Back of the envelope model to set target for team batting second in an interrupted limited overs match

Wicket Adjusted Run Rate (WARR)

Introduction to Rain Rules
Height Comparison – Part 1
Dealing with No Results
Height Comparison – Part 2
Setting Targets for Team 2
Determining value of a Wicket
UPDATE 1: Answers to Questions Asked Once

Introduction to Rain Rules

On 16th Jan 1982, Pakistan facing West Indies in a must win match to qualify for the finals scored 177 in 50 overs at Wolloongabba, Brisbane. There was a storm. Match was reduced after the rain. This basic arithmetic was the basis of the first par score in limited overs cricket:

177 runs / 50 overs * 30 overs = 3.54 runs per over * 30 overs = 106.2 runs

West Indies were set 107 to win in 30 overs. Sarfraz removed Greenidge, Haynes and Richards followed by Bakht who accounted for Gomes and Lloyd. After 19 overs, Windies were reduced to 61/5. Bacchus was reprieved on 3 who went on for his strokes. Wickets tumbled at the other end and when Mudassar removed Holding and #11 Joel Garner walked in, 2 more runs were still needed to win. The notable fact is that both Imran and Sarfraz had finished their quota. Guess how many overs? Obviously 10 each in a scheduled 50 over match. West Indies won with 7 balls to spare by scoring 107/9 in 28.5 overs out of the allocated 30.

In a 50 over match, 107/9 after 30 would not qualify as a win chasing 178. But this was a 30 over match for West Indies. According to the rules of limited overs cricket, wickets are not proportionately removed when the match is truncated. In subsequent matches, a 30 over game ensured no more than 6 overs per bowler. Pakistan used only 4 bowlers. Remaining nine overs were split 7-2 between Sikander Bakht and Mudassar Nazar. Getting a possible 28 out of the 30 overs between 3 main bowlers was good captaincy indeed. A minor rotational change could have ensured all 30 split between 3 bowlers.

There are some obvious problems in setting the target for team batting second under different conditions. Maintaining a run rate for higher number of overs is more difficult than maintaining it for fewer. And reduction of overs does not mean fewer wickets at disposal. Team batting second can plan the chase taking more risks. Team batting second has an advantage.

Team batting second always has the advantage of knowing in advance how many runs are enough to win. This is not considered to be an unfair advantage. Both sides bat for the same number of overs with identical fielding restrictions for an equal number of overs in each power play. But with reduced overs, we have a game of two halves.  How to arrive at a fairer target?

Ten years later on 1st Mar 1992, India faced hosts Australians in the 12th match of 1992 World Cup at the same ground. Batting first, Australia scored 237 in 50 overs starting slowly. Kapil Dev bowled 2 maidens and kept things tight with Prabhakar at the other end in the opening spell. Indian reply was sedate too. After 16.2 overs India was 45/1 when rain interrupted the match for 15 minutes. 3 overs were reduced. A revised target of 236 runs in 47 overs meant that only 2 runs were reduced for the 18 balls lost. India lost the match by 1 run. The revised rain rule was an improvement to negate the unfair advantage for teams batting second.

The solution, drawn up by experts including Richie Benaud, was that when rain interrupted the second innings of a match the reduction in the target was to be proportionate to the lowest scoring overs of the side batting first, a method that took into account the benefits of chasing, as opposed to setting, a target.

There were signs early on that all was not well. England bowled out Pakistan for 74 at Adelaide. The loss of three hours created a much stiffer target than the Pakistan batsmen had set. For the match to stand, a minimum of 15 overs had to be available to England; but as Pakistan’s most successful 15 overs had yielded 62 of their 74 runs, under the rain rule the minimum target had to be 63. After a further shower it was set at 64 from 16, and England still needed 40 from eight when play was abandoned and the points shared.

