Duckworth–Lewis–Stern method


The Duckworth–Lewis–Stern method previously known as the Duckworth–Lewis method is a mathematical formulation designed to calculate the target score for the team batting second in a limited overs cricket match interrupted by weather or other circumstances. The method was devised by two English statisticians, Frank Duckworth and Tony Lewis, and was formerly known as the Duckworth–Lewis method. It was introduced in 1997, and adopted officially by the International Cricket Council in 1999. After the retirement of both Duckworth and Lewis, the Australian statistician Steven Stern became the custodian of the method, which was renamed to its current title in November 2014. In 2014, he refined the model to better fit modern scoring trends, especially in T20 cricket, resulting in the updated Duckworth-Lewis-Stern method. This refined method remains the standard for handling rain-affected matches in international cricket today.
The target score in cricket matches without interruptions is one more than the number of runs scored by the team that batted first. When overs are lost, setting an adjusted target for the team batting second is not as simple as reducing the run target proportionally to the loss in overs, because a team with ten wickets in hand and 25 overs to bat can play more aggressively than if they had ten wickets and a full 50 overs, for example, and can consequently achieve a higher run rate. The DLS method is an attempt to set a statistically fair target for the second team's innings, which is the same difficulty as the original target. The basic principle is that each team in a limited-overs match has two resources available with which to score runs, and the target is adjusted proportionally to the change in the combination of these two resources.

History and creation

Various different methods had been used previously to resolve rain-affected cricket matches, with the most common being the Average Run Rate method, and later, the Most Productive Overs method.
While simple in nature, these methods had intrinsic flaws and were easily exploitable:
  • The Average Run Rate method took no account of wickets lost by the team batting second, but simply reflected their scoring rate when the match was interrupted. If the team felt a rain stoppage was likely, they could attempt to force the scoring rate with no regard for the corresponding highly likely loss of wickets, meaning any comparison with the team batting first would be flawed.
  • The Most Productive Overs method not only took no account of wickets lost by the team batting second, but also effectively penalised the team batting second for good bowling by ignoring their best overs in setting the revised target.
  • Both of these methods also produced revised targets that frequently altered the balance of the match, and they took no account of the match situation at the time of the interruption.
The D/L method was devised by two British statisticians, Frank Duckworth and Tony Lewis, as a result of the outcome of the semi-final in the 1992 World Cup between England and South Africa, where the Most Productive Overs method was used. When rain stopped play for 12 minutes, South Africa needed 22 runs from 13 balls, but when play resumed, the revised target left South Africa needing 21 runs from one ball, a reduction of only one run compared to a reduction of two overs, and a virtually impossible target given that the maximum score from one ball is generally six runs. Duckworth said, "I recall hearing Christopher Martin-Jenkins on radio saying 'surely someone, somewhere could come up with something better' and I soon realised that it was a mathematical problem that required a mathematical solution." The D/L method avoids this flaw: in this match, the revised D/L target of 236 would have left South Africa needing four to tie or five to win from the final ball.
The D/L method was first used in international cricket on 1 January 1997 in the second match of the Zimbabwe versus England ODI series, which Zimbabwe won by seven runs. The D/L method was formally adopted by the ICC in 1999 as the standard method of calculating target scores in rain-shortened one-day matches.

