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The Duckworth-Lewis Method (1999)
An introduction to the D/L (Duckworth/Lewis) method of resetting targets in interrupted one-day cricket matches by Frank Duckworth & Tony Lewis
The D/L method of resetting targets in rain-affected one-day cricket matches has now been in use for well over two years and has been adopted for use in most test playing countries of the world. And it is the 'rain-rule' in use for the 1999 World Cup competition played in May and June and hosted by the England & Wales Cricket Board.
The method is the invention of Frank Duckworth and Tony Lewis. Frank is a consultant statistician and editor of the Royal Statistical Society's monthly news magazine, RSS NEWS. Tony is a lecturer in mathematical subjects in the Faculty of Computer Studies and Mathematics at the University of the West of England, Bristol and chairman of the Western Branch of the Operational Research Society
The method that they have invented is simple to apply provided one is prepared to take a few minutes to understand its logic. The calculations can easily be performed using nothing more than a single table of numbers and a pocket calculator although a purpose built computer program is available to undertake the calculations accurately and quickly in match situations (see further information on this below). With a little practice, however, there is no reason why anyone should not be able to calculate the revised target and in quick time. The authors firmly believe that the method is simple enough for it to be adopted for use at all levels of limited-overs cricket. This article provides a summary of the way the method works.
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 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 and 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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Worked 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 and so to give the target Team 1's score 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
For competitions commencing April 1999, the next lower whole number, 181, is the score to tie, or the 'par score' for the match situation at the stoppage.
As there is to be no further play, the winner is decided according to whether or not the par score has been exceeded. With 199 runs on the board, they have exceeded this by 18 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 ICC (1995) 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.]
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Example 2: Interruption to Team 2's innings
A one-day match has been shortened to 40 overs per side before it commenced. 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 and so to give the target Team 1's score 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 which rounds down to 175 to tie with a revised target of 176. They then require a further 36 runs to win from 5 overs with 5 wickets in hand.
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Example 3: Interruption to Team 1's innings
In an ODI, Team 1 have lost 7 wickets in scoring 190 runs in 40 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 40 overs. What is the target for Team 2?
Because of the different stages of the teams' innings that their 10 overs are lost, they represent different losses of resource. Team 1 have lost 7 wickets and had 10 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 20.6% resource percentage they had remaining. Having started with 100% they have used 100 - 20.6 = 79.4%; in other words they have had 79.4% resources available for their innings.
Team 2 will also receive 40 overs. With 40 overs left and no wicket lost you will see from the table that the resource percentage which they have available (relative to a full 50 over innings) is 90.3%. Team 2 thus have 90.3 - 79.4 = 10.9% greater resource than had Team 1 and so they are set a target which is 10.9% of 225, or 24.53, more runs than Team 1 scored. [225 is the average in 50 overs for ODIs]
Using the sum 190 + 24.53 = 214.53 rounding down gives 214 to tie and Team 2's target is 215 in 40 overs.
Note: All other target resetting methods currently in use make no allowance for this interruption. They set the target of 191 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 40 overs and have been handed an unfair advantage. D/L neutralises this by setting Team 2 a higher target than the number of runs Team 1 actually scored.
