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Duckworth-Lewis — Decoding a brilliant, complex method through graphical representation

MS Dhoni's men scored more runs in the 2nd ODI against New Zealand but still lost the match under the D/L method © AFP (File Photo)
MS Dhoni’s men scored more runs in the 2nd ODI against New Zealand but still lost the match under the D/L method © AFP (File Photo)

The Duckworth-Lewis method once again provided a counter-intuitive result during the second ODI between India and New Zealand at Hamilton. However, it remains a scrupulously fair technique and the best target revision method available today. Arunabha Sengupta argues that much of the heart burn is due to complexity and can be tackled with visual data representation.


Counter-intuitive, but fair


On January 22 at Hamilton, when New Zealand ended their shortened innings on 271 for seven from 42 overs, there were sparks of protest kindled in the fan world. India were asked to score 293 from 41.3 overs, 22 runs more than the Kiwis had made.

This was counter-intuitive. Why would the side batting second suddenly need to score more than the side batting first in almost exactly the same number of overs? The agitation erupted in a massive flame of furore when MS Dhoni’s team lost after scoring 277 for nine.

So, the end result was most difficult to justify. The team that scored more runs in a slightly less number of overs ended up losing the match. This, for all intents and purposes, redefined counter-intuitive. The verdict from a large section of the public was ‘nonsensical’. “Could anything be more atrocious?” – demanded some highly outraged voices.

So, once again Messrs Duckworth and Lewis were branded as illogical, and in the rather less edifying language that pronounce the judgement on the social media – incompetent, worthless, drunk and so on.

Well, the truth is that the target revision was as logical as it could get and as fair as it has ever been since the start of the limited-overs formats. And yes, it is obviously a bit difficult to wrap one’s brain around the technique.

There is actually a perfectly logical explanation

The apparent lack of logic can be explained even without going into the complicated mathematics that beautifully chisels the Duckworth-Lewis method into the best rain-rule we have had till now. In fact, it speaks volumes about the immense improvements that have been brought into the science of revising targets since the days South Africa were asked to make 22 runs from one ball in 1992.

To put it very simply, New Zealand had started out with the aim of batting a full 50 overs. They were on 167 for two in 33.2 overs when rains interrupted the innings. They had set up their innings perfectly for acceleration in the remaining 16.4 overs. The interruption ensured that in the end they got just 8.4 of those 16.4 overs for their final burst.

If the Indian target had been set to 272 in 42 overs – and in the pre Duckworth-Lewis days that would have been the case – it would have been grossly unfair to the Kiwis. After the excellent launchpad that they had set up, they would have lost out on eight overs budgeted for acceleration, overs that generally produce runs at a very high rate if one has wickets in hand. Thus, before Duckworth-Lewis, interruptions in the first innings almost always aided the side batting second – unless we are talking about the ridiculous most-economic overs rule used in the 1992 World Cup.

In the 50-over game, the Duckworth-Lewis method does an excellent job in ensuring a fair match in such cases of interruption. To put it heuristically, the technique considers the two resources available to the team at the time of interruption –wickets and overs. Using these two parameters which are most important factors for posting totals, fitting past the same to past data, the effects of interruption are rendered negligible.

The revised target of 293 was arrived at to ensure that New Zealand was not disadvantaged because of batting the first 33.2 overs assuming that there were 50 overs in the game – and similarly India was not advantaged by knowing from the start of their innings that they had just 41.3 overs to bat. The Duckworth Lewis technique looked at New Zealand’s score at interruption – the overs left and wickets lost, the score posted at the end, and tables based on past data.

The effect was to adjust the score to a target which New Zealand would have been most likely to set if they had set out with the knowledge that they would bat 42 overs.

This is the fairest method used in cricket so far.

The Kiwi bowlers never let the Indians settle down during their steep chase © Getty Images
The Kiwi bowlers never let the Indians settle down during their steep chase © Getty Images

A simple example

Let us explain it with a simpler example, which explains the concept but does not deal with the additional complication of interim interruption and resumption.

Suppose Team A starts to bat for 50 overs and reaches 205 for one in 40 overs when they are interrupted by rain. The match is reduced to 40 overs a side. If Team B is asked to score 206 to win from 40 overs it will be grossly unfair to Team A – because the remaining 10 overs with nine wickets in hand, Team A would have been most likely to accelerate with great effect. That is what past data tells us. Additionally, Team B will know from the beginning that they will have 40 overs to bat and can be much less cautious about losing wickets than Team A had been.

Therefore, the percentage resources that was unused by Team A is computed – say the combined value of nine wickets and 10 overs give a figure of 31.6 percent in the Duckworth-Lewis table (the figure is illustrative). And hence revised target is set extrapolating the number of runs Team A would have scored if they had used up 100% their resources.

Let us suppose this target is 290 in 40 overs. In that case it is indeed fair if Team B scores 265 in their 40 overs – more than Team A – and still end up on the losing side. (The figures are illustrative)

The system also has enough features to take care of interim interruptions during the innings.

Problem for the spectators


The only problem with this system of target revision is that the general fan, who may not be that keen on numbers and not too willing to invest time and thought in understanding the method, is very likely to get confused.

The primitive method of revising a target of 250 in 50 overs to 125 in 25 overs was obviously unfair. It considered the run rate, but ignored the glaring fact that maintaining the same run rate in a smaller number of overs was much easier if the number of wickets remained the same. But, fans could work this out because it involved one simple division. Yet, the simplicity of the technique could not really conceal the unfairness.