A fairer rule was introduced in the next few years. The first ODI decided by the method designed by Frank Duckworth & Tony Lewis was played on 30 Oct 1998 between Sri Lanka and South Africa in Dhaka. South Africa scored 240 in scheduled 39 overs. Due to further delay five more overs were cut from Sri Lankan chase. Duckworth/Lewis method produced a target of 224 from 34 overs. It may be considered confusing but the method is fairer than both the original (average run rate method) and the revised 1992 rain rule drawn up by cricket experts (most productive overs method). The most recent FAQ can be found here. Another method developed by V Jayadevan that offers similar benefits was used only by ICL which operated between 2007 and 2009.

* * *

Height Comparison – Part 1

I had a random number generator made up of two parts. When initiated, Part A, randomly generates a height value between 145 and 190 for a typical adult. Part B spawns any value between 45 and 190 along with a corresponding age value between 0 and 100. If the age value is 18 or more a direct comparison of two values is made to decide a winner between A & B. There is no winner when the age value is under 6. For all the middle values, a function creates a scaled down value of A proportional to age value spawned by B. Winner is decided between derived value of A and actual value of B.

* * *

Dealing with No Results

I never liked three separate lists for the margin of victory in limited overs cricket – by runs, by wickets and by balls remaining. Two of these are natural – by runs for team batting first and by balls remaining for team batting second. The one by wickets also gets listed but that is an offshoot from the longer two innings format. The objective of a limited overs match is to score an extra run within specified deliveries. The innings will always come to close irrespective of 0 or 10 wickets remaining at the end of scheduled overs. Result is determined by the runs scored in maximum allocated overs irrespective of how many batsmen contributed towards the total. The search for a uniform margin of victory took some time but I eventually stumbled on Ball Difference during World Cup 2011. Once discovered, it was fairly straightforward to determine the Ball Difference for any result match. This normalised value of Ball Difference was the basic building block to allocate total number of points to a team which was further distributed into batting, bowling and fielding based on every measurable value in a scorecard. This is a Relative Value model that allows comparison between say a 100 and 4/35 in the same match and also across matches. It took a long time to get the final breakdown of values for various match types but a framework for result matches was in place since early 2013. The intermediate modelled values for last two IPL seasons were published on this blog. Fortunately none of those IPL matches involved a No Result.

We can get a result if Team 2 bats for a specified minimum overs. But there are matches where the first innings is still in progress or second innings has just started. Neither D/L nor VJD has to deal with such matches. But the Relative Value career figures were incomplete if I discarded all performances in such ‘drawn’ matches.

So I had to find a way not only to determine who was ahead in a no-result match but also to quantify the difference. Fortunately this does not involve any assumptions about what may or may not happen. Relative performance is evaluated based on how far the match has progressed along with comparing match data against the par value for the corresponding era.

The shortest non-abandoned ODI was played just 2 days before the rain interrupted World Cup match between India and Australia mentioned earlier. Srikkanth scored a single in the two deliveries that he faced off Ramanayake in match reduced to 20 overs. The model divides 10.417 points for the match placing Sri Lanka marginally ahead of India (5.223 – 5.194). Kenya leads 37.8 – 26.2 in a 50 over match that was abandoned after Ireland scored 18/2 in 8 overs.

* * *

Height Comparison – Part 2

I modified that random number generator. Most of the functionality remains same except the middle values. Now a function estimates adult height of B. Winner is decided between actual value of A and derived value of B.

* * *

Setting Targets for Team 2

How to set a fair target for team batting second with reduced number of overs? Remember that first truncated ODI between Pakistan and Windies? Once the rain stopped, it was decided that a maximum 30 overs were possible and a revised target was set for 30 overs. Then and there the rules of engagement between Team 1 and Team 2 changed. Pakistan batted for 50 overs protecting 10 wickets but West Indies had to protect its 10 wickets for 30 overs only.  In that match 2 bowlers managed to bowl their full quota of 10 overs and a third one managed to bowl 7. Current regulations prohibit this. Essentially two different matches, each only one inning long, are played in an interrupted match.