Theory

Calculation summary

The essence of the D/L method is 'resources'. Each team is taken to have two 'resources' to use to score as many runs as possible: the number of overs they have to receive; and the number of wickets they have in hand. At any point in any innings, a team's ability to score more runs depends on the combination of these two resources they have left. Looking at historical scores, there is a very close correspondence between the availability of these resources and a team's final score, a correspondence which D/L exploits.
The D/L method converts all possible combinations of overs and wickets left into a combined resources remaining percentage figure, and these are all stored in a published table or computer. The target score for the team batting second can be adjusted up or down from the total the team batting first achieved using these resource percentages, to reflect the loss of resources to one or both teams when a match is shortened one or more times.
In the version of D/L most commonly in use in international and first-class matches, the target for Team 2 is adjusted simply in proportion to the two teams' resources, i.e.
If, as usually occurs, this 'par score' is a non-integer number of runs, then Team 2's target to win is this number rounded up to the next integer, and the score to tie, is this number rounded down to the preceding integer. If Team 2 reaches or passes the target score, then they have won the match. If the match ends when Team 2 has exactly met the par score then the match is a tie. If Team 2 fail to reach the par score then they have lost.
For example, if a rain delay means that Team 2 only has 90% of resources available, and Team 1 scored 254 with 100% of resources available, then 254 × 90% / 100% = 228.6, so Team 2's target is 229, and the score to tie is 228. The actual resource values used in the Professional Edition are not publicly available, so a computer which has this software loaded must be used.
If it is a 50-over match and Team 1 completed its innings uninterrupted, then they had 100% resource available to them, so the formula simplifies to:

Summary of impact on Team 2's target

  • If there is a delay before the first innings starts, so that the numbers of overs in the two innings are reduced but still the same as each other, then D/L makes no change to the target score, because both sides are aware of the total number of overs and wickets throughout their innings, thus they will have the same resources available.
  • Team 2's target score is first calculated once Team 1's innings has finished.
  • If there were interruption during Team 1's innings, or Team 1's innings was cut short, so the numbers of overs in the two innings are reduced, then D/L will adjust Team 2's target score as described above. The adjustment to Team 2's target after interruptions in Team 1's innings is often an increase, implying that Team 2 has more resource available than Team 1 had. Although both teams have 10 wickets and the same number of overs available, an increase is fair as, for some of their innings, Team 1 thought they would have more overs available than they actually ended up having. If Team 1 had known that their innings was going to be shorter, they would have batted less conservatively, and scored more runs. They saved some wicket resource to use up in the overs that ended up being cancelled, which Team 2 does not need to do, therefore Team 2 does have more resource to use in the same number of overs. Therefore, increasing Team 2's target score compensates Team 1 for the denial of some of the overs they thought they would get to bat. The increased target is what D/L thinks Team 1 would have scored in the overs it ended up having, if it had known throughout that the innings would be only as long as it was.
  • If there are interruption to Team 2's innings, either before it starts, during, or it is cut short, then D/L will reduce Team 2's target score from the initial target set at the end of Team 1's innings, in proportion to the reduction in Team 2's resources. If there are multiple interruptions in the second innings, the target will be adjusted downwards each time.
  • If there are interruptions which both increase and decrease the target score, then the net effect on the target could be either an increase or decrease, depending on whether Team 2's resource loss is large enough.

    Mathematical theory

The original D/L model started by assuming that the number of runs that can still be scored, for a given number of overs remaining and wickets lost, takes the following exponential decay relationship:
where the constant is the asymptotic average total score in unlimited overs, and is the exponential decay constant. Both vary with . The values of these two parameters for each from 0 to 9 were estimated from scores from 'hundreds of one-day internationals' and 'extensive research and experimentation', though were not disclosed due to 'commercial confidentiality'.
Image:DuckworthLewisEng.png|right|Scoring potential as a function of wickets and overs.
Finding the value of for a particular combination of and , and dividing this by the score achievable at the start of the innings, i.e. finding
gives the proportion of the combined run scoring resources of the innings remaining when overs are left and wickets are down. These proportions can be plotted in a graph, as shown right, or shown in a single table, as shown below.
This became the Standard Edition. When it was introduced, it was necessary that D/L could be implemented with a single table of resource percentages, as it could not be guaranteed that computers would be present. Therefore, this single formula was used giving average resources. This method relies on the assumption that average performance is proportional to the mean, irrespective of the actual score. This was good enough in 95 per cent of matches, but in the 5 per cent of matches with very high scores, the simple approach started to break down. To overcome the problem, an upgraded formula was proposed with an additional parameter whose value depends on the Team 1 innings. This became the Professional Edition.