 100 runs in 25 overs from an expected 50 when extended rain leads to Team 1's innings being terminated and Team 2's innings is also restricted to 25 overs.
What is the target score for Team 2?
Because of the different stages of the teams' innings that their 25 overs are lost, they represent different losses of resource. Team 1 have lost 2 wickets and had 25 overs left when the rain arrived and so from the table you will see that the premature termination of their innings has deprived them of the 61.8% resource percentage they had remaining.
Having started with 100% they have used 100 - 61.8 = 38.2%; in other words they have had only 38.2% resources available for their innings.
Team 2 will also receive 25 overs. With 25 overs left and no wicket lost you will see from the table that the resource percentage which they have available (compared to a full 50 over innings) is 68.7%. Team 2 thus have 68.7 - 38.2 = 30.5% greater resource than had Team 1 and so they are set a target which is 30.5% of 225, or 68.63, more runs than Team 1 scored. [225 is the average in 50 overs for ODIs]
Team 2's revised target is therefore set at 168.63, or 169 to win in 25 overs, and the advantage to Team 2 from knowing in advance of the reduction in their overs is neutralised.
Note: Most of the other target resetting methods in use make no allowance for this interruption. They set the target of 101 to win simply because both teams are to receive the same number of overs. This is clearly an injustice to Team 1 who were pacing their innings to last 50 overs when it was curtailed, whereas Team 2 knew in advance of the reduction of their innings to 25 overs and have been handed an unfair advantage. D/L allows for this by setting Team 2 a higher target than the number of runs Team 1 actually scored, as described above.
SO FROM ABOVE EXAMPLES ONE MUST SATISFIED WITH THIS LAW..?
NO WAY
Here are some famous bad examblesQ1.
Suppose we are playing a 50-overs-per-side game where only 10 overs per side
are needed for the match to count. Team 1 send in pinch hitters and get off
to a wonderful start making 100 for no wicket after 10 overs.
There is then a prolonged stoppage and when play can resume Team 1's innings is closed and there is only just time for Team 2 to face the minimum 10 overs.
The D/L calculation (Standard Edition) gives Team 2's target as 151 in 10 overs.
How can this practically impossible target be justified?
Q2
Same playing regulations as in Q1 Team 1 make the excellent score of 350 in their 50 overs
and Team 2 start their reply cautiously and reach 40/0 in 10 overs.
The heavens now open (or the floodlights fail) and further play is ruled impossible.
Under the Standard Edition of the D/L system Team 2 are declared the winners by 3 runs.
They were clearly already falling behind the run rate they needed even allowing for the
fact that they had all their wickets intact, so how can this result be justified?
The above represent the two worst-case scenarios for treatment by the Standard Edition of the D/L method. They could only give such extreme consequences with playing regulations that allow a minimum of 10 overs per side for the match to count. But a similar, though less exaggerated, injustice could still arise even with a minimum of 20 overs per side required.
The Standard D/L method was devised so that anyone could perform the calculations with nothing more than the single table of resource percentages and a pocket calculator. This was regarded as an essential requirement for the method. It was considered that to be totally dependent on a computer would mean that the method could not be used universally, it would be vulnerable to computer failure and it would be more difficult to explain how the targets were calculated.
The use of the simplifying single table of resource percentages meant that actual performance must necessarily be assumed to be proportional to average performance. In 95% of cases this assumption is valid, but the assumption breaks down when an actual performance is far above the average, as is the case in the scenarios of Q10 and Q11 and in the record-breaking match between South Africa and Australia (March 2006) in which South Africa scored 438/9 to beat Australia's 434 in 50 overs.
This problem has now been overcome by use of the Professional Edition and this has been in general use for most matches at the top level of the game, including ODIs, since early in 2004. It can only be operated by using a computer program.
FREQUENTLY ASKED QUESTIONS
1. What is the difference between the Standard Edition and the Professional Edition?
At the top level of the game, the Professional Edition of the D/L method is now used.
This requires use of a computer program. At lower levels of the game, where use of a computer cannot always be guaranteed, the Standard Edition is used. This is the method which was used universally before 2004; it is operated manually using the published tables of resource percentages.
2. How do the results of the Professional Edition differ from those of the previous
(Standard) Edition?
For innings when the side batting first (Team 1) score at or below the average for top
level cricket (which would be about 235 for an uninterrupted 50-over innings),
the results of applying the Professional Edition are generally similar to those from
the Standard Edition. For higher scoring matches, the results start to diverge and the
difference increases the higher the first innings total. In effect there is now a
different table of resource percentages for every total score in the Team 1 innings,
and so a computer is essential to operate the system.
3. How do we know whether to use the Professional Edition or the Standard Edition?
The decision on which edition should be used is for the cricket authority which runs
the particular competition. The Professional Edition can only be operated by running
the computer software CODA.
Playing conditions for ODIs and for most countries' national competitions require that
the Professional Edition is used where a computer can be guaranteed to be available
for all matches; otherwise, or in the unlikely event of the computer failing to be
available and operable, the Standard Edition is used (see Q1).