Frank Duckworth and Tony Lewis took this into consideration and computed the revised target as a function of remaining resources – both runs and wickets. Since it is a non-linear function of two input variables, it is not that easy to understand the algorithm. The said method is a brilliant application of mathematics that uses exponential decay functions. However, it is nowhere near as easy as runs divided by overs.

This difficulty in comprehension, added to the sometimes counter-intuitive results, gives rise to suspicion, scepticism and misgivings – and ultimately the far easier alternative of dismissing the method as impractical and atrocious. After all, Mathematics is the only subject that a huge proportion of the general populace are proud to ‘not know’.


However, it has to be admitted that the mental gymnastics required in absorbing this method of resetting targets place a lot of demand on the mental faculties of the cricket fan – much more than any other sport with the exception of perhaps the cerebral games like chess and bridge. Interruptions are not unknown in other sports, but one never sees a tennis match between A and B ending with a score of 6-3, 5-4 in favour of A, but B being declared the winner. Or in a football game, rain cannot bend a scoreline reading Team A 3 : Team B 1 into victory for Team B.

It is the multiple parameters, number-richness and intricacies of cricket that leads to this situation. And thus, fair and mathematically rigorous though the method is, the problems in understanding the workings of the method lead to a lot of heartburn.

Visual data representation to increase awareness and viewing pleasure


The problem does not lie with the method. As stated earlier, the Duckworth-Lewis system is the best available solution of finding a fair result in foul weather. The hitch is that it is not meant for easy interpretation and understanding of a majority of fans. They tend to feel left out and cheated when confronted with sudden changes in the game situation that they had not anticipated and did not see coming. And indeed, that is a very important factor in a spectator sport.

A recent study on a representative sample of cricket fans on social media showed that only 9.2% could explain the mechanism governing net run rate in limited overs competitions with some degree of accuracy. And that is a computation technique involving nothing more than addition, subtraction and division. Hence, expecting the majority to understand the workings of exponential decay function is too far-fetched.

The solution cannot be to resort to a simpler and flawed target revising measure. That would be unfair to the teams and ultimately to the game.

The answer to this intrigue actually lies in visual data representation and easy explanations.

To make this point, let me provide an everyday example of very complex mathematics used in cricket matches.

[S] (xt, yt, zt) = S(x0, y0, z0) + ʃ 0,t (vv, vh, vl) t is an often used formula, where  S (xt, yt, zt) are coordinates of the ball in three dimensions at time t, and vv, vh, vl are the vertical, horizontal and lateral velocity components of the ball at elapsed time. These equations are used to extrapolate the trajectory of the ball to the wicket, predicting the probability of it hitting the stump.

This is definitely more complex than the Duckworth-Lewis formula, but the fan finds no problem with it. It is because the output of the computations is represented in the Hawkeye graphic, showing how a ball moved before hitting the pad and carrying on to show the probable direction after that.

Since the graphical simulation almost recreates the game as people see it, the complex background equations have not made fans term Hawkeye as illogical, worthless or incompetent. However, according to some experts Hawkeye is much more prone to failure than D/L is.

Similarly, the science behind Thermal Imaging that generates the hotspot is similarly complex and takes into consideration equations like Planck’s law and the Wien Displacement Law. But, the fan has no problem with this either, since they see the relatively simple images of the bat coming down and a spot appearing on it.

In fact, Hawkeye and Hotspot are aspects of the game that the fan eagerly waits for. It is the Board of Control for Cricket in India [BCCI] who is targeted for abuse instead, when they voice the rather justified doubts that some of these technologies are far from perfect. The non-acceptance of Decision Review System [DRS] is seen as another of the many crimes in the charge-sheet against the Indian Board.

As explained, much of the gripe with the Duckworth Lewis target revision is the abruptness with which it is announced, leaving fans at a loss for comprehension. Perhaps if the excellent modern graphical tools can be used to track the D/L revised target with passing overs, it will go a long way in making it more accessible to fans.

Figure 1: Suggested Duckworth-Lewis Graphic for 1st match innings (the figures are illustrative)


There are plenty of opportunities to build spectacular animated graphics that can actually make Duckworth-Lewis both accessible and interesting to the fan. When showed on television, the giant electronic scoreboards and cricket websites providing live scores, this can add to the interest in the game. The opportunity of using visual data representation is especially high in the second innings where the change of target for different number of overs with wickets lost can be demonstrated with use of clever animations. The complication of the equations can be converted to an advantage by converting them into brilliant graphics.


Figure 2: Suggested Duckworth-Lewis Graphic for 2nd match innings (the figures are illustrative)


With this kind of graphical display, the spectator will constantly be in the game, aware of the effect of rain, knowing what to expect, with the complexity reduced through understandable charts. The match can be followed by the general fan secure against the sudden shocks and surprises that are sometimes the unpleasant side-effects of the method.

Perhaps this is the way ahead for the wider and popular acceptance of an excellent and fair method.

(Arunabha Sengupta is a cricket historian and Chief Cricket Writer at CricketCountry. He writes about the history and the romance of the game, punctuated often by opinions about modern day cricket, while his post-graduate degree in statistics peeps through in occasional analytical pieces. The author of three novels, he can be followed on Twitter at http://twitter.com/senantix)

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