Duckworth/Lewis Explained by Daniel Kingdom

Duckworth/Lewis Explained by Daniel Kingdom

Frank Duckworth and Anthony Lewis

Somerset fan Daniel Kingdom explores the much-debated Duckworth-Lewis method and explains how it’s affected Somerset’s recent Royal London One Day Cup games in this exclusive blog for The Incider. You can follow Daniel at@DanKingdom96 on Twitter.

The Duckworth/Lewis (D/L) method. In most limited overs matches, the names of these two statisticians, who lend their names to one of the most interesting and bizarre elements of sport, are irrelevant. The D/L ‘par score’ may be displayed on the scoreboard for the second innings, but in the majority of games, no one pays attention to it as there are no stoppages to bring it into play. But in a minority of matches, it rains. That is where D/L comes in.

First, let me go back to the time before D/L. Rain-reduced matches used to be solved using the ‘average run-rate’ method, where the winner was the team with a higher run-rate. This was unfair because it took no account of wickets lost. A team could be 9 down while chasing and 100 runs from the target, but still win after rain if they have a better run-rate than the team who batted first. It is easier to score 150 from 25 overs than 300 from 50 overs, even though the run rate is the same.

A slight improvement on the ‘average run-rate’ rule was the ‘most productive overs’ method. The gist of this was that in setting a revised target, only the team batting first’s highest scoring overs would be taken into account. So if a second innings was reduced to 25 overs, the first innings’ 25 highest scoring overs would be added together to create the new target (with 1 run added for the win). This created a situation in the 1992 World Cup where South Africa required 22 off 13 balls before rain, and they returned to the field to find that they now needed 22 off 1 ball. Need I say more?

It was after this debacle that Frank Duckworth and Tony Lewis created a more mathematical method of settling rain-reduced matches. Contrary to popular belief, you don’t need a mathematics degree to understand it – just a good understanding of how limited overs cricket works. Essentially, it makes an educated guess at what would have happened had rain not intervened. Its basic principle is that a batting side has ‘resources’ to score runs – overs and wickets. Therefore keeping wickets in hand and scoring fast is the best way to ‘beat’ D/L. One of the advantageous elements of the formula is that it can cope with a rain delay in either the first or second innings, and can set, supposedly, a mathematically fair target for the side batting second. It also takes into account the fact that a run rate of, for example, 7, is harder to sustain over 50 overs than over 25 overs. This was the major flaw with the ‘average run rate’ method. Additionally, it ensures that farcical situations such as the 1992 World Cup instance are unlikely to be created.

If there is first innings rain, the team batting first usually receives ‘compensation runs’ to make up for the fact that they were batting as though there was not going to be a reduction. For example, a team could reach 100 for 0 after 20 overs before rain. It would be unfair for the team batting second to have a target of 101 off 20 overs because the team batting first, in theory, didn’t know it was going to rain, so batted as though it was a 50 over game, therefore scoring slower. D/L adjusts the first innings score. I would guess that in this example the chase would probably be around 170. It is effectively saying ‘this is what you would have scored if you had known from the start that it was going to be a 20 over innings.’

If there is second innings rain, D/L gives a ‘par score’ which changes after every ball. If a par score is 100 after 20 overs, this means that if a team has 101 after 20 overs before rain, they would win because, according to D/L, they would have won anyway had there been no rain. Here it is saying ‘if there had been no rain, the side batting second would have won had the match been played to its natural conclusion.’ The par score is adjusted when wickets are lost, because the loss of wickets obviously means that the team batting second has a reduced chance of winning.

It is important to note a few things – firstly, D/L is a mathematical guess based on runs, overs and wickets at what would have happened had rain not interrupted. A hat-trick may have been about to happen but D/L only takes into account what has happened in the match before rain arrived. Secondly, D/L assumes that teams bat as though there is not going to be any rain. This is explained above. Lastly, D/L takes into account that scoring faster increases the likelihood of lost wickets, but remember also that in a match reduced to 30 overs, teams score faster than in a 50 over game – this will be important later.

Now onto the main point of this article – why were the D/L calculations seemingly different in Somerset’s recent games against Glamorgan and Sussex? I will start with the Glamorgan match on the 12th August. Glamorgan batted first and, for the first 41.5 overs, thought that it would be a 50 over match. Therefore, in theory, they scored at a rate looking for a competitive 50 over score. It then rained and the match was reduced to 47 overs a side. Glamorgan scored 289. But Somerset’s target was not 290 – it was increased to 303. This adjustment compensates for the fact that for most of the innings, Glamorgan scored slower than they would have done had they known it would be 47 overs a side. D/L is basically saying ‘if you had known from the start it would be 47 overs a side, you would have scored 302.’

