# The Duckworth-Lewis Method

In first-class cricket matches, each team has two-innings, ten wickets to lose in each inning and no limit on the number of overs in which to use those ten wickets. However, there is typically a four or five day limit on the length of the match. The aim is to score as many runs as possible and a team is declared the winner if they have scored more runs than the other team over both innings' provided that their opposition has lost all their wickets. Thus if one team has scored less runs but has not lost all their wickets when time runs out, the match is declared a draw.

A one-day cricket match consists of one innings per team with 10 wickets to lose and a 50-over limit on the innings. So unlike first-class cricket, a one-day cricket match must have a winner unless each team's score is the same at the end of the match, in which case the result is a tie.

Cricket cannot be played in inclement weather (such as rain or bad light). A first-class match can simply resume from where it left off after a rain delay. Adopting this philosophy in a one-day match could easily prevent the match from completing in a single day due to time constraints. Thus there must be rules that shorten a rain-affected match so that the rain delay doesn't advantage either team.

This provides a real life problem for statisticians and mathematicians to solve. In this first of a series of articles we outline some of the methods used to calculate revised targets in rain-affected matches and we describe in detail how the current method, the Duckworth-Lewis method, is used. We then give some real life situations of rain-affected matches where the Duckworth-Lewis method was or could have been used.

Let's use the 1992 World Cup semi-final between England and South Africa as an example of a rain-affected match. Due to rain before the match commenced, each team was restricted to 45 overs. England scored 6/252 batting first and South Africa in reply were 6/231 after 42.5 overs before rain interrupted play with South Africa requiring 22 runs from 13 balls with 4 wickets in hand. With a place in the World Cup final at stake, it was crucial that the overs to be lost by South Africa did not advantage or disadvantage either team. We show how this match proceeded later in the article.

Over the years there have been many different methods used to recalculate targets after rain delays. A simple method is the Average Run Rate (ARR) method in which the team with the higher average number of runs per over faced is declared the winner. In the Most Productive Overs (MPO) method, the target for the team batting second, when their innings is reduced to x overs, is the sum of the x highest scoring overs of the team that batted first. Both these methods, along with the others, have flaws and would regularly disadvantage one of the teams.

When play resumed in the 1992 World Cup semi-final, enough time had been lost that 2 overs had to be removed from South Africa's innings, leaving them with just 1 ball to face. The MPO method in use at the time controversially calculated that South Africa's target was only reduced by one run and thus they required 21 runs from the remaining 1 ball, which left them in an impossible position, and England advanced to the World Cup Final. This match clearly highlighted the need for a better method for calculating revised targets in one-day matches.

Frank Duckworth and Tony Lewis, two statisticians in England, together invented a method for fairly calculating revised targets and it has been the official method used in one-day international matches since 1997. It is based around the fact that batting sides have two types of resources with which to score their runs - wickets and overs - and that their potential for run scoring depends on both these resources.

They define a function P(u,w) that denotes the average proportion of runs yet to be scored by a side in an innings with u overs left to be bowled and w wickets lost. Extensive research and statistical methods led to the creation of this function, which cannot be disclosed due to commercial confidentiality. However, Table 1 is an excerpt of the table containing P(u,w) values for a few select values of u and w. Click here for the full fifty over 2002 Duckworth-Lewis table.

#### Table 1: An excerpt of the 2002 Duckworth-Lewis table

Overs left \ Wkts lost0257910
50100.085.149.022.04.70.0
4089.377.847.622.04.70.0
3075.167.344.721.84.70.0
2056.652.438.621.24.70.0
1032.130.826.117.94.70.0
00.00.00.00.00.00.0

The statistics in the table implicitly show the tactics and the way the game is played. At the start of a full innings, the batting side has 100% of its resources still available, so P(50,0)=100%. Every over bowled and each wicket lost results in a loss of resources for the batting team. Every time there is an interruption that causes overs to be lost, the batting side unfairly loses resources. Fairly or unfairly, how much resource they have lost depends on the stage of the innings and the number of wickets lost.

Stating this formally, if rain interrupts play with a team w wickets down with u1 overs remaining and play resumes with u2 overs remaining, then they have lost P(u1,w) - P(u2,w) of their resources and whether they are batting first or second, the Duckworth-Lewis method needs to reset their score (if batting first) or their target (if batting second) to reflect this loss of resources.

Let us first take the case where team 1 has completed its innings uninterrupted, that is they had 100% of their resources available, scoring S runs and suppose that team 2 has its innings interrupted and overs are lost. We can easily calculate the proportion of resources team 2 has at its disposal after a rain delay. The new target for team 2 will be set so that the ratio of team 2's target to team 1's score is the same as the ratio of team 2's resources available to team 1's resources available. That is,

Put simply, if team 2 only has available x% of the resources that team 1 had, then they only have to score x% of the runs team 1 scored. Note that the revised target is always rounded down and this is the target for a tie. To win the match, the team batting second needs to score at least one more run than the target.

