## Revisiting the Duckworth-Lewis method

Srinivas Bhogle

When it rains, remember Messrs. Duckworth and Lewis. Their method of calculating targets in rain-affected ODIs has always been a little complex to understand. Our expert sheds some light on the same.

The D/L rationale

To win an ODI match, the team chasing requires two ‘resources’: ‘wickets in hand’ and ‘overs remaining’. Everyone knows this. We knew this even 20 years ago!

But the rain rules in 1989 were strange: they only considered the ‘overs remaining’ resource.

So if India scored 250 in 50 overs (O1=50) and then a rain interruption left India’s opponents with only 25 overs (O2=25), they had to score 250*(O2/O1)+1 = 250*(25/50)+1 = 126 runs to win with all 10 wickets in hand!

The option of using all 10 batsmen even when the overs were reduced offered a considerable advantage. No wonder all captains wanted to chase if they won the toss on a cloudy day.

The Duckworth-Lewis (D/L) method corrected this anomaly. D/L came up with a mathematical model that combined these two resources into a single ‘combined resource’. D/L said that it’s more appropriate to use the ratio of the ‘combined resource’ instead of just using the ratio of ‘overs remaining’.

In our example, therefore, India’s opponents will be required to score 250*(R2/R1)+1 to win, where R1 is India’s combined resource, and R2 is the combined resource of the opposing team.

To understand this R1, R2 business, look at the illustration below:

The x-axis is a countdown of the 50 overs in an ODI game. You start the innings at the origin, and after every ball you take one step to the right, till you reach the right extreme with ‘0 overs remaining’ when the innings must end.

The y-axis shows the ‘combined resources’ percentage. When all 50 overs remain, you have 100% of the resource percentage. After every ball is bowled, the combined resource reduces a little, and when all 50 overs are bowled, the resource percentage drops to 0%.

We also see ten coloured curves in the illustration – all of them start off from somewhere on the y-axis and eventually merge into a single curve at the ‘south-east’ extreme corresponding to 0 ‘overs remaining’.

Each curve corresponds to the ‘wickets in hand’. The top most curve corresponds to ’10 wickets in hand’, while the bottom most curve (barely visible) corresponds to ‘1 wicket in hand’. These curves indicate how the combined resource depletes as the ODI innings progresses.

A team starts its innings by travelling right along the curve corresponding to ’10 wickets in hand’. When a wicket falls, the team drops to the curve below (corresponding to ‘9 wickets in hand’) and continues its 50-over journey. So after every ball, it is possible to track the combined resource percentage used by the batting team.

Illustrative examples

As an example, suppose Pakistan has scored 128/3 in 30 overs chasing India’s 250 when rain ends the match. Who wins? If we go by the 1989 rain rule – that only considered ‘overs remaining’ – India wins, because the Pakistan target was 250*(30/50) + 1 = 151. But would Pakistan win based on the D/L method? Let’s check.

Since India batted all its 50 overs, India used up 100% of its available combined resource (R1=100). How much of its available combined resource has Pakistan used up when rain ended the match? Let’s look at our illustration. Move your finger right on the x-axis till you reach ’20 overs remaining’ (remember 30 overs have been bowled). Then track the curve corresponding to ‘7 wickets in hand’ (because Pakistan lost 3 wickets). You’ll find that Pakistan at that point had used up about 50% of its combined resource (R2=50). So Pakistan’s winning target at this point was 250*(50/100) + 1 = 126 and, therefore, Pakistan wins!

Let’s now suppose that India was cruising at 200/2 in 40 overs, when rain ended the Indian innings. The 1989 rain rule would have required Pakistan to score 201 in their 40 overs. Was this fair?

If we look at the curves again, we’ll find that India had over 30% of its resource remaining (look at the yellow ‘8 wickets in hand’ corresponding to ’10 overs remaining’), so it had used only about 70% of its combined resource (R1=70). Pakistan, on the other hand, had the opportunity to use up to 90% (R2=90) of its combined resource (look at the ’10 wickets in hand’ curve corresponding to ’40 overs remaining’). Clearly this was unfair; that’s why the D/L target for Pakistan would have had to be higher than 200.

