A study on the Duckworth Lewis method

A study on the Duckworth Lewis method: So much has been talked about the rain-interrupted and suspended matches and with that comes the Duckworth Lewis method. When we see the sudden calculations of those projected scores, such revised targets, many of us would have certainly wondered and spent some time in scratching our heads about the nature how this method works and how scores are projected and totals are set. Is it some randomized or a lottery like thing? Or something involving rocket science alike calculations to be understood by the nerds only? Well, the answer to the above two questions is a BIG NO! Now, let us take some time to get command of this method with simplicity. Keep your pen and papers ready!

History and Evolution:

Prior to the evolution of Duckworth Lewis method, several methods were adopted but proved out to be buggy. Initially, a method called Average run rate was considered. It took into account run rate of the team batting first and based on that, team 2 had to score that many runs in the reduced overs. Obviously, this was a method that comprised of large levels of anomalies and the team batting second had too much advantages.

Keeping that in mind, an alternative method was figured out by some of the cricketing legends including Richie Benaud and it came into effect in 1992 World Cup for the first time. This method was called the Most productive overs method, it kept into account the advantages of the team batting second and had many factors. But this method’s fallacies got badly exposed in the semi final of the World Cup, 1992 between England and South Africa. South Africa were batting second and required 47 runs off the last 5 overs when play was suspended. When it resumed, South Africa required 22 off 13 but there was a further suspension due to the wet outfield. English captain Graham Gooch did not accept the playing conditions and further 12 minutes were lost. In the end, it reduced to South Africa requiring 21 runs off 1 delivery! This made the method a joke and was thoroughly criticized.

It was then, that Frank Duckworth and Tony Lewis, statisticians by profession, introduced this method or rather an algorithm so that rain intervened or suspended matches can be completed in the stipulated period of time to fetch an outcome. The method got introduced in 1996 into international cricket and the first match that had the method’s influence was on the match between Zimbabwe and England in 1996-97 and Zimbabwe won the match by 7 runs.

Terminologies:

Prior to the discussion, let us discuss about the terms we would be using, so that things will be easy to catch later on. Have a read-

• ‘Team1’ and ‘Team2’ are generally the two teams involved in the match. Simply, the team batting first will be regarded as ‘Team1’ and the team batting second will be regarded as ‘Team2’.
• N is the number of overs per innings as it was decided prior to the start of the match. Generally, for an ODI, it is 50 (stands for the 50 overs per innings) and for a T20, it is 20.
• S is Team1’s total score i.e. the total score made by the team batting first.
• G50 is the average score agreed upon by the two teams at the start of the game’s play. Generally, a score of 245 is taken for a 50-over game. However, this term is not used for matches where Professional Edition of the D/L method is used, which is generally used for matches where a computer program is used in setting up and deciding the scores. For the Standard Edition, this term is used alongside the published tables. Standard Edition was the only edition used prior to 2004.
• Resource Percentage: Now this is an important term! Basically, the resource percentage stands as the pivot for the calculation of the Duckworth Lewis method. Here, the number of overs left and the number of wickets remaining for the batting side are considered as resources for the teams. For instance, at the start of the innings, the number of wickets left is 10 and the number of overs left is 50, so the resource percentage available at the start of the innings is 100%.
  Duckworth and Lewis framed tables for estimating the resource percentages. The table has been prepared by keeping the wickets remaining ranging from 10 to 1 horizontally and the overs remaining ranging from 50.0 to 0.0 vertically.
  R1 is the Resource Percentage available to Team1, relative to a 50-over innings.
  R2 is the Resource Percentage available to Team2, relative to a 50-over innings.
  If you are a bit confused about this thing, don’t worry! All the doubts will be cleared while we discuss the examples.
• T is the target that ‘Team2’ have to score.

The Algorithm:

1. Calculation of the Resource Percentage Lost Due to A Suspension in the Play:
Just when the play stops or gets suspended temporarily, the resource percentage still available to the team is to be noted from the Resource Percentage Table. We name it as R(a).

