Monday, March 03, 2008

Cricket Simplified --> Duckworth-Lewis System

The Duckworth/Lewis system was invented by two statisticians, Frank Duckworth and Tony Lewis in an attempt to solve the perennial problem of creating a fair target for the side batting second in a limited-overs match shortened by the weather.
At its simplest, the problem is this. If the side batting first in a 50-over match makes 250 runs, the target for the side batting second is 250 to tie, 251 to win. Their average run-rate must therefore exceed 5 per over if they are to win in a full 50 overs. But if rain or bad light stops further play after only 20 overs, during which the second side score 101 runs, then comparing average run-rates would make the second side the winner. But if they had lost nine wickets in gaining those 101 runs, then few would give much for their chances in an uninterrupted 50 overs. So the old average-run rate system gave an unfair advantage to the side batting second if the number of overs was reduced. They could blaze away without worrying too much about losing wickets. Other variants were tried, such as basing the target on the best overs of the side batting first, but this was unfair to the side batting second.
Enter Messrs Duckworth and Lewis, who studied the performances in a great number of limited overs matches, and concluded that the factor to take into account was what they called the "resources" available to each side. The resources depend on the number of wickets lost and the number of balls remaining to be bowled.
So at the start of a 50-over innings, the batting side have
100% of their resources available. With 40 overs remaining, the resources would be 90.3% if no wickets had been lost, down to 7.6% if 9 wickets had fallen, as shown in the table below. The full Duckworth/Lewis tables go down to the resources available after each individual ball has been bowled, but for purposes of explanation we will stick to units of 10 complete overs here.
With 30 overs remaining, the table gives 77.1% of resources remaining if no wickets have been lost (in other words, on average the team would be expected to make 77.1% of their total score in the remaing overs), down to 7.6% again if nine wickets have fallen. So our team chasing 250, having scored 101 after 20 overs when rain stops play would lose 77.1% of their potential resources to rain if no wickets had fallen, leaving them with 100% - 77.1% = 22.9% resources used.So they should be expected to have scored 22.9% of the first innings score, and their target to tie the match would be 22.9% of 250, which is 57.25 rounded down to 57 to tie, and hence 58 to win. So 101 for no wicket after 20 overs would be a winning score. But if they had lost 4 wickets then their resources lost would be 54.9%, and their resources used would be 45.1%. So the target to tie the match would be 45.1% of 250 = 112.75, rounded down to 112. So 101 for four wickets or more down would be a losing score.
If only things, and the weather were that simple. Either innings could be interrupted, possibly several times, or cut short.But the Duckworth/Lewis system can cope with anything the weather can throw at a game.
If the game starts late and is reduced in overs, or if it was only scheduled to be of less than 50 overs in the first place, then each side has their resources available reduced by the appropriate amount. So a 40-over match starts with each side being reduced to 90.3% resources, so having lost 9.7%. If there is an interruption with 30 overs remaining, as a result of which the match is reduced by ten overs, then if four wickets had been lost, the resources available would drop from 54.9% to 46.1%, a loss of 8.8%. All the losses in an innings are totalled and then subtracted from 100% to find the resources available for that innings.
If the resources are the same for both sides, then no calculations are necessary, and the side with the highest score wins.
If the side batting first had more resources (R1) available than the side batting second (R2), the the target to tie the match is calculated as the first innings score multiplied by (R1/R2) rounded down.
If the side batting first had less resources (R1) available than the side batting second (R2) then the target is the first innings score plus the difference between the resources divided by 100 and multiplied by the typical score in a 50 over match (normally 235, but could be different - it was 190 in the 1997 ICC Trophy). So if the first side scored 150, and had resources available of 60%, and the second team had resources available of 70% when the innings finished, then the target to tie would be 150 plus (10/100)*235 = 173.5, rounded down to 173.
So the apparently unfair position can arise where if the first innings is truncated by rain after 40 overs with six wickets lost, then resources lost would be 24.6, resources available would be 75.4. The second side might start with 40 overs to play, no wickets lost, and so have 90.3% resources remaining, more than 75.4% so their target to tie would be 235*14.9/100 rounded down = 35 more than the first innings total, even though both sides would have to face 40 overs. The discrepancy would be because the first side would have been pacing themselves for 50 overs, and the second would have known from the beginning of their innings that they would only have to face 40 overs.
One final detail is that if the side bowling first take more than their allotted time to bowl their quota of overs, then they may be penalised by having overs deducted when their turn to bat arrives. The Duckworth/Lewis system compensates for this this by treating it as a reduction in the resources available for the first side as if they had been deprived of that many overs.
To avoid the spectacle of the batsmen meeting in mid-wicket and consulting Duckworth/Lewis tables and pocket calculators between every over, the scorers generally do the calculations at the end of each over and display the difference between the runs achieved and the score needed to tie if the match were to be finished at the end of the over as a plus or minus figure on the scoreboard (or sometimes just the target), so all can see whether they are ahead of or behind the target.

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