Duckworth, Lewis & Stern Method

Dear Readers,

It has been a while since I have been fascinated by the Duckworth and Lewis method. After that fiasco in the 1992 World Cup between England and South Africa, it was my belief that the rain-affected matches are skewed due to the working of the methods applied to calculate the target scores. Since their work has been shrouded in secrecy, its mystery value only increased with each passing day. The recent Asia Cup; India- Nepal tie egged me on to study the intricacies of the method. The more I read, the more it fascinated me. It is hard to cover the topic in a mere 2000 words of text. It requires the application of the mind and the formulas to a number of contingencies to grasp the shortcomings and the strengths of the method. What I have put in the piece is a simple narrative for all to understand the basic workings of the method so that next time you, as a cricket enthusiast, are more involved in understanding when a match gets rain delayed. I hope you will enjoy reading it. Any queries would be welcome.

Duckworth, Lewis & Stern Method

In the 1992 Cricket World Cup, in Australia, England and South Africa (SA) were battling in the semifinals for a berth in the Finals. A delayed start forced it to a 45-over match. England batted first and scored 252 for six wickets in 45 overs. SA required 253 to win in 45 overs. At 43.5(Point five denotes five balls) overs, the rain stopped play when SA had scored 231 for Six wickets. The rain delay lasted 12 minutes, the equivalent of two overs, i.e. 12 balls. Thus, two overs were removed from the total of 45 overs. The new equation meant only one ball was left for England to bowl. When the play was interrupted, SA required 22 runs in 13 balls. By modern T-20 standards, it was a cakewalk. But when SA returned to bat, the umpires, based on the previously agreed rules, revised the target for SA to score 21 runs in one ball. The entire cricketing world was laughing at the mockery of this rule. The SA team and the nation were angry and disgusted at the unfairness of the system. Then a BBC cricket commentator, Christopher Martin Jenkins, commented, “Surely someone somewhere can come up with something better.” This comment egged Frank Duckworth, a mathematician at Sheffield University England, to do something. He decided that someone would be him. He and his fellow mathematician Tony Lewis decided to work on a better formula. Together, they devised a Duckworth-Lewis method of adjudicating rain-delayed and rain-curtailed matches.

D/L Method

The first match played under the D/L method was the one-day international limited-overs match between England and Zimbabwe in 1996-97. The ICC accepted it as the official method for the 1999 Cricket World Cup. Since then, it has been in practice and proven far more effective and acceptable to all the cricketing nations.

What is the Rule?

It would be interesting to understand what existed in earlier times. There were two popular methods. These were the Average Run Rate (ARR) and Most Productive Overs (MPO) methods. The first averaged the run rate at the end of an innings and offered it for the second team to beat. The most productive overs looked at overs which produced the maximum number of runs to determine the run rate required in the remaining overs. While the first favoured the team batting second, the second favoured the team batting first. Both suffered from a severe deficiency of overlooking wickets as a resource in their calculations. It didn’t matter in determining the target runs, how many wickets were left or gone.

After the 1992 World Cup fiasco, the ICC graduated to the D/L system, as explained. The basic assumption of the D/L system is that there are two resources for any team to get runs: wickets in hand and the overs remaining. And both need to be factored into the equation for determining the target. Duckworth and Lewis looked at the scoring patterns of 100 one-day international matches. In addition, they did theoretical modelling, and together they arrived at a table. The table represented a team’s available combined resources in percentage terms at any point in the game. At the beginning of a match, the team has 50 overs and 10 wickets and thus has 100% combined resources. On the other end, at the end of 50 overs, whether the team has lost all ten wickets or less, it has zero combined resources available. The availability of resources in between is the crux of the table. The relationship between overs left and the wickets to determine the combined resources is not linear. If the team lost five wickets at the end of 25 overs, its combined resource was not 50% but less. This table is a product of a set of mathematical equations. There is a constant factored in for an exponential decay. Meaning the curve would not be uniform but skewed towards the X or Y axis. This fact was found after studying more than 100 international matches worldwide, as stated above. Cricket lovers would know the back-of-hand calculation which commentators so often parrot. A team is likely to double its score of what it has at the end of the 30th over by the end of their innings, provided it has wickets in hand. This exponential curve graph and the table are reproduced below.

How does the Table Work?

To explain the workings of the table, specific terms need to be explained.

·      Team batting first is: T1 

·      Team batting second is T2

·      Combined Resources of team one is: R1

·      Combined Resources of team two is: R2

·      Team one score: S1

·      Score required to be equalled by team 2: S, also called the Par score or score which ties the match. 

·      To win, Team 2 would require to score one more run, i.e. S +1. Also called ‘T.’

The formula used to calculate how much is needed to win by team two (T2) the match is : 

T score = Score of Team 1 X  R2/R1 + 1

The application of this formula is explained with the help of three examples. There could be many more contingencies when the rain comes down to cause a delay, disruption, or abandonment. For readers to understand the fundamentals, the three examples cover three contingencies. The rider for applying the D/L rule is that both sides should have played a mandatory number of overs for it to be a game. In a 50-over match, each side should play a minimum of 20 overs; for a T20 match, this limit is six overs. 