Should we estimate the height of an adult when he was say 7 years old or is it better to determine how tall a 7 year old is likely to be as an adult? What if West Indies had to play towards a target of 178 in 50 overs with the knowledge that only 30 overs are possible? This means that the target for the loss of 0 through 4 wickets is still 107 but it increases to 122/5, 139/6, 153/7, 163/8 or 172/9. In order to win the match by wickets, West Indies had to score the 50 over target in less than 30 overs and in that case the match would be deemed complete as a heavy victory saving more than 120 balls. To tie the match the tenth wicket falls with scores level.

By that logic, India would get a target of 224 in 47 overs for the 1992 WC match against Australia as long as it lost no more than 8 wickets else it must score 230/9. To win the match outright with wickets in hand, the target of 238 must be achieved in less than 47 overs. For a Tie, the tenth wicket must fall at 237.

What happens when the game is interrupted during the first innings? If there is time left to complete at least part of the first innings, then Team 1 starts where it left off. No adjustments to be made. Irrespective of number of interruptions, an equivalent 50 over score should be determined based on actual runs scored, wickets lost and overs played at the end of that innings. Target for Team2 should be devised using the 50 over equivalent of Team 1.

The rules for fielding restrictions/power plays will remain the same for both sides as determined before the start of the match. If the game is reduced to say 40 overs a side at the outset then all targets should be estimated using 40 overs and not 50. Needless to add that if both sides get to bat 40 overs then it is a normally completed game with an equal number of reduced overs played by both sides.

What about the number of overs per bowler? On 6th Dec 1991, India and West Indies played a low scoring tie at Perth. Defending only 126, India used its 4 main bowlers for the first 40 overs expecting the match to get over one way or the other. West Indies reached 121/9 in 40 overs so Tendulkar, the change bowler, came in for the 41st. This means that in an uninterrupted match, a captain is free to complete the full quota of any bowler as early as the team warrants. Miandad who captained that match against West Indies could have used just 3 bowlers in the 30 over affair. Team 2 has the advantage of scoring fewer runs at the same rate (as long as it does not lose too many wickets) so the Team 1 captain should have the option of using the services of main bowlers only.

Let us take a simple run-a-ball example for this ‘back-of-envelope’ model. Team 1 scores 299 in 50 overs (actual or projected) setting a target of 300 in 50 overs. If the match is reduced to 20 overs, we will not try to offset the relative advantage in power plays here, the target is 120 for the loss of 0, 1 or 2 wickets else it will increase to 131/3, 171/4, 206/5, 234/6, 257/7, 275/8 or 290/9. What happens if Team 1 reaches 120/0 in 15 overs? The match will continue till entire 20 overs are bowled because a win by wickets to stop the match earlier requires scoring 300 runs. Team 2 will try to take wickets right up to the 120th delivery, quietly confident that taking 4 wickets in remaining 5 overs will increase the target to 171 which means that the match is wide open.

* * *

Determining value of a Wicket

In 1999 World Cup final, Pakistan batted first scoring 132. Australia reached 129/2 after 19.5 overs overhauling the target with ease in just over 20 overs. What if the match had to stop due to unforeseen reasons after 19.5 overs? Do we know how to determine a winner in an interrupted match where team batting second did not bat for at least 40% of the scheduled overs? If you believe that logic prevails in the World Cup final, here is a reminder from the 2007 final between Australia and Sri Lanka:

The World Cup, the final of which began in spectacular fashion before descending into the unseemly realms of the bizarre, was awarded eventually to Australia in such farcical circumstances that it would have been no surprise to see Steve Bucknor drop his trousers to reveal polka dot underpants and inquire if there was anyone for tennis.

After more than seven weeks of cricket, the game’s showpiece, contested by indisputably the two best one-day sides and witnessed by 28,000 in the Kensington Oval and millions more around the world, was decided first by the weather and then by Messrs Duckworth and Lewis.