For example, suppose a team is 5 wickets down with 20 overs remaining when rain causes 10 overs to be lost. When they leave the field, the batting side still has P(20,5)=40.0% of their resources remaining. After the rain delay they only have P(10,5)=27.5% of their resources available, so they have unfairly lost P(20,5)-P(10,5)=12.5% of their resources. In other words, assuming there are no more rain interruptions, this team only has 87.5% of the resources team 1 had available. Thus if the team was batting second chasing S to win, their new target would be S x (1 - P(20,5) + P(10,5)) = S * 87.5%.

When rain interrupts and overs are lost from the innings of the team batting second it always disadvantages them and hence their target is reduced as we just described. When rain interrupts the innings of team 1, in which case match officials try arrange the loss of overs so that each team's innings is reduced by the same amount, it sometimes disadvantages the team batting first and sometimes disadvantages the team batting second.

For instance, if team 1 still has 8 wickets in hand when rain caused the premature closure to their innings with 10 overs remaining, they are significantly disadvantaged, since it is highly likely they would have scored a lot more runs, and hence the target for team 2 should be increased. While if team 1 was 9 wickets down when rain caused a premature closure to their innings with 25 overs remaining, this benefits them because it was unlikely they were going to score many more runs anyway, and hence the target for team 2 should be decreased.

These scenarios can still be modelled in terms of loss of resources. Let R1 denote the resources available for team 1 and R2 denote the resources available for team 2. If the reduction of overs meant that team 1 has more resources available than team 2 (that is R1 > R2), then team 1 has been advantaged and the target for team 2 will be reduced as per Equation 1. If the reduction of overs meant that team 1 has the same amount of resources available as team 2 (that is R1 = R2), then no team has been disadvantaged so the target will not be changed. If the reduction of overs meant that team 1 has fewer resources available than team 2 (that is R1 < R2), then team 1 has been disadvantaged and the target for team 2 will be increased (see Situation 3). In this situation, team 2's target will not be increased using Equation 1, because scaling up team 2's target by the ratio of resources available to each team can sometimes lead to distorted results. Instead, team 2's target is scaled up by applying the excess resource team 1 has (R1 - R2) to the average score achieved in a full uninterrupted first innings, G, which currently is 225.

In summary, if team 1 scores S runs, then the revised target T for team 2 to tie is as follows:

The newer Professional Version of the Duckworth-Lewis method which is currently used at the elite level and requires a computer for calculations, overcomes the problem of distorted results when team 2's target need to be increased. For amateur matches where the use of a computer is not guaranteed, the standard method we have outlined is used.

Since the Duckworth-Lewis method came into operation in 1997, their tables have been updated a few times to reflect changes in the game, specifically the overall increase in scoring rate and the change in scoring patterns.

### Real Life Examples

We have seen how to apply the Duckworth-Lewis tables for all different stages of the match in which a rain interruption might occur. Let us have a look at some real match situations and how the Duckworth-Lewis method would be applied. We will start with that infamous and controversial World Cup match in 1992.

Situation 1

In the rain interrupted 1992 World Cup semi-final, each side was reduced to a maximum of 45 overs. England batted first and scored 6/252. South Africa in reply was 6/231 after 42.5 overs when rain interrupted play causing 2 overs of play to be lost. By the Duckworth-Lewis method what should South Africa's revised target have been?

The first rain interruption caused each team's innings to be reduced by 5 overs. As the resources of each team have both been depleted to 95% no correction is required for this interruption.

However, the rain interruption that occurred during South Africa's innings came with 2.1 overs remaining and 6 wickets lost. The loss of two overs at this stage of the innings cost South Africa approximately 6.8% of their resources by the Duckworth-Lewis tables at the time. Using Table 2, South Africa's target to tie should be

Rounding down gives the score for South Africa to tie as 233. Thus South Africa's target for victory would have been 234 and they would have required 3 runs of the remaining delivery. Recall this compares to the 21 runs required under the MPO method, which was actually in place at the time.

Situation 2

In February 2006 India batted first in a one day international against Pakistan and scored 328 all out. Pakistan in reply was 7/311 after 47 overs when play was abandoned. Who won the match?

Losing the final three overs (at 7 wickets down) corresponds to a loss of 7.3% of resources by the Duckworth-Lewis tables at the time. This then makes Pakistan's target to tie

Thus Pakistan needed to be on more than 304 for victory. Since they were on 311, they were declared the winners by 7 runs.

Situation 3

In Perth in 1983, England had reached 3/45 after 17.3 overs before a heavy thunderstorm caused play to be delayed and each innings to be reduced to 23 overs. In their remaining 5.3 overs, England scored 43 runs for a total of 88 runs. The method at the time made no change to the target for New Zealand and they won easily. By the 2002 Duckworth-Lewis tables, what should New Zealand's target have been?