All @ SurreyDowns.Org thank Tony Lewis and Frank Duckworth who have granted permission to re-produce their D/L Method @ SurreyDowns.Org"

The D/L Method has of course been adopted by cricket authorities around the world as the way of determining revised match totals after rain. Paul@SurreyDowns.Org March 2003

Summary of Duckworth / Lewis method

The D/L method sets a revised target for the side batting second (Team 2) when overs have been lost by a suspension in play.  The revision is not in proportion to the numbers of overs the two sides can receive but is in accordance with the run-scoring resources the two sides have at their disposal.  These resources include both overs and wickets in combination.

• A single table gives the resources remaining at any stage of an innings for any number of overs left and wickets lost.  The resources are expressed in terms of the percentages of the resources of a full 50-over innings.

• If either innings is shortened after it has started then the balance of resources of the two sides is upset and a revised target needs to be set in accordance with the resources available to the two sides.  To find the resources available for either innings, you use the table to find out the resources lost from that innings and subtract this from the resources with which the innings started.  If the innings started with 50 overs to be received, the resources at the start of the innings are 100%.  But if the innings is shortened before it starts, or if the match is of less than 50 overs per side, then the resources at the start are less than 100%.

n      To find the resources lost from an innings due to an interruption:

      (i) note the numbers of overs left and wickets lost at the start of the suspension; use the table to find the resources remaining

      (ii) note  the same  at  the  resumption  of  play  and  from  the  table  read off  the resources now remaining

      (iii) subtract (ii) from (i) to give the resources lost.

n      To find the resources available subtract the resources lost from the resources that were available when the innings started.

n      When a revised target has to be set, find the resources available for both sides and calculate the revised target as follows, always rounding down to a whole number.

n      If the resources available to Team 2 (denote this by R2) are less than those for Team 1 (R1), then the target is revised downwards in proportion to the resources.

Thus Team 2's revised target = Team 1's actual score x R2/R1, plus one run.  (One run less than the target gives a tie.)

If the resources available to Team 2 are greater than those for Team 1, then Team 2's target must be revised upwards.  The excess runs required are calculated by applying the excess resource to the average 50-over total of 235 (or whatever number is decided upon for the appropriate class of game).

Thus Team 2's revised target = Team 1's actual score + (R2 - R1) x 235/100, plus one run.  (One run less than the target gives a tie.)

The Duckworth/Lewis method - General Information

Background
The Duckworth/Lewis (D/L) method of target resetting in one-day cricket was first used early in 1997 and has since spread to most of the cricketing world.  On 1 Sept 2001 the method became the official  'rain-rule' in use in all one-day competitions in ICC full, associate and affiliate member countries and is likely to be used also in domestic competitions in these countries.  Its great virtue is that, unlike other methods of correction previously in use, it maintains the balance of the game across a suspension in play.  If one team are in the ascendancy when play is halted and the match has to be shortened, they are still in the ascendancy, and by just as much, when play can resume with fewer overs remaining.

Concept of the method
The method is based on the concept that a match starts with both teams possessing the same resources to build their innings.  They have a fixed number of overs to receive, usually 50 in international matches, and they have ten wickets to lose.

As the innings progresses these two resources are gradually used up and a single table of figures gives the percentages of the run scoring resources of the innings that remain for all possible combinations of overs left and wickets lost.  The table has been constructed from a detailed study of the scorecards from many hundreds of one-day matches, mainly internationals, played over recent years.

If an innings is interrupted and has to be shortened, the table is used to find out what percentage of the resources are lost.

Any shortening of the match after it has started upsets the balance of resources and a revised target is necessary to compensate the team that have suffered the more.  This is the case even if the two teams both end up with the same number of overs to receive.

The target adjustment is based on the relative run scoring resources available to the two sides after the resources lost by each team have been taken into account.

The table reflects the fact that the resources lost by a loss of a certain number of overs depend on:-

• How many overs remained, and

• How many wickets were down at the time.

