# “If it rains, do you still have a sporting chance?”

## A critical review of target resetting methods as applied to One-Day cricket

Matthew J. V. Ovens BSc.
School of Mathematical Sciences
Monash University, Australia

Originally presented at: The 7th Australasian Conference on Mathematics and Computers in Sport, 29-31 August 2004, Palmerston North, New Zealand

### Abstract

With the introduction of One-Day cricket and the objective of producing a result within a predetermined time frame, many methods have been suggested and used to recalculate the target score after a stoppage. Currently, the Professional Edition of the Duckworth/Lewis method is used, via computer software called CODA. This paper critically reviews the current method of target resetting and discusses a potential alternative.

Keywords: cricket, Duckworth/Lewis, Jayadevan, target resetting

### Introduction

One-Day (or Limited Overs) cricket began in England in the 1960s and eventually spread to the member countries of cricket’s governing body, the International Cricket Council (ICC). Originally the two sides were restricted to between 40 and 65 overs, however the standard for One-Day International (ODI) matches is now 50 overs. The game is played between two teams of 11 players. When batting, each side receives 50 six-ball overs unless they lose 10 wickets before such time. The side batting first (referred to as ‘Team 1’) attempts to score as many runs as possible whilst trying to ensure it will receive the full 50 overs. The side batting second (referred to as ‘Team 2’) must then attempt to score one more run, within 50 overs, than Team 1’s total to win the game.

### The Problem

The rules of One-Day International cricket do not allow for play to extend over to another day and as such, any stoppages (i.e. rain, sandstorms or any event that causes play to be temporarily suspended) affect the match time available. In an effort to maintain fairness to both sides, the number of overs to be bowled is reduced. The problem then is, "How can the target for Team 2 be set in such a way as to not greatly advantage one team?"

### Objectives A Solution Should Satisfy

Any suggested solution to the problem of fairly setting the target in a reduced overs match should, in this author’s opinion, endeavour to meet (or attempt to come close to) as many of the following objectives, listed in order of decreasing importance, as possible.

#### Objective 1: Simplicity

The proposed method should be simple enough that any player, umpire, official, commentator or spectator can compute the new target, either mentally or with the aid of a "ready reckoner".

The proposed method should take into account all available variables (e.g. wickets in hand, balls remaining, etc.) with regard to the state of play at the time of the stoppage.

#### Objective 3: Lack of Constants

The proposed method should not require the use of arbitrary constants, except where these constants are derived from commensurate historical data and are used as a form of benchmarking. Constants derived from historical data (e.g. G50 in the Duckworth/Lewis method) should be updated after every commensurate game or at least after every series.

#### Objective 4: Repeatability

The proposed method should be capable of handling multiple stoppages within any one game. Additionally, each application of the method should be independent of how many stoppages have occurred previously (i.e. the method is invariant with respect to the number and manner of stoppages).

#### Objective 5: Fairness

The proposed method should set a fair and reasonable target with respect to the state of play at the time of the stoppage. The method should not be capable of being used as a playing strategy by either team (i.e. knowing that a stoppage is likely should not alter playing strategies, like choosing whether to bat or not after winning the toss).

#### Objective 6: Maintenance of Probability of Winning

The proposed method should attempt to maintain the probability of winning for each team across the stoppage. This objective assumes that the probability of winning is determinable at every stage of the match, which may not be true.

#### Objective 7: Maintenance of Excitement

The proposed method should maintain the competitive excitement for the spectators. The method should not be capable of producing a target that is clearly unachievable when a team is potentially capable of winning prior to a stoppage (e.g. Team 2 needing 22 runs off 13 balls prior to the stoppage revised to 21 runs required from 1 ball! {England v South Africa, World Cup 1992}. See Note 1).

### Duckworth / Lewis (D/L) Method:

The Duckworth/Lewis method comes in two varieties, Professional and Standard editions. The Professional Edition is to be used in all ICC recognised matches, according to ICC rules. The Standard Edition may be used in all other matches or in ICC recognised matches when the Professional Edition software is not working or unavailable. The mathematics and mechanics of the two methods are essentially the same. Since the full mathematical details behind the method are not available due to commercial reasons and the CODA software is currently only available to full ICC members, for the purpose of this paper, we refer only to the tables for the Standard Edition (hereafter known as the D/L method).

Although the full mathematical details of the current method are not available, the original version of the method was described in full mathematical detail in [1]. It is assumed that the derivation of the method remains unchanged and that only some of the parameter values have been altered to update it to the current method.

The central concept behind the method is that each team has a certain amount of resources with which to play the game. These resources consist of both the number of overs and the number of wickets available to each team. When a stoppage occurs, it causes the number of overs to be reduced; this is a reduction in resources. The amount of resources lost is dependent upon the state of the match at the time of the stoppage (i.e. which team is batting, how many overs remain in their innings prior to the stoppage and how many wickets have been lost).

