For a fair formula

Published : Dec 06, 2002 00:00 IST

V. Jayadevan, an engineer from Kerala, has devised an alternative to the Duckworth-Lewis method, used to calculate targets in rain-interrupted One-Day Internationals.

THE most frustrating aspect of Limited Overs International (LOI), or One Day International (ODI), cricket matches is the interruption of play, mostly due to rain. To ensure a result in such shortened matches the International Cricket Council (ICC) has, since September 1999, used the Duckworth-Lewis (D/L) method to calculate a revised target for the team batting second. Devised by two British statisticians, Frank Duckworth and Tony Lewis (not the former England Test player), the method was published in the Journal of the Operational Research Society in 1998 (Volume 49, No.3). It is an attempt to avoid bizarre situations such as the one that arose during the England-South Africa semi-final in the 1992 World Cup. South Africa had to score 21 runs off the final ball to win, under the "most productive overs" rule, which was applied in that World Cup.

The D/L method, according to Duckworth and Lewis, is based on an elegant mathematical model, and has till date been applied in over 350 LOIs, according to Duckworth and Lewis, and it has worked extremely well. Much of the criticism made against it comes from those who have failed to understand the D/L formulation fully.

However, the D/L method is not perfect. Srinivas Bhogle of the National Aerospace Laboratories (NAL), Bangalore, brother of the commentator Harsh Bhogle, has studied the D/L method extensively. He said that it has its "blind spots". Now an indigenous alternative to the D/L formulation has been devised by V. Jayadevan, a 39-year-old civil engineer who has an M.Tech degree from the Indian Institute of Technology (IIT) and who works at the Kerala Engineering Research Institute (KERI) in Thrissur. His procedure, called the J-method, is devoid of D/L's shortcomings and in fact seems to offer a serious challenge to it. Even though cricket's national and international bodies have not given Jayadevan's system due consideration, the J-method, based on sound mathematical principles, has been recognised in the September 10 issue of the research journal Current Science. Jayadevan's logic for arriving at the mathematical structure of his formulation is somewhat different from that of D/L. Bhogle calls it a "practical engineer's approach".

To appreciate this aspect of the J-method one must know how the D/L method works. It uses the notion that teams have two resources to make as many runs as possible: the number of overs left and the number of wickets in hand. At any stage in a team's innings, its run-scoring capability depends on a combination of these two resources. Duckworth and Lewis analysed data compiled from hundreds of matches and worked out a (ball-by-ball) table that gives the percentage of these combined resources that remain for the overs left and wickets remaining. These tabulated values seem to capture most of the situations that arise in LOIs. By evolving the concept of resources percentages (RPs), the D/L method overcame the shortcomings of other procedures that were being applied earlier.

One of the methods in use was the "most productive overs" rule evolved by the Australian Cricket Board (ACB). It is applied as follows: Suppose that the team batting first, Team 1, scores x runs in 50 overs and a rain interruption leaves only (50 - r) overs for Team 2 to play. The rule required that the total runs scored by Team 1 in every over to be arranged in the descending order. Also suppose that Team 1 scored y runs in the last r overs, where the overs are arranged in order of descending productivity. The revised target for Team 2 was taken to be x - y runs to be scored off (50 - r) overs. This method turned out to be blatantly unfair to Team 2. In the England-South Africa semifinal of the 1992 World Cup, South Africa needed 22 runs off 13 balls to enter the final. Thanks to a brief shower, two overs were lost. When the rule was applied, it merely knocked off the last two overs in which England had been restricted to just one run. The irony was that South Africa's tight bowling in the last overs of the English innings cost South Africa the match. In the D/L method, South Africa would have had to score three runs off the last ball. The J-method too gives the same value.