By then all semblance of cricket’s dignity had been lost, with the umpires, Bucknor and Aleem Dar, together with the off-field official, Rudi Koertzen, and the match referee, Jeff Crowe, making such a fundamentally blundering interpretation of the rules regarding interrupted matches that it is hard to see how Crowe, a good, intelligent, thoughtful man in overall control of the game, can survive in his post. It was sad almost beyond embarrassment.

The last rites were simply bemusing. With the cruise ships in Deep Water harbour long since lit up, a white Al Jolson mouth – Andrew Symonds presumably – bowled the last ball of the tournament gently to an unidentifiable Sri Lankan batsman, who may or may not have made contact. Few in the ground without the aid of night vision glasses would have been able to tell. It sparked manic Australian celebrations, the second such.

We may be drifting so let us talk about determining the state of a no-result match from the scorecard. If the second innings is in progress but not far enough to get a result under prevailing ‘rain rule’, we have to account for three variables: a) actual number of overs bowled out of scheduled maximum, b) runs scored relative to target & c) number of wickets that have fallen.

In the example above, Australia was just 4 runs short of 133, the target score. In other words, that match can be deemed to be about 99% finished despite the fact that more than 30 overs remained. We can also imagine a scenario where a team has lost 9 wickets under similar circumstances.

I reviewed the fall of wicket information for all the 3000+ ODIs to determine the state of any no-result match on the basis of wickets lost. It is clear that fall of 10th wicket represents 100% completion of innings irrespective of number of runs required and balls remaining. For all other wickets, a percentage value was determined empirically. For example, loss of 5 wickets indicates an innings completion factor of around 70% because top order will score bulk of the runs required.

Going back to the simple run-a-ball example – how do we determine the number of wickets that a team may lose without affecting its original required run rate? If the revised match lasts only 20 overs, in other words if team bats for only 40% of total scheduled overs, we analyse the wickets percentage data. Loss of 2 wickets is less than 40% but loss of 3 wickets is more than 40%. This means that we have to determine the number of runs required in exactly 40% of scheduled overs which is 120 runs. So if the team score is 120/0, 120/1 or 120/2 we can say that overs used is more than the wickets lost. But if the 3rd wicket falls before 20th over, the innings completion factor is higher than 40%. In that case, we increase the number of runs that need to be scored to stay ahead of Team 1. That is how we arrive at 131/3, 171/4, 206/5, 234/6, 257/7, 275/8 or 290/9 as the target score after 20 overs for Team 2. 

UPDATE 1: Four questions/suggestions Answered

I should have started a new post to incorporate suggestions/queries but decided to retain all the related information at one place.

Table below is about the first suggestion requesting rounded ‘over or half-over wicket values’. I have followed the rule barring 9th wicket for T20 which is equated to 19.2 overs (instead of 19.0 or 19.3 overs)

Wickets Lost ODI T20
Overs %age Overs %age
1 7 14% 3 15.0%
2 14.3 29% 6.3 32.5%
3 22 44% 9 45.0%
4 28 56% 12 60.0%
5 34 68% 14 70.0%
6 39 78% 16 80.0%
7 42.3 85% 17 85.0%
8 46 92% 18.3 92.5%
9 48.3 97% 19.2 96.7%
10 50 100% 20 100%

Assume that Team 2 gets 20 overs (or 40%) to overhaul the target in an ODI. From the above table we can see that losing wickets 1 and 2 is below that limit. Hence the initial target is set to 40% when achieved by losing no more than 2 wickets. But if the 3rd wicket is lost, then the Target score is increased to 44%. 56% if 4th wicket is lost and so on.

In a T20 match reduced to just 5 overs, Team 2 can lose only 1 wicket to chase initial 25% target. The target will go up to 32.5% if second wicket falls.

The second suggestion was to give it a name. The match between Pakistan and West Indies was won by West Indies according to the Average Run Rate (ARR) rule. This method retains most of the things from that match except increasing the target for losing extra wickets – hence ‘Wicket Adjusted Run Rate’ or WARR is the designated name.

Now let us look at examples from T20 Internationals where Team 2 batted for fewer than the 20 overs played by Team 1.