The D/L method (as described in [1]) sets the revised target by using the percentage of resources available to each team, the resources available to Team 1 are called R1 and similarly for Team 2 they are called R2. The values of R1 and R2 are obtained either from the software or from the D/L method table. When using the table, the resources are determined by the following procedure:

1. Note the resource percentage, from the table, corresponding to the number of overs left and the wickets lost. Call this Ra.
2. Note the resource percentage, from the table, corresponding to the number of overs remaining after the stoppage and the same number of wickets lost. Call this Rb.
3. The difference Ra – Rb gives the resource percentage lost. Call this Rc.
4. The resource available to Team A (R1) is then the resource percentage available to Team A at the start of their innings less Rc (e.g. R1 = 100 – Rc, in the case where the first stoppage occurs during Team 1’s innings)

In the case of multiple stoppages, the resource percentage lost due to a stoppage is subtracted from the resources available prior to that stoppage to get the new value of resources available to the current batting team.

These are then used in the following formula:

(See Note 2)
Standard Edition D/L Target Formula

where T is the target for Team 2 (rounded down to an integer), S is the number of runs scored by Team 1 and G50 is the agreed average total score in a 50-over innings (currently equal to 235).

With reference to the objectives defined in section 3 of this paper, the D/L method satisfies, to some extent, almost all of these objectives. Duckworth and Lewis ([2], [3]) identify that the objective of simplicity is of paramount importance to ensuring the acceptance of any method by the cricket-going public. Thus, Duckworth and Lewis appear to prefer that this objective must be met first and the Standard Edition meets this criterion. The method certainly satisfies the objective of adaptability and repeatability. The method does have one "constant" in it, namely G50, which is the agreed average score for that level of play. With regards to the remaining three objectives, namely Fairness, Maintenance of Probability of Winning and Maintenance of Excitement, this method is often criticised. Many spectators and commentators believe that the method favours the team batting at the time of the stoppage. This may be true, but difficult to prove without substantial evidence. To the outside observer though, it does not appear that the potential use of the D/L method alters the playing strategies of opposing teams (i.e. choice of batting or bowling first, made at the toss).

Duckworth and Lewis [2] identify that the "Probability of Winning" is difficult to define and determine, but suggest that using "degree of difficulty" in place of probability is reasonable. This allows the D/L method definition of resources to be used to maintain the probability of winning across a stoppage. The main difficulty associated with this objective still comes from stoppages during Team 1’s innings, since the "probability" of winning is even harder to find.

The Maintenance of Excitement objective, whilst important in this increasingly commercial age, should only be used as a final criterion when evaluating methods. The D/L method’s main aim is to maintain the margin of advantage across a stoppage. This sometimes leads to a game being already won when play could be recommenced, a hardly exciting prospect for the spectators who have waited around in case play could recommence. In some cases, the D/L method has maintained the excitement for the fans by setting an achievable, but difficult target. Thus, this method does, on occasion, satisfy the last objective.

### Proposed Alternative Solution

An alternative to the Duckworth/Lewis method is that proposed by V. Jayadevan in his paper [4] published in Current Science. Jayadevan, a civil engineer, took the novel approach of looking at the scoring patterns of teams playing in closely fought matches. These scoring patterns were then expressed as percentages of innings and runs to create normal (or par) scores. Regression analysis performed on the data produced a cubic equation (Equation 1, where R is cumulative percentage runs and O is the cumulative percentage overs) for the normal scores. This cubic equation resembles the match development much better than the parabola used in the Parab method. A second cubic equation (Equation 2) is found, again using regression, to provide target scores. Figure 1 shows the two curves plotted on the same axis. It should be noted that the normal curve corresponds to the expected number of wickets lost at each stage of the match, whereas the target curve corresponds to nine wickets down, since the match can still be won.

Equation 1: Jayadevan Normal Score Equation

Equation 2: Jayadevan Target Score Equation

Figure 1: Jayadevan method cubic curves

As Figure 1 shows, the "normal" development of an innings is: a quick start followed by a "survival and accrual" period and ending with a "slogging" session. Jayadevan provides details of how these equations were derived and the assumptions underlying the method. Jayadevan also criticises the D/L method for inclusion of the G50 constant. He states, The history should be used only up to the stage of arriving at a suitable model. It should not again be pulled in (as it is done in the D/L system), while the model is applied to the prevailing situation. [4]

Jayadevan states that any stoppage in a match can be classified into one of three mutually exclusive possibilities. The three possibilities are: a stoppage between innings; a stoppage during Team 2’s innings; or a stoppage during Team 1’s innings. Stoppages that occur prior to commencement of play do not affect the application of the method. Multiple stoppages are handled by repeated application of the method.