When Somerset’s target was announced at the ground, many fans around me were questioning why the D/L formula had seemingly worked differently from two days previous, where Sussex were given a reduced target after a similar situation. The answer is simple: it didn’t work differently. In an odd way, it makes sense. Somerset reached 10 for 2 after 5.1 overs, thinking it was 44 overs a side. After a rain delay, the match was reduced to 33 overs a side. Somerset scored 193, and I think everyone at the ground, including me, expected Sussex to have an increased target, similar to the Glamorgan match. However, Sussex’s chase was less than what Somerset scored – 189. Now you may be thinking, ‘in what world can a side have the same number of overs but need less runs?’ Well, the answer is: in the weird world of D/L. Now bear with me here, this is about to get slightly complex.

Remember, D/L was effectively saying ‘if you had known from the start that the innings would be 33 overs, you would have scored 188, which is less than what you actually scored.’ This is despite the fact that, in theory, Somerset batted those 5.1 overs slower than they would have had they known it would be a 33 over match, so you would think that D/L would give Somerset more runs to compensate for this. The reason for the supposed anomaly is this (this goes back to a previous point – scoring faster means you lose more wickets): if Somerset had known it would be a 33 over match during that 5.1 over period, in theory the batsmen would have tried to score faster, and in theory, lost more wickets, causing Somerset to score less runs – losing wickets means that run-scoring is more difficult. Somerset batted so poorly in that 5.1 overs, losing 2 wickets while scoring slowly, that D/L thinks that if the batsmen tried to score quickly, more wickets would have been lost, having a negative effect on the part of Somerset’s innings after the 5.1 over period, therefore decreasing Somerset’s final score.

Essentially, in the case of first innings interruptions, if you bat poorly before the rain break, you are punished, and if you bat well before the rain break, you are compensated. In the Glamorgan match, D/L thinks that Glamorgan batted well enough in the 41.5 overs that they would have scored more runs. In the Sussex match, D/L thinks that Somerset batted so badly in the 5.1 overs that trying to score faster would have lost them wickets, causing a decreased target.

The problem with D/L is that it is a guess. It does not definitively say what would have happened, merely what is likely to have happened based on wickets lost, runs scored and overs remaining. In reality, during that Sussex game, Nick Compton and Colin Ingram would probably have knuckled down and wouldn’t have lost more wickets in that first period. In reality, Somerset would most likely have scored more than 188 in the first innings had the batsmen known all along that it would be 33 overs a side. But D/L cannot take this into account. It can only take into account the data that is available.

A few changes can be made to make D/L more robust. Although the point is probably exaggerated a little, D/L does favour the side batting second – you can read more about it in the first link at the bottom of this article. There are several ways that the formula could be adjusted.

Within the formula is the ‘G50’, which is an abbreviation for the expected score for a team in a 50 over innings. Currently G50 is 245, and it is used worldwide. D/L could be improved by adjusting the score to reflect recent scoring rates in the country that it is being used in. For example, G50 should be different in India to England because 50 over scores are, in general, higher there. The data used in the ‘resources remaining’ tables (I won’t go into those here), used in the calculation, could also do with more regular updating. Another flaw in the formula is that it does not take into account powerplay overs.

But what really annoys me is the occasionally needless use of D/L. Earlier this season, Somerset played Kent in a 50 over match. Kent scored 383. Just as the second innings was about to start, it rained for 10-15 minutes. The umpires then deducted two overs from Somerset’s chase and set a new target of 374. Somerset would probably have won if D/L had not been applied. So my question to the ECB is this: why is there not leeway before overs start being lost? The match finished at about half past six. There was at least an hour and a half from that time before it started getting dark. So why couldn’t the match have finished fifteen minutes later? In the County Championship play can be extended at the end of the day; why not in the one day cup? Why allow the result to be needlessly influenced by a mathematical formula? D/L cannot take into account the natural fluctuations of a cricket match, so why make an utterly pointless adjustment to a target?

In conclusion, a lot of the animosity for D/L stems from t20. Here, the formula is mathematically incorrect. You can read more about that in the second link at the bottom of the article. However D/L, in 50 overs, works fairly well. Small adjustments are probably needed, but the formula is on the right lines, as it takes into account most of the variables and can cope with rain at any time. It is unlikely that any future rain rules will be very different from it.

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