MtID Result Summary Match Target WARR Target
I0011 Sri Lanka won by 18 runs (D/L method) Nzl 162/8(20); SLK 62/1(5.5) 45 48 for 0 to 1 wkts, 53/2, 74/3, 98/4, 115/5, 131/6, 139/7, 151/8, 158/9 in 5.5ov
I0111 West Indies won by 5 wickets (D/L method) Eng 161/6(20); WIN 82/5(8.2) 80 73 for 0 to 3 wkts, 98/4, 114/5, 130/6, 138/7, 150/8, 157/9 in 9ov
I0124 England won by 1 run (D/L method) ENG 202/6(20); Saf 127/3(13) 129 132 for 0 to 4 wkts, 143/5, 163/6, 173/7, 188/8, 197/9 in 13ov
I0157 Sri Lanka won by 14 runs (D/L method) SLK 173/7(20); Zim 29/1(5) 44 44 for 0 to 1 wkts, 57/2, 79/3, 105/4, 122/5, 140/6, 148/7, 161/8, 169/9 in 5ov
I0158 West Indies won by 8 wickets (D/L method) Eng 191/5(20); WIN 60/2(5.5) 60 58 for 0 to 1 wkts, 63/2, 87/3, 116/4, 135/5, 154/6, 164/7, 178/8, 186/9 in 6ov
I0159 New Zealand won by 7 runs (D/L method) Zim 84(15.1); NZL 36/1(8.1) 30 35 for 0 to 2 wkts, 39/3, 51/4, 60/5, 68/6, 73/7, 79/8, 83/9 in 8.1ov
I0242 South Africa won by 11 runs (D/L method) SAF 219/4(20); Ind 71/0(7.5) 83 87 for 0 to 2 wkts, 99/3, 132/4, 154/5, 176/6, 187/7, 204/8, 213/9 in 7.5ov
I0270 Australia won by 17 runs (D/L method) Win 191/8(20); AUS 100/1(9.1) 84 88 for 0 to 3 wkts, 116/4, 135/5, 154/6, 164/7, 178/8, 186/9 in 9.1ov
I0300 Sri Lanka won by 2 runs (D/L method) SLK 161/4(20); Aus 119/3(15) 122 122 for 0 to 5 wkts, 130/6, 138/7, 150/8, 157/9 in 15ov
I0340 South Africa won by 4 runs (D/L method) SAF 153/7(20); Pak 60/2(9.1) 65 71 for 0 to 3 wkts, 93/4, 108/5, 124/6, 131/7, 143/8, 149/9 in 9.1ov
I0373 Ireland won by 21 runs (D/L method) Uae 123/6(20); IRE 103/3(14.2) 83 89 for 0 to 5 wkts, 100/6, 106/7, 115/8, 120/9 in 14.2ov
I0380 New Zealand won by 9 runs (D/L method) Eng 172/6(20); NZL 52/1(5.2) 44 47 for 0 to 1 wkts, 57/2, 78/3, 104/4, 122/5, 139/6, 148/7, 161/8, 168/9 in 5.2ov
I0398 Sri Lanka won by 27 runs (D/L method) SLK 160/6(20); Win 80/4(13.5) 108 112 for 0 to 4 wkts, 113/5, 129/6, 137/7, 149/8, 156/9 in 13.5ov

Paul Collingwood was the captain of England for T20I#111 played against West Indies on 15th Jun 2009. Batting first, England set a target of 162 that can be deemed as par. But the chase was reduced to 9 overs. An extract from the match report:

England’s stop-start progress in the tournament came to a permanent halt after the Duckworth/Lewis method seemed to weight the game in West Indies’ favour. Rain in the interval caused the reduction of their task from 162 in 20 overs to just 80 in nine. West Indies slipped to 45 for five against feisty bowling, but then Sarwan teamed up with Chanderpaul, and the task of taking 35 from 22 balls proved simple for the tournament’s most experienced sixth-wicket pair.