If we use the formula we used earlier, the reset target would be 200*(90/70) + 1 = 258, which, at first sight, appears unreasonably high. Duckworth-Lewis knew that such scale-ups could give very unreasonable results; that’s why they introduced the G50 turn-around, which set a more reasonable target of about 230-240. The G50 criterion says that if R2 is greater than R1, it is better to use an additive formula to determine the elevated reset target instead of a multiplicative scale-up. This made the D/L formula less elegant mathematically, although it still gave very reasonable targets in most cases.

As these examples illustrate, the D/L method indeed appears to be fair, with the reset targets appearing reasonable in most cases.

D/L glitches

However, there were two glitches that some of us had identified (that Duckworth-Lewis were aware of, even if they were a little reluctant to publicly say so in the early years).

#1 While chasing big targets, the D/L method seemed to favour the chasing team if they didn’t lose too many wickets.

Even a cursory glance at the D/L curves tells the story. The top three curves (corresponding to 10, 9 and 8 wickets lost) show that a team is sitting with just under 60% of its combined resource intact even when just 20 overs remain. So a team with a score of 160/2 after 30 overs wins even if the target is about 350.

Put another way, the top three D/L curves don’t come down quickly enough during the first 30 overs, and this anomaly can be successfully exploited by the chasing team.

#2 In low-scoring games (less than 150 runs scored), if a lot of overs (20 or more) were lost, the D/L target was unreasonably high for the chasing team.

This anomaly was linked to the ‘G50’ criterion of the D/L method. The G50 criterion, which assumed an average ODI score of 225 or 235, was a flawed instrument; Duckworth-Lewis knew this, but couldn’t fix it as long as their Standard Edition was being used.

Towards perfection

In the early years, the ICC’s mandate apparently was that the D/L method must be simple (back of the envelope calculations only), and must not require a computer.

But as ODI scores started climbing (300 is pretty common today), Duckworth-Lewis realized that glitch #1 could create a potentially difficult situation, especially in big World Cup matches – there was a brief moment during the 2003 World Cup final when India appeared to have a small chance of gaining a thoroughly ill-deserved victory against Australia, but India then lost a couple of wickets and the big rain never came.

The way to correct the glitch was to ‘play’ with the shapes of the D/L curves – especially those corresponding to 10, 9 or 8 ‘wickets in hand’ – so that they drop sooner and steeper. While the curves would still start from the same point on the y-axis, we now have a kind of #### at the south-east extreme (corresponding to ‘0 overs remaining’) that can be turned to make the curves tauter. As long as the team batting first scores 235 or less (G50=235), we don’t turn the ####. As soon as the score crosses 235, we start turning the ####: the higher the score above 235, the more we turn the #### and the tauter the curves get.

This seems to be the general idea in the new Professional Version that is used for the international matches. The Professional Version also does away with the G50 criterion by successfully managing scale-ups, although I haven’t understood how Duckworth-Lewis have achieved this – and they aren’t sharing this information in the public domain yet.

With these corrections, it would be fair to say that D/L is now very close to the ‘perfect’ rain rule for ODI cricket, although – ironically – this seems to be happening just when the 50-over version seems threatened.

Sadly, the D/L method for T20 cricket is still far from perfect. The general idea is to pretend that a T20 match is actually a 50-over match with the first 30 overs lost: we operate only with the segments of the curves to the right of the ’20 overs remaining’ vertical line.

This has to be considered to be a make-shift ‘fix’ because the dynamics of a T20 game are very different from the last 20 overs of an ODI game; for example, field restrictions during the first 6 overs will make quite a difference to the run-scoring and wicket-losing pattern.

It is also easy to see that all curves start strongly overlapping beyond the ’10 overs remaining’ stage and are practically indistinguishable in the last five overs. This means that the ‘wickets in hand’ resource is practically ignored from this point.

The real solution is to completely redraw the curves for 20 overs, with the combined resource diminishing from 100% to 0%. To do this, Duckworth-Lewis need more data points – this will come only when more T20 matches are played. Till then, we must wait … and not grieve too much if England is wrongly declared a loser in a match it should have won against the West Indies.

Posted by Srinivas Bhogle on 12/15 at 04:36 AM

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