To apply the D/L method, all that is required is to read the RP values of the two teams and do some simple arithmetic. If the RPs of the two teams are R1 and R2 respectively and R2 is less than R1 - which is the case if there is an interruption in between the innings or during Team 2's innings - then Team 1's score scaled down by the factor R2/R1 + 1 gives the target required for Team 2. If, however, R2 is greater than R1 - which happens if the play is interrupted during Team 1's innings - the target is revised upwards for Team 2. In this case, the scaling is a little more complicated and Team 2's target is not merely revised upwards by a factor of R2/R1. In order to win, Team 2 has to score (R2 - R1)/100 x G50 more than Team 1's score, where G50 is the average score in a 50-over match. G50 is currently placed at 235 given present scoring patterns. This figure of 235 is subject to revision based on the trends. Incidentally, the D/L table of RP values was revised recently.

It is this concept of upward revision in the D/L method that has attracted a great deal of criticism, which is misplaced. The rationale for the upward revision is simple. When Team 1's innings is curtailed, it is at a disadvantage because it had begun the innings with a certain strategy of pacing its play over the entire 50 overs. This strategy could not be realised due to interruption, resulting in a certain unutilised RP. In such a case, Team 2 can then deploy its full resources accordingly.

Bhogle points out that when Team 1's innings is severely reduced in low scoring matches (150 runs or below), the upwardly revised target set by D/L for Team 2 can become quite unrealistic, essentially because of the G50 factor. This happened during a match between New Zealand and South Africa in Durban in November 2000. New Zealand batted first in a rain-interrupted match and made 114/5 in 32.4 overs. By the D/L method South Africa was required to score 153 in 32 overs, which would appear quite unfair. South Africa eventually won because of Lance Klusener's hard hitting. The J-method too requires an upward revision in situations where Team 1's innings is curtailed but it leads to more realistic targets since it does not make use of the G50 factor. For instance, in the New Zealand-South Africa match, the J-method would have set a more realistic target of 128.

Another weakness of the D/L method is that it allows the team batting second, Team 2, to play according to a strategy in situations when the target is large but there is the prospect of rain curtailing the match. All that the team has to do is keep its wickets intact, even if the scoring rate is less than required. Consider the abandoned India-England match in the Natwest tri-series in Durham in July 2002. India's score of 286 was a stiff target for England and rain was imminent. England's innings was abandoned after it had played 12.3 overs and made a score of 53/. If England had been able to play for 25 overs, the D/L rule says that it would_-have won if it had managed to make any of the following scores - 99/1, 110/2, 123/3, 140/4 or 161/5.

From its position of 53/1 in 12.3 overs, England would have had to make 46 runs in the next 12.3 overs without losing any more wickets. Had England's innings been reduced to 30 overs and had it lost onlyone wicket, it would have had to maintain a rate of a little more than 4.5 runs an over. Only if the match had gone on for 35 overs would it have become somewhat difficult. The J-method, on the other hand, sets fairer "par" scores at the point of abandonment. In the above example, the J-method would have required England to score 119/1 or 123/2 or 148/4 or 172/5 at the end of 25 overs to win.

These are rare instances where D/L seems to falter. For most situations in interrupted LOIs, D/L sets fairly realistic targets. More importantly, D/L is able to handle situations involving multiple interruptions in a fairly straightforward manner. If a method is to replace D/L, it should be comparable with D/L in all respects and do better than it in situations where D/L seems to be weak. The J-method, as initially devised, had some anomalies in handling multiple interruptions. But the version that has been published, it is anomaly-free. Bhogle, who has studied the refined J-method too in some detail, says: "All things considered, the method does marginally better than the D/L in normal situations and, most significantly, does not seem to have the blind spots of the D/L method."

The J-method, like the D/L method, takes into account information on the resources remaining at any time, namely the overs to be bowled and the wickets remaining, to arrive at a fair target. While Duckworth and Lewis based their "look-up" table on a sophisticated mathematical model, Jayadevan sets up two curves to arrive at this table. The `normal' curve shows the scoring pattern when there is no interruption, arrived at by the analysis of data from a sample of LOI matches, in particular some close finishes. The second curve (`target curve') shows how the batting team needs to `speed up' after an interruption.