For instance, a loss of overs near the end of an innings, especially when there are plenty of wickets still in hand, is usually a far greater loss of resource than the same loss of overs at the beginning of an innings.

How it works
Suppose the team batting first (Team 1) have completed their innings and an interruption occurs during the second innings.   The team batting second (Team 2) will then have less resources for their innings than Team 1.  In this case Team 2 will be set a reduced target based on the smaller amount of resources they had for their innings compared with Team 1.

But if an interruption occurs during Team 1's innings, it often happens that Team 2 end up with more resources for their innings than Team 1 have had and in this case their target is revised upwards to compensate Team 1 for the way they were more disadvantaged by the timing of the stoppage. 

This is best understood by considering the case where Team 1's innings is curtailed after 40 of an intended 50 overs and Team 2 have just time to face the same 40 overs.  Team 1 had been pacing their innings to last 50 overs and provided they had wickets in hand might have expected to have made 60 or 70 runs from the final 10 overs. 

Team 2, on the other hand, knew they had only 40 overs to face from the moment they started their innings and were able to take greater risks to achieve a higher scoring rate right from the start.  On average a team only make about 20 - 25 fewer runs from a 40-over innings than they make from a 50-over innings.

Par score
Provided that the minimum number of overs have been faced by each side, if a match has to be terminated before a result has been obtained, this is decided by comparing Team 2's score at the termination with the ‘par score' for their target.  The par score is calculated in the same way as a revised target and is the score that Team 2 would have to have exceeded to win if the game were abandoned at that point.  Like a revised target it depends on how many overs remain and wickets are lost.

If Team 2's innings has to be abandoned, Team 2 will win if they are ahead of the par score, by the amount that they are ahead.  If they are behind par then Team 1 win by the corresponding amount.  If they are equal to par, the match is tied.

During the progress of Team 2's innings, and regardless of whether or not rain is threatening, the par score provides a useful guide as to how Team 2 are performing relative to their target.  This information is sometimes displayed on scoreboards, usually  as the par score for the end of the  over in progress.

The revised target
Match regulations dictate that a tied match will always be possible even when a revised target has been set.  The revised target is the minimum number of runs that Team 2 require to win the game.  The calculated score to beat is rounded down as necessary (i.e. ignore any figures after the decimal point) to give the score needed for a tie.  The target is one more run than this.

The D/L table
The D/L table as supplied to the scorers fits on a single sheet and covers every over from 50 down to 1 and each number of wickets lost from 0 to 9.  A longer version of the table covering every individual ball is also supplied for dealing with cases where the stoppage occurs mid-over.

Below is an abbreviated version of the table for use with the examples following.  The figures given are the resource percentages remaining relative to those for a full 50-over innings.

Resource Percentage Remaining...

Example 1 (interruption during Team 2’s innings)
Team 1 score 250 in 50 overs.  Team 2 reach 80 for 2 wkts after 20 overs when rain causes the loss of 10 overs.

• When play was stopped, there were 30 overs remaining and 2 wkts down.  The table tells us that 67.3% resources remained.

• When play resumed there were only 20 overs remaining and still 2 wkts down and the table tells us that 52.4% resources now remained.

• So the resources lost due to the rain were 67.3 – 52.4, or 14.9%,

• Team 2 had 85.1% (100 - 14.9) available for their innings compared with Team 1's 100%.

• Team 2 had less resource than Team 1 so the calculation 250 x 85.1/100 gives 212.75 and the revised target is 213 with 212 needed to tie.

Example 2 (interruption during Team 1’s innings)
Team 1 score 143 for 5 wickets in 30 overs and then rain takes away 10 overs from each side's innings reducing it to a 40 overs-per-innings match.  They resume their innings and go on to make a total of 200 in their 40 overs.  [The number of wickets lost is irrelevant.]

• When play was stopped, Team 1 had 20 overs remaining and 5 wkts down.  The table tells us that 38.6% resources remained.

