Today’s blog is slightly different. It is about the fusion of statistics and cricket, the Duckworth-Lewis Method. This method is used ,to set a target for team batting second if any interruptions (most frequently due to rain) happen and overs are lost during the match. Some examples where the DLS method was used are the India vs. England Champions Trophy Final, the CSK vs. GT IPL 2023 Finals, and many more. But why did we need this method, and how was it devised?

## Need for DLS

In 1992, the ICC cricket world cup was held in Australia-New Zealand. The South African side returned to ICC tournaments after 21 years of ban due to the country's ‘apartheid policy’. South Africa reached the semi-finals, surprising everyone!! South Africa was to play against England in the semi-finals. When South Africa won the toss,they decided to bowl first. Due to time constraints, both innings were reduced to 45 overs each. England scored 252 runs at the end of the first inning, setting a target of 253 for South Africa. South Africa, while chasing the target, collapsed first, but after an electrifying knock from Jonty Rhodes, followed by McMillan and Richardson, South Africa had to score 22 runs in 13 balls. But then it started raining,and overs were to be reduced again.

In that World Cup, the absurd rule of ‘Most Productive Over’ was implemented. According to this rule, the least productive overs of Team 1 should be subtracted from the target.

#### Most productive overs method:

Let’s say it is 5 over match. Team batting first scored runs in the following manner:

OVER | RUNS |

1 | 4 |

2 | 2 |

3 | 6 |

4 | 8 |

5 | 0 |

TOTAL | 20 |

Now, due to interruption in the second innings, only 3 overs to be bowled. Thus the target for second team will be:

Number of least productive overs lost: 2

Target =Total runs- Least Productive Overs+1

From the table, we can see that over 5 went for zero runs and over 2 went for 2 runs. Thus Target for team 2 to chase in 3 overs is:

Total Runs- Run scored in 2nd and 5th over

20-(0+2)+1

=19 runs

Thus the team needs to score 19 runs in 3 overs to tie the match. The absurdity of this rule is that ,team has to pay for their best bowled overs. Due to this method, South Africa needed 22 runs in 7 balls instead of 13 balls. (How it went from 7 balls 22 runs to 1 ball 22 runs required is whole another story, which is not related to mathematical formula. Thus that part is omitted). Everyone, including the English side, was disappointed with this.

Frank Duckworth, who was listening to this match on the radio, heard commentators say, 'Surely someone, somewhere could come up with something better’ and realized that it was a mathematical problem and needed a mathematical solution.

## Duckworth-Lewis Method

So, the method goes like this: In order to score maximum runs, the batting team has two 'resources', namely ‘overs in hand’ and ‘wickets in hand', to make maximum runs. At any point in the innings, a team’s ability to score runs depends on the combination of these resources they have left.

Duckworth and Lewis went through historical data and made a chart of available resources in percentage figures. For example, 50 overs and 10 wickets mean 100% available resources.

Calculation for par score will be as follows.

If the answer comes in decimals , the number is rounded off to the next integer. For example, if a rain delay means that Team 2 only has 90% of resources available, and Team 1 scored 254 with 100% of resources available, then 254 × 90% / 100% = 228.6, so Team 2's target is 229, and the score to tie is 228. How was this relation derived?? Let's talk about the mathematical theory behind the D/L method in brief.

#### Mathematical theory

This method assumes that, number of runs that can be still scored for given number of overs and wickets lost follows exponential decay relationship

To show this using formula,

Let Z be the runs that can be scored.

Let ‘u’ be the overs remaining and ‘w’ be the wickets lost

Then, Formula is

Here Z0 denotes average total score expected as overs tend towards infinity and b is exponential decay constant. Yep…,the method is pretty much mathematical. Number of wickets(w) will vary from 0 to 9 . For each wicket, ’hundreds of one-day internationals’ data was analyzed and the value of Z0 and b were determined. Now after getting values of these two parameters they are assumed to be standard. So proportion is found by formula,

Here Z value will be calculated based on current match scenario which means how many overs are remaining and how many wickets are left in hand.

These proportions are shown in the graph given below

In 2014, Steve Stern made modifications to the method to optimize it for T20 games, and that’s why it is now known as the ‘Duckworth-Lewis-Stern’ Method. Yes, there are still flaws in this method, and it is not 100 percent accurate, but maybe we will discuss them in our next blog.

New concepts, like machine learning, can play an important role in the development of this method. There are research papers published in this field to correlate DLS with machine learning. One of the papers that I found interesting is attached at the end of the article. It’s really fascinating to see how mathematics and science give us a way to solve complicated problems and make us more comfortable in life.

Research paper Link:-

K. Abbas and S. Haider, "Duckworth-Lewis-Stern Method Comparison with Machine Learning Approach," 2019 International Conference on Frontiers of Information Technology (FIT), Islamabad, Pakistan, 2019, pp. 197-1975, doi: 10.1109/FIT47737.2019.00045.

Great👍

Very informative and interesting blog, definitely worth the read!

The blog we(cricket fans) didn't knew we wanted, but we needed. Amazing read!