Example 1: Second Innings Curtailed due to Rain

India vs. Nepal Game in Asia Cup, 04 September ’23.

Nepal batted for 48.2 overs in their allotted 50 overs to score 230 runs. Rain interrupted play for over two hours, and the innings were curtailed to 23 overs. (The thumb rule is five minutes to an over.) How much was the Winning score for India (T). The following are the calculations:

S1 = Nepal 230 runs

R1: Nepal batted its entire quota as it lost all ten wickets. So, it utilised 100% of its combined resources. Note: Even if Nepal had 1.4 over left to bat, it did not have any wickets; hence, it was deemed to have utilised its entire resource. Therefore R1= 100

R2, as per the table given above, India had ten wickets and 23 overs = 62.7% ( against 10 wickets column, look at 23 overs row)

Applying the formula T = 230 X 62.7/100 +1. It comes to 144 plus one =145 runs. 

Result India won by 10 wickets by D/L method. Whenever the D/L comes into play, the result has to reflect its use.

Example 2: First Innings Complete, Second Innings Abandoned due to Rains

The 2003 World Cup game Sri Lanka vs. South Africa. The game was abandoned due to rain. Who was the winner?

Sri Lanka batted first full 50 overs and scored 268 for nine wickets. South Africa batted till 45 overs and scored 229 for six wickets when the rain came and play was abandoned. Who was the winner as per D/L method? 

Team 1: Sri Lanka 

S1= 268 ( Note that the number of wickets remaining does not count as the quota of 50 overs is complete)

R1= 100 %

Team 2: South Africa

R2 = 100 %– 14.3% ( Look up five overs and four-wicket in the table, the unutilised part of the combined resource)= 83.7%

Formula for T = 268 X 85.7/100 = 229.676 + 1=230

The par score was 229 ( Note that the number in the Par score calculation is usually a non-integer. It is brought down to the lower integer for calculating the value of S). Since South Africa had scored 229 in 45 overs, the match was a tie. 

Example 3: When Rain Interrupts Play and First Innings is cut Short.

In January 2001, the West Indies was playing Zimbabwe. In a 50-over game, West Indies reached 235 for six in their 47th Over when rain stopped play. Due to the loss of time, the match was declared a 47-over game. What was the target for Zimbabwe to chase? It would be evident to the readers that if Zimbabwe were to chase 235 for six in 47 overs, it would be unfair to the West Indies as they did not know their innings would be curtailed at 47 overs. They would have accelerated their scoring rate if they had known in advance, as they still had four wickets in hand. Zimbabwe thus needed an enhanced target. How is it calculated?

Team 1 ; West Indies

S1= 235 for six wickets

R1= 100%- 10.2% ( Look at resources available for three overs and four wickets from the table given above) = 89.8%

Team 2: Zimbabwe

R2 = 97.4 % ( 47 overs, 10 wickets from the table)

Whenever R2 is greater than R1, the concept of G50 comes into play. G50 is the number of average runs scored during a 50-over innings. Since it is different for different countries and venues, its value is mutually agreed to by the two captains at the beginning of the series. In case it is a multinational tournament, its value is calculated by experts based on average scores in different venues and announced before the tournament starts. This value fluctuates between 225 and 250. For the a/m series, its value was 225, for men’s One Day Internationals.

The formula for T or S becomes slightly different when R2 is greater than R1. It is: 

T = S1 + G50 X ( R2 – R1)/100

T in the a/m case = 235 + 225 ( 97.4-89.8)/100

= 235 + 17.1 = 252.1 + 1 = 253 runs to win and 252 to tie. 

In this match, Zimbabwe were all out for 175, thus giving West Indies a win by 252- 175= 77 runs by D/L method. 

DLS Method

The readers would have realised the combined resources’ role in determining the Par Score or the winning score. It was observed that the formula did not hold very well when scores were very high. Professor Steven Stern of Queensland University worked on the D/L formula to refine it. He submitted the refined formula to ICC in 2014 before the 2015 World Cup. ICC accepted it and made Professor Stern the custodian of the new method, rechristening it as the DLS method. 

The DLS method calculations are not available to the public. They are in the form of software, which is available only on computers authorised by the ICC. The software is constantly updated. It has a rolling four-year data, as new yearly data is fed into the software yearly. This new data allows the DLS method to sync with the latest scoring patterns. The DLS method of calculations for T20 remains the same as in the case of one-day international matches. But the table is different, as T20 requires a more aggressive run rate. 

The DLS model is a complex set of equations for numerous permutations and combinations. It is difficult for the ordinary person to understand the calculations. Yet it is not hard to know how the Par Score is calculated by referring to the Table (Given in the beginning). The subject makes fascinating study as numerous scenarios could come into play. For a reader who is an avid cricket fan, it is a eureka moment once he understands the workings of the DLS. With the Asia Cup in progress in the thick of rains and the One Day Cricket World Cup starting in October 2023, it would create a new buzz in a cricket enthusiast if he understand how Duckworth-Lewis-Stern rules come into play.