#### A stoppage between innings:

In the case of a stoppage occurring after the end of Team 1’s innings and prior to Team 2 commencing their innings, Jayadevan’s method is applied as follows:

1. Determine the percentage of overs for Team 2.
2. Look up the corresponding target score percentage in the target table.
3. Multiply this percentage by Team 1’s final score to obtain the new target score for Team 2.

#### A stoppage during Team 2’s innings:

In the case of a stoppage occurring during Team 2’s innings, Jayadevan’s method is applied as follows:

1. Determine the percentage of overs already completed by Team 2.
2. Look up the corresponding normal score percentage in the normal table for the number of wickets fallen.
3. Multiply this percentage by Team 1’s final score to obtain the par score for Team 2 (called PAR1).
4. Determine the percentage of remaining overs to be played at recommencement.
5. Look up the corresponding target score percentage.
6. Multiply this percentage by the result of Team 1’s score minus the par score (PAR1) found in step 3.
7. Add PAR1 to the result of step 6 to find the new target score for Team 2 (net target-1).
8. In the event of another interruption to Team 2’s innings:

9. Using the table, determine N1, the normal score percentage corresponding to the first stoppage with respect to the total number of overs available to Team 2 prior to this stoppage. Then find N2, the normal score percentage corresponding to the current state of play with respect to the total number of overs available to Team 2 prior to this stoppage. PAR2 is then given by:
10. Determine Ta, Tb and Tc. Ta is the target score percentage corresponding to the number of overs to be bowled after the stoppage (C) divided by the number of overs remaining when the first stoppage occurred (A). Tb is the target score percentage corresponding to the number of overs to be bowled before the stoppage (B) divided by the number of overs remaining when the first stoppage occurred (A). Tc is the ratio of Ta and Tb. Subtract PAR2 from net target-1 and multiply the result by Tc.
11. Add PAR2 to the result of step 9 to find net target-2. This is the new target for Team 2.

#### A stoppage during Team 1’s innings:

In the case of a stoppage occurring during Team 1’s innings, Jayadevan’s method is applied as follows:

1. Determine the percentage of overs completed by Team 1.
2. Look up the corresponding normal score percentage in the normal table for the number of wickets fallen.
3. Determine the percentage of remaining overs after the stoppage with respect to the original number of overs remaining.
4. Look up the corresponding target score percentage in the target table.
5. Multiply the target score percentage by the difference between 100% and the normal score percentage.
6. Add this percentage to the normal score percentage obtained in step 2 to get the Effective Normal Score (ENS) in the total percentage of overs played.
7. Look up the target score percentage for the total percentage of overs played.
8. Multiply this target percentage by the ENS from step 6 to get the Multiplication Factor (MF).
9. Multiply the score made by Team 1 with MF to get the target for Team 2.
10. In the event of another interruption to Team 1’s innings:

11. Compute PAR2 as:
12. Compute target percentage for the remaining overs as:
13. Add the result from step 11 to PAR2 to get the new ENS.
14. Look up the target score percentage for the total percentage of overs played.
15. Multiply this target percentage by the new ENS to get the new Multiplication Factor (MF).

Whilst, at first glance, Jayadevan’s method appears complex, it is actually relatively simple to apply and is only slightly more complicated than the D/L method. The use of computer software makes it even easier, but this would not be required to employ the method (c.f. the Professional Edition of D/L).

### Comparison of Solutions

Comparing both of the methods reviewed in this paper, we can construct Table 1, which provides a quick summary of the objectives, as defined in section 3, met by each method.

The true test of any method, though, is in the ability of the method to set fair and reasonable targets in real world (or theoretical) situations. To compare the methods, the examples used are taken from various papers by Jayadevan and Duckworth and Lewis. These examples are listed, in no particular order, and summarised in Table 2.

Example 1:
Team 1 scores 300 in their allotted 50 overs. Team 2 has lost 2 wickets after 25 overs when play is abandoned. What is the target for Team 2 to be declared the winners?

Example 2:
Team 1 scores 300 in their allotted 50 overs. A stoppage then occurs causing Team 2’s innings to be shortened to 25 overs. What is the new target for Team 2?

Example 3:
Team 1 scores 250 in their allotted 50 overs. Team 2 in reply are 75 with no wickets down after 20 overs when a stoppage occurs and their innings is shortened to 30 overs. What is the winning target for Team 2?

Example 4:
Team 1 scores 250 in their allotted 50 overs. Team 2 in reply are 75 with 2 wickets down after 20 overs when a stoppage occurs and their innings is shortened to 30 overs. What is the winning target for Team 2?

