An insight on Duckworth Lewis method

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In this changing world, everything around us changes rapidly, so is it with the rules of the sports. In the game of cricket, Duckworth and Lewis method (better known as D/L method) is a mathematical calculation employed now to calculate the revised target for the team batting second in a One-Day cricket match interrupted by weather or by other means.

Before we proceed to give details about D/L method, it is better to have some knowledge of old rain rules and their criticism with examples. Let me start with the “Better Run rate Rule”. As per this rule in a truncated match, the team, which possesses better run rate than the other team, would be declared winners. Consider the India-Pakistan match in 1990 in which India scored 300/2 in their full quota of 50 overs while in the Pakistan’s reply, India bowled magnificently and reduced Pakistan score to 151/9 in 25 overs. Due to rain, match stopped at that juncture and could not resume afterwards. Having scored 151 in 25 overs, Pakistan had slightly higher than the required run rate of 6.0 and was declared winners.

This “Better Run rate Rule” continued until 1992. After realizing that this rule tends to favour the team batting second and that it does not take into account the number of wickets fallen, in an attempt to correct this deficiency, Australia came with the idea of “Most Productive Overs Rule”.

According to this rule, assume Team 1 scored 275/4 in 50 overs, and due to unforeseen circumstance, overs are reduced to 45 for Team 2. In order to revise the target, first arrange the runs scored by Team 1 in each over in the descending order. Then cut off the runs scored in the last 5.0 overs. For example after arranging in descending order, the runs scored by Team 1 in last 5.0 overs is 15. Then, the target for Team 2 is 260(275-15) in 45 overs. This rule was unfair to the Team batting second.

If you can recall the semifinal match between South Africa and England in 1992 World Cup where South Africa needed 22 runs from 13 balls to enter the World Cup finals, rain interrupted and then the target was revised to 21 runs from 1 ball. This rule undermines the good bowling performance of Team 2.

In order to curb all the above defects in the rain-truncated matches, International Cricket Council (ICC) introduced D/L method from 1999 onwards. So far, this method has been applied to more than 350 matches successfully with some criticism. Much of the criticism against it comes from those who failed to understand the D/L formulation fully.

D/L method was devised by two English statisticians, Frank Duckworth and Tony Lewis. D/L recognises the resources that are available to batting side to set the target for the interrupted matches - overs, wickets and the combination of both.

To understand the D/L method, one needs to understand the concept of Resource Percentage (RP). If you could look at the D/L table below to find out RP (The table which is used here is only an extract of the original table that list the RP corresponding to each wicket lost and each ball in every over).

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 7.6
5 18.4 17.9 16.4 14.0 7.0

The row corresponds to number of overs left and a column corresponds to number of wickets lost. In order to calculate RP, for instance look at the table for 50 overs left and 0 wicket lost and the corresponding value in the table is 100 (Resource remaining is 100%). Now calculate RP, after batting 30 overs and losing 5.0 wickets, which will be 40% (resource remaining is only 40%), in other words 60% of resources are already utilized. The RP values in the table are evolved after a detailed analysis of over hundred matches and observed most of the variation in one-day cricket matches.

After calculating RP lost and available, it is easy to find out revised target. Here we take two examples to find out the RP value and revised target.
Consider,
R1 is the RP available of Team 1 (The team that bats first)

R2 is the RP available of Team 2 (The team that bats second)

First example:
Consider Team 1 scored 250/5 in their full quota of 50 overs. Team 2 were 130/5 in 30 overs, rain interrupted at this stage , match started after 50minutes due to which 10 overs were reduced for Team 2 . Then what will be Team 2’s winning target?

Since Team 1 finished their innings normally, they used their entire 100% RP. Initially, Team 2 had 20 overs left and five wickets lost. At that stage, Team 2 RP remaining would be 40%, so they have used 60% of their RP. Then 10 overs were lost. Therefore, when play resumed Team 2 had 20-10 =10 overs left and 5.0 wickets lost, corresponding RP value remaining would be 27.5%.

Due to interruption, Team 2 RP diminished to 40%-27.5% = 12.5% .Team 2’s RP available (R2) has become less than 100% i.e. 100%-12.5%=87.5%. Being lower than Team 1’s 100%, R2 is less than R1. Therefore, Team 2’s wining target is,

RP available (R1) of 100% Team 1 scored 250 then,
RP available (R2) of 87.5% Team 2 score is 250*87.5/100 = 218.75 ≈ 219 (par score), add 1 to get winning target. Therefore, Team 2’s winning target is 219 runs.

Second example:
In a 50-over match, Team 1 makes 120/2 in 20 overs when rain stops play. Team 1’s innings is closed at that point and, when play resumes, Team 2 starts its 20 over innings. What is the RP remaining values of Teams 1 and 2 and the winning target for Team 2? Team 1’s RP remaining (had 30 overs left and 2 wickets lost) is 68.2%.i.e. Team 1 utilised only 31.8% of their resources.

Team 2’s RP remaining (20 overs left and 0 wicket lost) is 58.9%

We therefore have a situation where Team 1 used only 31.8% of their available resources while Team 2 has the opportunity to use 58.9% of their available resources. Obviously, Team 2 has the advantage. In order to neutralise this advantage, increase the target for Team 2 in their 20 overs. RP available (R1) value was 31.8% and for R2 was 58.9%. Since R2 is greater than R1, we need to raise the target.

For that, calculate the difference of R1 and R2 i.e. 58.9%-31.8% = 27.1%. On an average, a typical one-day score in a 50-over match being 235, we estimate the ‘run equivalent’ of 27.1% to be 27.1/100 x 235=63.69 i.e. 64 Therefore, Team 2 must score an additional 64 runs to equal Team 1’s RP and their victory target is therefore 120+64+1=185 runs to win.

To put it simple, after calculating R1 and R2

If R2 is less than R1, Team 2’s revised target is obtained by reducing Team 1’s score S in the ratio of R2 to R1.

i.e. Target = (S*R2/R1) + 1

If R2 is equal to R1, no revision would be needed and Team 2’s target is one more run than Team 1’s score,

i.e. Target = S + 1

If R2 is greater than R1, calculate the amount of excess resources, R2-R1, and take this percentage of the average 50 over total, G50, to give extra runs needed.

i.e. Target = S + (R2-R1)* G50/100+ 1

Much of the criticism for the D/L method is, it gives undue weight to wicket resources than overs, which leads to team batting second not losing many wickets and maintaining average run rate enough to win the match. This method does not consider fielding restriction and power plays and because of its complex calculations, it can be misunderstood.

One of the best way to defeat rain interruption is to convert “one-day internationals” (ODI’s) to “limited-over internationals” (LOI’s) and continue the matches the following day. It is good for cricket to have results on cricketing credentials than on mathematical calculations.