Cricket Interruptus

One of the most contentious questions of the game is what a new target should be for a rain-shortened match. Is there a fair way to arrive at this figure?

A theoretical physicist will doubtless quibble about the illogic of cricket. A central idea of quantum mechanics is that the act of observing an experiment determines its outcome. A free particle, when not being observed, has a finite probability of being anywhere in the universe. It is, therefore, impossible and unreal to predict what would happen to a particle if it is not obstructed in its path. The rationale behind the LBW rule is blind to the objections of quantum mechanics and chaos theory. In a purely theoretical sense, cricket is inherently flawed.

But this is an obscure gripe. I use this to illustrate the difficulties associated with a much bigger problem in limited overs cricket: the problem of what to do when there is an interruption during the match on account of rain, crowd trouble or anything else. No other sport is as centrally dependent on conjecture as cricket. The ‘what if’ moments in other sports are confined mostly to post-game analyses and bar fights. In cricket, they are at the core of the rules of the game. The best we can do is minimise error.

The errors associated with the LBW rule are relatively minor. There are many factors that influence the trajectory of the ball, but the biggest complication is what happens after the ball lands and deviates. There are many other factors such as wind, swing, variable bounce on the pitch and chaotic dynamics. Chaos theory tells us that even the slightest difference in initial conditions between two experiments leads to outcomes that are increasingly different as time passes. The reason Hawkeye technology is not entirely trustworthy is that it cannot take into account the convoluted mathematics of all these possibilities. The umpire is still in a better position to judge, because many of these calculations happen unconsciously in the human mind and are conveniently grouped under the blanket of ‘intuition’. The calculations are impossible, but they influence one delivery at a time, unlike those associated with interruptions.

Rain interruptions have played crucial roles in key matches, curtailing play and forcing an outcome based on a formulaic comparison of the reduced overs’ performance. But, as with LBW predictions, there are many factors that complicate the formula—the point of which is to project a full match from what score data is available, and then scale it down aptly. With about as much certainty as a meteorologist caught in a cyclone, people have tried to come up with methods to deal with these factors. The Duckworth-Lewis, Carter-Guthrie, Stern and Jayadevan methods (to name a few), do their best to take them into account. As a result, they seem morbidly complicated, but yield more reasonable predictions than some of the famously shocking methods used earlier.

Like with an LBW decision, the success of a rain rule can be determined by how close it comes to what an experienced and impartial observer watching the match would believe is fair.

In the 25 years between the first ever One-Day International and the appearance of the Duckworth-Lewis method, cricket matches went through many different ideas. From the obvious but disastrous idea of simply comparing the run rates at the corresponding over in the first innings (which heavily favoured the team batting second), to more adventurous ideas like the ‘highest consecutive scoring overs method’, over a dozen rain rules were devised.

The most famous example of a match ruined by a stupid rain rule, one which will forever be used as a case study for why such a rule should be developed by a mathematician, not Richie Benaud and other cricketers, is the semi-final of the 1992 World Cup between England and South Africa. The revised target after a rain delay when South Africa was chasing, was determined by the ‘highest scoring overs’ rule. The overs that the first team had played were arranged in decreasing order of the number of runs scored off each. The second team needed to get one run more than what the first team scored in X number of overs in the newly arranged sequence of overs. What this meant was that after 48 overs, South Africa needed one more than England had in their 48 highest scoring overs. South Africa was actually penalised for having bowled a maiden over. Needing 22 to win from 13 deliveries before the delay, South Africa returned after the delay to discover the revised target was 21 runs off one delivery! It was absurd. Something needed to be done.

The Duckworth-Lewis (D/L) method was developed in the early 1990s by Frank Duckworth and Tony Lewis, as a reaction to the outcome of that match. People are mistrustful of mathematical tables. Although the D/L method has never given a result as abhorrent as the one above, it has received its fair share of criticism. One argument is that it is difficult to understand, but that would be so of any future method. To assess other complaints, one needs to understand the method. The biggest problem is that runs are never scored in a linear fashion. In other words, every innings goes through phases of varying run rates, and the ability of a team to score quickly towards the end of an innings is generally, but not exclusively, determined by the number of wickets in hand. The D/L system disregards all factors other than the number of wickets in hand and number of overs left, based on which it assigns a ‘resource score’ to each team. When the interruption occurs, the score is fed into a formula and a revised target is obtained. In addition to the resource scores, the formula depends on the average 50 over score of cricket matches around the world (which is currently 235).

A little spare time and the online D/L calculator (on their website) yields some interesting results (although the one on the website only works for scores below 235). The formula assumes that there is always a compromise between scoring quickly and keeping wickets in hand. This simplistic quantifying of available resources has led to some counterintuitive results, most obviously in instances such as the match between the West Indies and Zimbabwe at the Adelaide Oval in 2001. Rain interrupted play after the West Indies had scored 235/6 in 47 overs. The target for Zimbabwe in 47 overs was raised to 253. The idea that Zimbabwe’s target was as many as 18 runs more than what the West Indies had scored is perplexing at first. To quote Duckworth and Lewis, ‘The enhancement is a logical and fair way of compensating West Indies for the unexpected shortening of their innings and neutralising Zimbabwe’s advantage of knowing in advance of their shorter innings.’ Zimbabwe lost the match by 77 runs, being dismissed for 175 in 40 overs.