To arrive at the `normal curve', Jayadevan breaks up a 50-over innings into seven stages: first 10 per cent of the overs (settling down), the next 20 per cent (exploitation of field restrictions), the next 20 per cent (stabilising of innings-I), the next 10 per cent (stabilising of innings-II), the next 20 per cent (beginning of acceleration), the next 10 per cent (secondary stage of acceleration) and the last 10 per cent (slog overs). The curve represents the regression equation for cumulative percentage of overs and the cumulative percentage of runs scored.

To set up the `target curve' Jayadevan employs a variant of the Australian `most productive overs' concept, which, he says, "was a good idea with flawed execution". The scores in each of the seven stages are arranged in the descending order of run productivity and a regression curve is obtained for the cumulative percentage of overs against the cumulative percentage of runs scored.

The resource corresponding to the wickets fallen is taken into account in the following manner. From his match database, Jayadevan sets up a correspondence between the percentage of overs and the percentage of wickets in the normal scoring pattern. For any fall of wickets that differs from this average pattern, the normal curve gets modified and is obtained by a technique of interpolation. The table is built up from the values of normal scores (as read from the normal curves) for different patterns of loss of wickets. Thus, the J table comprises 12 columns: overs percentage, target run percentage (which is independent of the wickets fallen), and normal run percentage for each wicket lost (0 to 9 wickets). However, while the D/L table exists for a ball-by-ball target evaluation, the J-table has so far been compiled only at intervals of 1 per cent of the overs, which works out to half an over. A more complete table could, easily be generated in the future.

The basic method of application is as follows: For overs played before interruption, values are read off from the normal scores columns (this depends on the number of wickets that have fallen); after the interruption, values are read off from the target column depending upon the percentage of overs remaining. In most cases, D/L and J estimates are close to each other. The essential difference arises from the premise on which the progression of a normal innings is based. The D/L method assumes an exponential type of function to describe how the "resources" are used up as wickets fall and offers, therefore, a uniform curve. In the J-method, such an exponential increase occurs only after 60-70 per cent of overs (see graph).

"Since the curve is developed on data at different stages of the match, this curve is closer to the actual match situations than the D/L curve. This is why the results obtained in the proposed system are more acceptable," says Jayadevan. "Another difference is the absence of the G50 factor, the average or typical score in a 50 overs innings. If a model cannot fix a target based on the performance of a team in more than 25 overs, it should be treated as a weakness of the model," he points out in his paper.

As the editorial comment in Current Science points out, unlike the D/L method, "Jayadevan's method, in the finest tradition of open-source software, has been described as completley as possible, within the constraints of the space available for an article in a scientific journal." On the other hand, the exact mathematical functions that form the key elements of the D/L method, namely the exponential equation that expresses runs as a function of the twin resources of overs and wickets, are not available in the open domain. "Commercial confidentiality prevents the disclosure of the mathematical definition of these functions," state Duckworth and Lewis in their paper. The application software, CODA, is, however, sold at a price.

To an e-mail query on Jayadevan's formulation, the two statisticians responded: "The basic problem with the method stems from the use of two curves. This leads to internal inconsistency. One such manifestation of this was that across a stoppage a team could move from a losing to a winning position. Jayadevan introduced his PAR concept to cope with this, copying the D/L concept... so what may have started out as a logical concept turns out to be seriously flawed, requiring many patches to cover its deficiencies that make the instructions to use the method virtually incomprehensible. Because of these problems of internal inconsistency one can have little faith that still more illogical situations do not exist which are impossible to predict or now to unearth due the method's complexity."

"Jayadevan's claim that his method gives more acceptable results to D/L is based on eight carefully selected examples, three of which are hypothetical. In the five real examples he gives, in general we would disagree that our targets are too high or too low... in the three hypothetical examples, Jayadevan has picked examples similar to those we have already debated both in academic fora and with the cricket authorities, where an upgraded D/L model of the one-day game gives D/L targets that are more in keeping with cricketing sense. This upgraded model would need to be implemented entirely by computer. So D/L can cope with well-above average scores but with the loss of the simplicity of the current modus operandum. At present, the cricket authorities have preferred to stay with the simple method that can be operated quickly and efficiently with nothing more than a table of numbers and a pocket calculator."