• When play resumed there were only 10 overs remaining and still 5 wkts down.  The table tells us that 26.1% resources remain.

• Team 1's resources were reduced by 12.5% (38.6 – 26.1).  They have 87.5% (100 – 12.5) resources available.

• Team 2’s loss of the same 10 overs from the start of their innings means that they have 40 overs and a resource percentage of 89.3% available.

• So Team 2 have 1.8% (89.3 – 87.5) more resource than Team 1 and need to be set a higher target to compensate Team 1 for the relative disadvantage they would otherwise suffer.

• The target is raised by applying the excess to the average score made in a 50-over innings which has now been updated to 235.  The excess runs is calculated to be 4.7 (1.8% of 235).

• So the revised target is set at 205 (in order to beat 200 + 4.23 = 204.23). 204 ties.

Example 3 (delay to start and premature termination of Team 2’s innings)
Team 1 are all out for 160 after 38 overs in a match reduced to 40 overs per side before its start. The start of Team 2's innings is delayed by rain and they are allocated only 30 overs.

• Team 1 had 40 overs available (the fact that they were all out before they were used up is irrelevant) so the table tells us that they had 89.3% resources (compared with a 50-over innings) available.

• Team 2 go out to bat with 30 overs allocated.  The table tells us that they have 75.1% resources available.

• Team 2 have less resource than Team 1 and need to be set a reduced target.  The calculation 160 x 75.1/89.3 gives 134.56, so Team 2 are set a target of 135.

Now suppose that after 25 overs they have reached 120/8 when heavy rain causes the match to be terminated.

• At that time, with 5 overs remaining and 8 wkts down, Team 2 had 9.4% resources left, and these have been lost by the termination.

• They started with 75.1% and have lost a further 9.4% so they have only a total of 75.1 – 9.4, or 65.7%, available compared with Team 1's 89.3%.

• The calculation 160 x 65.7/89.3 gives 117.71, which is rounded down to give the par score as 117.  At 120/8 Team 2 are 3 runs ahead of par and so Team 2 are declared the winners by 3 runs.

Further points

• To correct a misconception that has arisen in recent years, following an interruption and shortening within the first innings, provided there are no further interruptions, the target for Team 2 is NOT affected by the subsequent loss of wickets.  It is only Team 1's total that is then relevant to the target calculation as can be seen in Example 2.

• The method can be applied to multiple stoppages by updating the resources lost on each occasion.

• They met in 1994 following a statistics conference at which Frank presented a formula for a fair method of target correction and Tony decided that the topic would make a good student research project.  Since then they have worked together and developed the method so that it can be applied using nothing more than the table and a pocket calculator, though they have also developed a computer program for speed and accuracy.

Contact details :

email: f.duckworth@rss.org.uk

email: ajlewis@brookes.ac.uk

History
Prior to D/L the major methods in use were as follows.

Average run rate with the target based on adjusting Team 1's score in proportion
of overs available. (eg half the overs means you have to beat half the runs).
This generally favoured the team batting second

The 1992 World Cup used Most Productive Overs, whereby in the example above the
best half of the overs would be used.  This was very much biased to the team
batting first.  Australia modified this very slightly and was used until D/L
took over

The parabola method and a derivative into percentages were used in the 1996
World Cup (but no rain there) and in West Indies and Asia,  and South Africa used the
Clark Curves.  D/L took over from both of these ahead of the 1999 World Cup

D/L arose post World Cup 1992 after the infamous semi final.  We began to approach ICC/ECB in May 95 - whereupon over the next few years the method began to get adopted and used nationally and internationally.  The rest, as they say, is history!

D/L is now the international standard rain rule for ODI and first class one-day
matches and at many other lower levels of the game.

D / L Method reproduced with the permission of Tony Lewis - March 2003
Our Thanks to Frank Duckworth & Tony Lewis from SurreyDowns.Org

D / L Method Q & A by Frank Duckworth & Tony Lewis