A more complicated alternative was proposed by Carter and Guthrie of the Victoria University of Wellington, New Zealand, and published in April 2004, based on an idea by Preston and Thomas (2002). It suggests that in addition to resource scores, it is the probability that the team batting second will win that should be preserved after the interruption. (Their paper, ‘Cricket interruptus: fairness and incentive in limited overs cricket matches’ can be found online at https://bit.ly/CarterandGuthrie). The resource scores are now redefined to factor in the chasing team’s probability of winning. Without adding any further resources to the calculation, the Carter-Guthrie (C/G) method addresses what is seen as a failing of the D/L system—that the D/L system heavily favours the team that is ahead when the interruption occurs, often leading to boring resumptions where the result is a foregone conclusion. The probability that the team chasing will win is never 100 per cent, no matter how well they are playing. So the match resumes in the same position that it was interrupted in, rewarding the chasing team for a good start (or penalising it for a bad one), and, as a happy consequence, making it more enjoyable as a spectator sport, unlike the original D/L method. The D/L system had already been modified in 2004 to take this into account, though in a different way, without conserving probability. This is why the modified D/L target needs to be calculated by computer.

Messrs Duckworth and Lewis responded to Carter and Guthrie’s method, accusing it of being ‘socialist’ in its approach, claiming that ‘maintaining probability has the effect of taxing the run-rich to aid the run-poor’ citing a hypothetical example of two simultaneous adjacent matches interrupted by rain (the complete description of which can be found in Duckworth and Lewis’ paper, imaginatively titled ‘Comment on Carter M and Guthrie G (2004). Cricket interruptus: fairness and incentive in limited overs cricket matches’ online at https://bit.ly/commentC-L ).

The example is reminiscent of the old joke that pokes fun at scientists and their obsession with unreal, idealised situations. A farmer approaches a theoretical physicist to figure out why his cows are not producing any milk, to which, after many calculations, the physicist replies, “I have a solution, but it only applies to spherical cows in a vacuum.”

Carter and Guthrie’s polite but unyielding rebuttal (‘Reply to the comments of Duckworth and Lewis’ which can also be found online at https://bit.ly/C-Lrebuttal) in a style typical of mathematicians feuding through published papers, further complicated the example, by adding a third simultaneous match and showing that it was, in fact, the modified D/L method that was ‘socialist’. What we have here is a case of ‘my cows are less spherical than yours’. Duckworth and Lewis identified and pointed out what they considered to be a flaw in the C/G system, not realising that their recently modified model would lead to a similar revised score.

The one alternative to the D/L method that has been the best received (or at least the most publicised) is the V Jayadevan method (VJD). The method will get its first big break in the upcoming IPL-4, after having been rejected by the ICC in favour of the modified D/L method. Jayadevan is a civil engineer from IIT-Madras who devised a system that borrows from two previous methods, including the D/L system, but incorporates more ideas about the structure of a run chase. It is also more robust in dealing with multiple interruptions. VJD’s method is also built on the theory of a resource score, but includes a break-up of the structure of an innings into seven stages, based on analysis of hundreds of ODIs. The twist is that it is the percentage of total runs scored in these segments, rather than actual runs scored or wickets lost, that come into play.

As explained in detail in Dr Srinivas Bhogle’s article on VJD,  the stages of the innings are ‘settling down (first 10% of the overs, i.e. the five overs), exploiting field restrictions (next 20% of the overs, i.e. overs 6–15), stabilising the innings-I (next 20%; overs 16–25), stabilising the innings-II (next 10%; overs 26–30), beginning the acceleration (next 20%; overs 31–40), secondary stage of acceleration (next 10%; overs 41–45) and final slog (last 10%; overs 46–50).’

A table is constructed based on this format of run scoring, where the percentage of the total number of runs to be scored is expressed in terms of the percentage of overs completed and number of wickets lost. The table also takes into account the change in style of batting after an interruption inasmuch as it is based on two different curves, a normal curve (depending on both the percentage of overs played and wickets lost) and a target curve (depending only on the percentage of overs played, used only to set revised targets).

For example, if the team batting first scores 300 in 50 overs, and the team batting second has to chase it after an interruption that shortens their innings to 30 overs (that is, to 60 per cent), the corresponding percentage of runs scored in that time, according to the table, is 72.3. Therefore the revised target is now 217 in 30 overs. The table also includes a provision for wickets lost. Whether or not this method is better is impossible to say. Even with its introduction into IPL-4, its efficacy relative to the D/L method will not really be verifiable. It is simply another alternative. It will, no doubt, be replaced by another improved system in a few years.

Let us return to the disastrous World Cup semi-final between England and South Africa in 1992, and compare the targets that would have been set by later methods. According to the D/L system, the revised target would require four runs to be scored off the last ball. Carter and Guthrie, no doubt incensed by this suggestion, would suggest that the probability of winning before and after the interruption be conserved. Their method asks South Africa to score two runs off the last ball. Finally, Jayadevan’s method, almost predictably, sets three runs to win off the last ball.

Any improvements to cricket’s rain rules will be incremental and will have to be even more complicated, taking additional criteria into account. Perhaps someday someone will develop a system that takes the earth’s rotation, angle of the sun’s rays, and a butterfly flapping its wings in Brazil into consideration. That too will have its critics.

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