Respond to Duckworth and Lewis' comments, Jayadevan says: "They essentially say that my method should be compared with their computerised version, and not the present version. Are they not indirectly accepting that my results are better than what is given by their present procedure?" Indeed, as one scientist pointed out, "what better evidence is needed for the superiority of the J-method than the fact that the revised D/L procedure now predicts values which are closer to the values given by the original J-method?"

Shane Booth, an Australian expert who has done extensive analysis of the D/L method, commented on Jayadevan's paper: "The J-method indeed appears to give sensible results in cases where the D/L tables do not. The D/L tables, unlike the J-method, do not distinguish between the team batting first and the team chasing a target. D/L also does not consider the effect of the relaxation of fielding restrictions in the first 15 overs. Jayadevan's paper indicates several cases where the D/L method produces seemingly low or high targets. In all cases the J-method generates more realistic targets. It appears that the only disadvantage of the method is that it is slightly more difficult to apply than the D/L method. However, the method can be implemented on any computer, so this should not be a drawback for high-level competition. I imagine it would not be too difficult to train someone to use it with only a calculator."

The response of the ICC to Jayadevan's procedure has, however, been indifferent, if not dismissive. According to Jayadevan, Sunil Gavaskar extended an e-mail invitation to him to make a presentation on July 11,2000 at a conference of the Board of Cricket Control in India's (BCCI) Technical Committee in Pune. Impressed by some of the results of the method, the committee asked him to make a presentation at the September 2000 Umpire's Seminar in Jamshedpur. As suggested at that seminar, a computer program was also developed subsequently to allow quick calculations. In a meeting held on April 7, 2001 the BCCI decided to forward this proposal to the ICC. But for unknown reasons, the ICC in a committee meeting held on May 23, 2001, did not take up this proposal and decided to continue with the D/L system. The BCCI could at least begin to use the J-method in domestic one-dayers and generate wider publicity for the procedure through non-ICC domestic and regional matches. Eventually, this might encourage the ICC to adopt Jayadevan's system.

Step-by-step procedure for the case when the interruption comes after Team-2 has batted through some overs in its innings.

1. Find out the percentage of overs played up to the interruption.

2. Find out the normal percentage of runs corresponding to #1 and the wickets fallen.

3. Find out the PAR score (say PAR-1) as normal score percentage multiplied by the score of Team-1.

4. Find out the percentage of remaining overs with respect to the total overs remaining.

5. Find out the corresponding target percentage.

6. Multiply the target percentage of #5 with `the total score of Team-1 minus PAR-1' to get the target score in the remaining overs.

7. Add PAR-1 with the target obtained in #6 to get the net target.

LOI# 1442: Australia vs West Indies

Australia 252 in 50 overs; WI, after 29 overs, 138/1.

Ten overs are lost. What is the target for WI in 40 overs?


Percentage of overs played by WI at the time of interruption = 58.

Corresponding normal score = 48.3%.PAR-1 = 48.3 252 = 121.7 (1).

Percentage of the remaining overs with respect to the total remaining overs = 11/ 21 100 = 52.4.

Corresponding target percentage = 65.6.

Target score for the remaining overs = 0.656 (252 - 121.7) = 85.5 (2).

Net target in 40 overs (1) + (2) = 121.7 + 85.5 = 207.2 = 208 runs.


In the recent ODI at Rajkot between India and the West Indies (WI), the match was abandoned after Team 2 (India) had batted for 27.2 overs chasing a total of 300 of Team 1 (WI). In the television discussion programme WISDEN 20:20, one of the remarks made was that the PAR score of 119 in 27.2 overs according to the D/L method was too low. According to Jayadevan, this PAR score is as per D/L's revised table. As per their original table it is still lower at 115.

By the J-method it is 137, which seems more reasonable.

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