After the suspension, when the play resumes, again, we are to make a note of the resource percentage available to the team at that point of time, owing to the reduction of overs. This can be done similarly as the above. We name it as R(b).
Substract, R(a) – R(b) to get R(L1). Thus, R(L1) = R(a) – R(b). R(L1) means the Resource Percentage Lost due to interruption or suspension of play.

Now, in case the innings does not resume after the suspension, the resource percentage lost is the resource percentage available at the time of suspension. In that case, Since R(b) = 0, R(L1) = R(a).

If there happens to be more than one suspensions, then all the resource percentages losses are taken in the order R(L1), R(L2), R(L3) and so on.

The final Resource Loss Percentage, R(L) = R(L1) + R(L2) + R(L3) ……

In the initial stage, Resource Percentage Available to any team at the start of the innings is 100%.

Firstly, we need to evaluate Thus, R1 = [ 100 – R(L) ]%
Now, we are to evaluate R2, which can be estimated simply by looking from the table. If Team2 gets a full 50 overs, R2 will be 100%. If not, it has to be read from the table based on the number of overs remaining and 0 wickets lost.

Now, values of R1 and R2 are to be compared next.
If R1 = R2, NO alterations are made to the target i.e. no revised target for Team2.
T = S + 1

If R1 > R2, Team2’s revised target is made by reducing Team1’s total, S, in the ratio R2/R               1. Any decimal figures should be neglected and the result should be obtained in integer form.
T = (S x R2/R1) + 1

If R2 > R1, Team2’s revised target is calculated and it is greater than Team1’s total, owing to the excess of resources.
The excess resources is calculated by R2 – R1 and it is added to the total as a multiple of the average total, agreed by the two teams, i.e. G50.
T = S + (R2 – R1) x G50/100 + 1

However, the above method holds for Standard Edition. In case of Professional Edition, there is basically no involvement of G50, or the average score.
Hence,   T = (S x R2/R1) + 1

Hence, this is the algorithm.

Calculation of Loss of Play:

There is a mention in the rule book of the ICC that if the playing conditions are not deemed suitable by the umpires owing to weather, light and security related concerns, the umpires possess the right to suspend the play temporarily. In case of rain interruptions, play can resume only after the rain stops completely. And play can resume, only after the on-field umpires inspect the fitness of the ground conditions. The umpires shall also consult the captains of the both teams for the resumption of the play. The Match referee finally amends the resumption of play. There is also a reserve time made for the match. Suppose if the reserve time is 1 hour and the play gets interrupted for 45 minutes, then there will be no loss of play and thereby no over reductions in the match.

However, if the interruption surpasses the reserve time, there will be loss of overs and the loss will be evenly distributed to the innings of each of the team. Generally, 4 minutes delay makes a one over loss after the reserve time has surpassed. Suppose, if there is a loss of 20 overs, 10 overs from each of the team’s innings is deducted and the play is resumed then.

Examples:

Example 1:

In an ODI match, Team 1 reaches 79/3 after 20 overs and then there is a suspension in play. It is decided that 20 overs of the match should be lost. Team 1 resumes to reach a final total of 180 in its revised allocation of 40 overs.

Solution – Since 20 overs are lost, 10 overs are lost for each team.

S = 180
When play was suspended, Team1’s resource percentage available(30 overs left and 3 wickets lost), R(a)             =                61.6
When play was resumed, Team1’s resource percentage available(20 overs left and 3 wickets lost), R(b) =             49.1

R(L)        =             R(a) – R(b)  =  (61.6 – 49.1)%  = 12.5%
R1           =             100  – R(L)  = (100 – 12.5)%  =  87.5%

Resource percentage available to Team2 at the start of the innings(40 overs left and 0 wickets lost), R2 = 89.3%

R2>R1
So, by Standard Edition, the revised target will be,

T = S + G50 x (R2 – R1)/100 + 1 = 180 + 245 x (89.3 – 87.5)/100 + 1 = 185

Also Read – A Microscopic Look into WASP

*Note: The Resource Percentage Table can be Found from the ICC’s official website and calculations are based on the standard edition of the method. The table could not be added due to issues of copyright.