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Story that numbers tell us

The logic of Duckworth-Lewis

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No one has control over inclement weather. Neither are we good at weather forecasting!

A cricket one day match that usually takes eight hours of play, has to withstand the whims of rain and bad light. To counter it, to have a result at end while being fair to both teams has been challenge for cricket administrators. More so because, it is like forecasting the result of matches somewhere in the middle of action, and this goes against the grain of sports whose charm lies in it’s unpredictibility.

Cricket tried few methods before settling in with D/L method. Most notorious of methods is maximun-score over concept. This costed SA the berth in WC 1992 final.

  • Maximum-score over concept: This method asked the second team to chase down the best overs ( in terms of runs) by first team in stipulated over.  Consider the popular match, SA vs Eng, WC 1992. England scored 252/6 in 45 overs, in reply SA scored 231/6 in 42.5 overs. Rain interrupted the match, and 2 overs were reduced.  Now since, out of 45 overs that England played, in best 43 overs they scored 251 runs.

Then in 1997 came D/L method, most scientific method developed by two statisticians, Duckworth and Lewis.

The essence of the D/L method is resources. The two resources are ‘overs left’ and wickets in hand. Feeding these two values, P(O,W) is calculated. The function p(o,w) has been kept confidential for commercial reasons. They have reached at this function applying statistical techniques on historial data.  They keep updating it, Most recenly it was updated on 2004 to accomodate the fact the one day cricket had become high scoring compared to last decade. The p values signifies resources left.

               p(50,10) = 100%          p(0,10) = p(0,0) = p(48,0) = 0%

Below chart plots p values.


 Here is it in tabular format. It’s in concise form; the complete table can be found hereDLtable



 How to use the table?

Broadly, there can be three kind of possibilities.

The most simple 3 in 1 scenario would be team batting first, bats for 20 overs for 110/3, then there is an interruption, and the game is now curtailed to 30 over game. Team 1 bats for 30 overs with final score of 210/6. On reply team 2 is on 90/4 in 10 overs when the again rain plays its part. For bad light game is again curtailed to 20 over game, now what would be target for team 2.


From above table,  Team 1 on its 20th over would have left with p(30,3) resources.  It’s   61.6% .                                                                                                     Now  when play resumed,  team 1 had p(10,3) resources left. it ‘s  29.8% . So total resources utilized by team 1 is 100-60.6+29.8=69.2%.  At lunch time the game was still of 30 overs, so resources available to team 2 is 75.1% . 

So at lunch the target for team 2 would be 210+225(75.1-69.2) = 223 . The value 225 is G value. When the team batting first looses some overs, sometimes it’s favours  batting side or bowling side, depending on wickets lost by batting side. For target calculation for team 2 in these situation we use

T = S*R2/R1 or S or S+G(R2-R1) for R1>R2 or R1=R2 or R1<R2 respectively. The value of G is 225.

Now back to our imaginary problem, after ten overs resources left for team 2 is p(20,4) = 44.6%, but there is a rain and they loose 10 overs more so resources left after rain 28.8%. Because of rain they loose 44.6 – 28.8 = 15.8% resources.  Hence revised target would be (210+225(75.1-69.2))*(100-15.8) = 188.

 There is one more method put forward by an Indian Vasudevan to set target. ICL has adopted Vasudevan’s method, and it seems to give more practical target compared to D/L. The criticism of D/L has been that it gives too much weightage to wicket, and wicket of , say Tendulkar and Ishant is given equal importance.




Written by SK

August 5, 2009 at 7:56 pm

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