------------------------------------------------------------------------------
HISTORY
Over the years if a one day cricket match
was shortened by poor weather calculations
were made about how many runs the team chasing the total
needed. This total was based on the average runs scored by
the other side over all their overs. this wasn't always fair
so another way was needed. One that was fair and could work
ball by ball as the target total, one that could be known all
the way through the game. The solution was the famous and sometime
infamous Duckworth-lewis system, a system not everyone understands.
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 seven 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 used previously, 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 these methods have flaws that are easily exploitable.
For example,
run-rate ratios take no account of how many wickets the team batting second have lost,
but simply reflect how quickly they were scoring when the match was interrupted;
so, 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 one ball is generally six runs).
Before 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 (a reduction of only one run)
to be scored off just one ball (a reduction of 12 balls).
The D/L method avoids 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.
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 still 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 single table gives the percentage of these combined resources that remain for any number of overs left and wickets lost. An extract of the over-by-over table is given in Table 1. (A ball-by-ball version of the table has also been produced to enable scorers to deal with instances when play is interrupted 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 run-scoring 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 result in Team 2 having more resources available, then their target is revised upwards to compensate for the extra resources they have at their disposal.
HOW DOES IT WORK?
For example: a team have lost five wickets after receiving 25 of their 50 overs
when rain stops play.
At this point, using the table produced by the Duckworth-Lewis method, the team's
remaining resources are valued at 42.2%.
If 15 overs are then lost because of the weather, the innings will be completed
after only 10 more overs.
The D/L method says that, with 10 overs left and five wickets lost, the team has
26.1% of their resources left.
To compensate for the lost overs, we must calculate the resource
% lost.
This works out to 42.2 - 26.1 = 16.1.
If the team had been chasing a total of 250 runs, their new target is calculated in
the following way.
Resources available at the start = 100%
Resources lost = 16.1
Resources available after rain interruption = 83.9%
Then reduce team one's score in the following way. Multiply team one's runs scored
by the recalculated resources divided by the resources available at the start.
That is: 250 x 83.9/100 = 209.75.
The target is then rounded to the nearest whole number, so the team batting second
would be set a target of 210 to win.
Simple!
STILL YOU HAVE SOME DOUBTS..?
See this examples
Example 1:
Premature curtailment of Team 2's innings
Team 1 have scored 250 runs from their 50 available overs and Team 2 lose 5 wickets in scoring 199 runs in 40 overs. Play is then stopped by the weather, the rain refuses to relent and the match is abandoned. A decision on the winner is required.
Team 1's innings: this was uninterrupted, so the resource percentage available is 100%.
Team 2's innings: resource % available at start of innings = 100%
After 40 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at suspension of play = 27.5%
As play is abandoned all this remaining resource is lost.
Hence resource % available for Team 2's innings = 100 - 27.5 = 72.5%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 72.5/100
Team 1 scored 250, so Team 2's 'target' is 250 x 72.5/100 = 181.25
As there is to be no further play, the winner is decided according to whether or not this target has been exceeded. With 199 runs on the board, they have exceeded their required target by 17.75 and so are declared the winners by 18 runs.
Note : The above result is quite fair as Team 2 were clearly in a strong position when play was stopped and would very likely have gone on to win the match if it hadn't rained. Most other methods of target revision in use would, unfairly, make Team 1 the winners. The average run rate method gives 201 to win, the Current ICC method gives 227 and the parabola method gives 226. [Setting the target by the method of Discounted Total Runs - the Australian rain-rule - requires knowledge of the runs made by Team 1 from their most productive overs but the target would almost certainly be no lower than that required under average run rate and would probably be much higher so that Team 2 would very probably lose by this method as well.
Example 2:
Interruption to Team 2's innings
In an ECB Axa Life (Sunday) League match Team 1 have scored 200 runs from their 40 available overs and Team 2 lose 5 wickets in scoring 140 runs in 30 overs. Play is then suspended and 5 overs are lost. What is Team 2's revised target?
Team 1's innings: At the start of 40 over innings resource percentage available = 90.3%
Team 2's innings: resource % available at start of 40 over innings = 90.3%
After 30 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at start of suspension = 27.5%
5 overs are lost, so when play is resumed 5 overs are left.
From table, resource % left at resumption of play = 16.4%
Hence resource % lost = 27.5 - 16.4 = 11.1%
so resource % available for Team 2's innings = 90.3 - 11.1 = 79.2%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 79.2/90.3
Team 1 scored 200, so Team 2's target is 200 x 79.2/90.3 =175.42, or 176 to win, and they require a further 36 runs from 5 overs with 5 wickets in hand.
Example 3:
Interruption to Team 1's innings
In an ODI, Team 1 have lost 2 wickets in scoring 100 runs in 25 overs from an expected 50 when extended rain leads to Team 1's innings being terminated and Team 2's innings is also restricted to 25 overs.