Example 5:
Team 1 scores 250 in their allotted 50 overs. Team 2 in reply are 180 with 4 wickets down after 30 overs when a stoppage occurs and play is abandoned. What is the winning target for Team 2?

Example 6:
Team 1 scores 100 in 25 overs when a stoppage occurs, terminating their innings. Team 2’s innings is also shortened to 25 overs. What is the winning target for Team 2?

Example 7:
Team 1 has scored 226 for the loss of 8 wickets in 47.1 overs when a stoppage occurs, terminating their innings. Team 2 have their innings shortened to 33 overs. What is the winning target for Team 2?

Example 8:
Team 1 has scored 176 for the loss of 5 wickets in 36.5 overs when a stoppage occurs, shortening their innings to 46 overs. Team 1 scores a further 5 runs from 6 balls (i.e. 181/5 after 37.5 overs) when a stoppage shortens their innings to 40 overs. Team 1 finish their innings at 193 for the loss of 6 wickets. What is the winning target for Team 2?

The above table illustrates the two methods and the targets produced under various circumstances. In most cases investigated by this author, the two targets produced differ by only a few runs; however, examples 1, 2, 3, 6 and 7 have targets with differences of 10 runs or more. In each of these cases, the target set by the D/L method (Standard Edition) appears either too easy or too difficult in this author’s opinion. Quantifying exactly what would be a reasonable target is difficult and we must give kudos to any persons capable of inventing a method that produces targets that are so widely accepted by the players, umpires and spectators of cricket.

### Conclusion

Whatever the method chosen for resetting the target in a one-day international match, there will always be criticism. The fans will usually claim the method is unfair to their team, regardless of the true state of the match. Commentators will complain that the method is too complicated to understand or that it is too simple and fails to account for one factor or another. The method, whatever it is, should be transparent and open to scrutiny from any interested party. This is one of the main criticisms of the current Duckworth / Lewis method, in that the mathematical details and CODA software are not widely available due to commercial reasons, which leads to some fans believing that the method relies upon black magic rather than logic. Jayadevan method, on the other hand, has detailed the methodology and the mathematics behind the creation of his method, allowing everyone to scrutinise the details. According to Jayadevan, “a large number of players, umpires, cricket administrators, critics and cricket enthusiasts,” have found the method superior to that of Duckworth and Lewis [4].

This paper set out to critically evaluate the current method and Jayadevan’s proposed alternative method for resetting the target in a one-day international cricket match after a stoppage. The results of this evaluation lead to the conclusion that Jayadevan’s method is superior to that of Duckworth and Lewis, even though the method does not meet all of the objectives as defined in section 3. Like the Duckworth / Lewis method before it, this method is the best of what is currently on offer and should therefore be quickly analysed and adopted by the ICC and its member nations as the new rain-rule.

It is interesting to note that the Jayadevan method was presented to the BCCI Technical Committee Conference on 11 July 2000 and to the Umpire’s Seminar in September 2000 as well as at the BCCI meeting on 7 April 2001. The BCCI forwarded the proposal to the ICC, but, as yet, the ICC have not taken up the method.

### References

[1] F. Duckworth, T. Lewis, A Fair Method For Resetting the Target in Interrupted One- day Cricket Matches, MathSport 3, N. de Mestre, Ed., Bond University, Queensland, Australia, (1996), 51-68.

[2] F. Duckworth, T. Lewis, Can the Probability of Winning a One-day Cricket Match be Maintained Across a Stoppage?, MathSport 6, G. Cohen, T. Langtry, Eds., Bond University, Queensland, Australia, (2002), 141-154.

[3] F. Duckworth, T. Lewis, Review of the Application of the Duckworth/Lewis Method of Target Resetting in One-Day Cricket, MathSport 6, G. Cohen, T. Langtry, Eds., Bond University, Queensland, Australia, (2002), 127-140.

[4] V. Jayadevan, Current Science 83, 577-586 (2002).

[5] F. Duckworth, T. Lewis, Developments in the Duckworth-Lewis (D/L) Method of
Target – Resetting in One-Day Cricket Matches
, MathSport 4, N. de Mestre, K. Kumar, Eds., Bond University, Queensland, Australia, (1998), 131-151.

### Notes

Note 1: England vs South Africa, World Cup 1992. This application of the rain-rule has been much debated. Many have questioned why the full 13 deliveries weren’t bowled given that there was time available. If the D/L method were used then the target would have been 28 runs from 1 ball. By comparison, if the Jayadevan method were used then the target would have been 23 runs from 1 ball. Both of these targets are higher than the one set by the rain-rule in place at the time. This extreme case should not be used as the basis for selecting a method, but does highlight other issues that should be looked at with regards to when a rain-rule should be applied. Details of the match can be found on the web at CricInfo or via the URL: