Duckworth-Lewis Method: A mathematical conundrum?

A dejected Shaun Pollock and teammates after South Africa were knocked out of the 2003 World Cup following a tie against Sri Lanka

It was the 3rd of March in the year 2003 on a damp evening in Durban. South Africa was engulfed in a close encounter against Sri Lanka in a must-win World Cup match. But destiny had something else in mind as Mark Boucher denied a single off the last ball of the 45th over.

Little did he know that the heavens would open up and end the home team’s run at the quadrennial event. The Duckworth-Lewis method was used to decide that the game was a tie, send the Proteas crashing out of their home tournament.

Since that eventful day a lot has happened on the cricket field, and the D/L method has been questioned more than once.

Why do we need a rain-rule?

How would you feel if the team you are supporting is in a winning position only to be denied a victory by the weather?

The founders of cricket were luckily not faced with the issue of determining who the winner was, as a draw was a fair result in a Test match. But with the advent of one day cricket, a drawn result did not seem fair to a team which had performed considerably well in the game.

So the cricket pundits had to gather their mathematical abilities to develop a tool which could effectively decide a winner in case of a curtailed encounter. However, the going was not easy for the statisticians as there were quite a few futile and ill-conceived attempts in the past.

Failed attempts in the past

The farcical rain rule used in 1992 is a fine example of these failed attempts. Unfortunately, the Proteas were again at the receiving end of rain rule calculations. The African team needed 22 off 13 balls to enter their first ever World Cup final. But 12 minutes of rain was all that was required to shatter the dreams of a nation returning to the cricketing fold.

From 22 off 13 balls, the target became 21 off 1 ball! This rain rule calculated the runs scored in the most economical over of the opponents’ innings. Since the South Africans had given a minimum of 1 run in 2 overs (these being the most economical overs), the target was reduced accordingly.

As times have changed, the ICC has decided to stick with a rain-rule which has been tried and tested – the Duckworth-Lewis Method.

What is the Duckworth-Lewis method?

On following cricket commentary on the internet, I was shocked to see qualified engineers (or at least they claimed to be qualified engineers) calling the D/L method more complicated than calculus! I assure the reader that the D/L method may be hyped to be an extremely complicated tool, but in reality it is based on a simple understanding of the amount of resources a team has.

The resource percentage of a team depends on the number of wickets fallen and the amount of overs remaining. The final aim is to ensure that both teams use the same amount of resources. The winner is decided based on the team which scores more runs with the available resources.

For instance, if a team has 50 overs and 10 wickets remaining, it has 100% resources available. The resource percentage a team has varies with the number of overs left and the number of wickets lost.

Stuart Broad reacting after the 2014 World T20 match between England and New Zealand. New Zealand won by 9 runs via the D/L method.

Now, the big question is how to use this to calculate the target.

Calculating the target (read on only if you’re interested in mathematics; otherwise, skip this segment):

If team A has used more resources than team B, then reduce the target for B according to the extra resources used and vice-versa.

So how do we know how many resources a team has? Well, we do not have to break our heads on this as statisticians have already prepared a table for the same. The table can be found here.

The Duckworth-Lewis table has a column for the wickets remaining and a row for the number of balls left, a simple matrix corresponding to each state of the match.

There are two cases which need to be considered:

Resources of team batting first > Resources of team batting second

Let is take the example of the Sri Lanka- South Africa game from the 2003 World Cup:

Sri Lanka batted first and scored 268 runs. Therefore, R1 = resources used by Sri Lanka = 100%

In reply, South Africa was 229/6 after 45 overs when rain stopped play.

Resources left for South Africa = value obtained from D/L table for 6 wickets and 5 overs remaining.

This value according to the D/L table is 14.3.

R2 = resources used by South Africa = 100 – Resources left

Therefore, R2 = 100 – 14.3 = 85.7

Since R1>R2, the target for South Africa = (R2/R1) x (score made by Sri Lanka) +1

Target = (85.7/100) X 268 +1 =229.6 + 1 =230 (rounded down)

Since the South Africans had one run less than the target, the match ended in a tie.

Resources of team batting first < Resources of team batting second

For this case, let us consider a match between India and Sri Lanka during the tri-series in West Indies last year.

India batted first and were 119/3 after 29 overs when rain stopped play. The match was reduced to a 29 overs-a-side match. Here’s how the target for the Sri Lankans would be decided:

Resources India had at start of innings = 100

Resources at the time of interruption = 100 – 50.6 (corresponding to 3 wickets fallen and 21 overs remaining) = 49.4

Resources used by India = R1 = 49.4

R2 = Resources available for Sri Lanka = 73.5 (corresponding to 0 wickets fallen and 29 overs in hand)

Since R2 > R1

Target = (R2-R1) x Average score /100 + Team 1 score + 1 (where average score = 245)

= (73.5 – 49.4) x 245/100 +119 + 1 = 178 (rounded down)

Therefore, the target was 178 for Sri Lanka to win.

Criticism of the method

A mathematical technique based on some assumptions has its pros and cons.

The Duckworth-Lewis method has often been criticized for favouring the team batting second. A simple look at the table shows that the reason for this advantage is because the chasing side has an idea of what it needs to do in the required time frame.

For instance, consider the world T20 game between England and New Zealand in Chittagong. After a splendid effort, the English scored the highest score on the ground, only to be denied by a few meaty blows from Brendon McCullum in the 5th over. In a T20 match, the chasing side can go hammer and tongs against the opposition as there is no fear for lack of wickets.

So why do they still use this method?

Well, here is the crunch part. With the loss of each wicket, the D/L par score moves upwards. What if McCullum had got out trying to accelerate? The match could very well have been headed the other way.

The same was the case during India’s tied game at Lord’s in 2011. The wicket of Rohan Bopara turned the game back in India’s favour and helped them eke out a tie. So the chasing side knows what to do but if a wicket falls their task gets even tougher.

There has also been widespread criticism on the applicability of the method to the T20 format. With only 5 overs constituting a completed match, the margin for error is at an absolute minimum. To add to that, teams scoring over 50 runs in 5 overs is pretty common, and the table has to be modified to take the high scoring into account.

I believe a better solution would be to have 8 overs constitute a completed T20 match. That way more number of overs would have to be played and to some extent the lottery behaviour of a curtailed match would be reduced.

Whatever be the case, an extrapolation of the different factors on a cricket field to determine a winner is not an easy task. That said, in more cases than not, the Duckworth-Lewis Method has successfully helped to find the worthy victor in a rain interrupted game.

We can only hope that the method evolves with the changing times and if the weather plays spoilsport a true winner is rightfully determined.