What is the target score for Team 2?
Because of the different stages of the teams' innings that their 25 overs are lost, they represent different losses of resource. Team 1 have lost 2 wickets and had 25 overs left when the rain arrived and so from the table you will see that the premature termination of their innings has deprived them of the 61.8% resource percentage they had remaining.
Having started with 100% they have used 100 - 61.8 = 38.2%; in other words they have had only 38.2% resources available for their innings.
Team 2 will also receive 25 overs. With 25 overs left and no wicket lost you will see from the table that the resource percentage which they have available (compared to a full 50 over innings) is 68.7%. Team 2 thus have 68.7 - 38.2 = 30.5% greater resource than had Team 1 and so they are set a target which is 30.5% of 225, or 68.63, more runs than Team 1 scored. [225 is the average in 50 overs for ODIs]
Team 2's revised target is therefore set at 168.63, or 169 to win in 25 overs, and the advantage to Team 2 from knowing in advance of the reduction in their overs is neutralised.
Note: Most of the other target resetting methods in use make no allowance for this interruption. They set the target of 101 to win simply because both teams are to receive the same number of overs. This is clearly an injustice to Team 1 who were pacing their innings to last 50 overs when it was curtailed, whereas Team 2 knew in advance of the reduction of their innings to 25 overs and have been handed an unfair advantage. D/L allows for this by setting Team 2 a higher target than the number of runs Team 1 actually scored, as described above.
SO FROM ABOVE EXAMPLES ONE MUST SATISFIED WITH THIS LAW..?
NO WAY
Here are some famous bad examblesQ1.
Suppose we are playing a 50-overs-per-side game where only 10 overs per side
are needed for the match to count. Team 1 send in pinch hitters and get off
to a wonderful start making 100 for no wicket after 10 overs.
There is then a prolonged stoppage and when play can resume Team 1's innings is closed and there is only just time for Team 2 to face the minimum 10 overs.
The D/L calculation (Standard Edition) gives Team 2's target as 151 in 10 overs.
How can this practically impossible target be justified?
Q2
Same playing regulations as in Q1 Team 1 make the excellent score of 350 in their 50 overs
and Team 2 start their reply cautiously and reach 40/0 in 10 overs.
The heavens now open (or the floodlights fail) and further play is ruled impossible.
Under the Standard Edition of the D/L system Team 2 are declared the winners by 3 runs.
They were clearly already falling behind the run rate they needed even allowing for the
fact that they had all their wickets intact, so how can this result be justified?
The above represent the two worst-case scenarios for treatment by the Standard Edition of the D/L method. They could only give such extreme consequences with playing regulations that allow a minimum of 10 overs per side for the match to count. But a similar, though less exaggerated, injustice could still arise even with a minimum of 20 overs per side required.
The Standard D/L method was devised so that anyone could perform the calculations with nothing more than the single table of resource percentages and a pocket calculator. This was regarded as an essential requirement for the method. It was considered that to be totally dependent on a computer would mean that the method could not be used universally, it would be vulnerable to computer failure and it would be more difficult to explain how the targets were calculated.
The use of the simplifying single table of resource percentages meant that actual performance must necessarily be assumed to be proportional to average performance. In 95% of cases this assumption is valid, but the assumption breaks down when an actual performance is far above the average, as is the case in the scenarios of Q10 and Q11 and in the record-breaking match between South Africa and Australia (March 2006) in which South Africa scored 438/9 to beat Australia's 434 in 50 overs.
This problem has now been overcome by use of the Professional Edition and this has been in general use for most matches at the top level of the game, including ODIs, since early in 2004. It can only be operated by using a computer program.
FREQUENTLY ASKED QUESTIONS
1. What is the difference between the Standard Edition and the Professional Edition?
At the top level of the game, the Professional Edition of the D/L method is now used.
This requires use of a computer program. At lower levels of the game, where use of a computer cannot always be guaranteed, the Standard Edition is used. This is the method which was used universally before 2004; it is operated manually using the published tables of resource percentages.
2. How do the results of the Professional Edition differ from those of the previous
(Standard) Edition?
For innings when the side batting first (Team 1) score at or below the average for top
level cricket (which would be about 235 for an uninterrupted 50-over innings),
the results of applying the Professional Edition are generally similar to those from
the Standard Edition. For higher scoring matches, the results start to diverge and the
difference increases the higher the first innings total. In effect there is now a
different table of resource percentages for every total score in the Team 1 innings,
and so a computer is essential to operate the system.
3. How do we know whether to use the Professional Edition or the Standard Edition?
The decision on which edition should be used is for the cricket authority which runs
the particular competition. The Professional Edition can only be operated by running
the computer software CODA.
Playing conditions for ODIs and for most countries' national competitions require that
the Professional Edition is used where a computer can be guaranteed to be available
for all matches; otherwise, or in the unlikely event of the computer failing to be
available and operable, the Standard Edition is used (see Q1).
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