The WARR Target for this match after fall of 5th wicket would have risen to 114 which means 69 were needed from 22 balls not 35. Of course such a comparison does not make sense because the match was played under different rules. If rules were different then West Indies may have promoted experienced Chanderpaul and Sarwan to reach 73 for less than 3 wickets whereas England would have used its best bowlers to take the 4th/5th wicket.

About 1 year later two matches were decided using D/L method on the same day. England lost once again to West Indies under Collingwood – Eng 191/5(20); WIN 60/2(5.5) – in one of the two games. An extract from The Guardian published next day on 4th May:

Frank Duckworth, one half of the duo of statisticians behind the Duckworth-Lewis method, was forced to defend his system for the second time in the space of a year today, after England once again lost to West Indies in a rain-affected Twenty20 match.

Duckworth said he, his partner Tony Lewis and the ICC were all happy with the use of D/L in Twenty20 matches, despite the criticisms made after yesterday’s match by both Paul Collingwood and the opposing captain Chris Gayle. “There’s a major problem with Duckworth-Lewis in this form of the game,” Collingwood said after the defeat, a view that Gayle supported.

The real question, though, is not with the D/L formula but the fact that the minimum number of overs that constitute a Twenty20 match is as low as five. “If people think that last night’s match was unfair, the question should be asked about the minimum overs,” said Duckworth, who was quick to add that he did not necessarily think that five overs was too few himself. “Rather than look at the D/L method which has to take account of the fact that West Indies were leading at the time the rain came, one might well ask whether five overs is sufficient for a valid match.”

In a one-day international 20 overs are required for a valid match, 40% of the innings total. In Twenty20 that would translate into eight overs, rather than the current five. It is this area that the ICC may have to address if it wants to avoid more scenarios like last night’s. If they do not want to extend the minimum number of overs required for a match, another option would be to settle the game by giving each side a super over.

So here is an attempt to deal with such very short matches within current rules of cricket. Read this and this for sophisticated alternatives presented by Dr. Srinivas Bhogle who is well versed with both D/L and VJD. The third suggestion was to retain the simplicity of this method where targets are not further adjusted for field restrictions etc. That bodes well with me – I do not need any additional paper besides the proverbial envelope used earlier.

The match was played under different rules but we can consider the scenario under WARR. The initial 6 over target  was 58/0 (or 58/1). West indies were 30/0 in 2.2 overs. England picked up two wickets after another rain break reducing Windies to 42/2 in 3.3 overs increasing the target to 63. At that stage, England would have pressed on to take the crucial 3rd wicket to for a significantly challenging 87 – a figure that was a fairer reward for scoring 191 in 20 overs.

The fourth and final recommendation was to add a visual to this post. I will present two charts. The first one is for a match where run-a-ball 300 were required in 50 overs. It is an opportunity to chart a successful chase using ball-by-ball data.

Sri Lanka calmly chase a big total

Sri Lanka calmly chase a big total

Two flat lines indicate the target of 120 and 300 for 20 and 50 overs. A thin dark line labelled ‘Target’ linearly moves from 0-300 except for a short ‘jump and step’ when Sri Lanka lost an early wicket. None of the remaining 4 wickets were lost ‘before due’ so the Target Curve remains linear. ‘Runs Scored’ is a colour coded line depicting which team is ahead based on the Target score for that delivery. A counter is started after the minimum 20 overs are completed. Blue and Red dots are counted for the remaining 180 balls, As we can see South Africa was marginally ahead between 27 to 40 overs but Sri Lanka did not lose too many wickets during the slog staying close to or above the target in the final ten overs.

Next we have a match where Sri Lanka was ahead of the target value till the last few balls but were trumped by a calculated chase featuring Dhoni.

Dhoni to the rescue

Dhoni to the rescue

Both sides nearly scored 300 in the earlier match where wickets were not easy to get. Here each team struggled to face 300 balls to score just over 200 in the final of a tri-series involving host West Indies. Sri Lanka checked Indian progress through 2 early wickets and a good economy. Partnership for 3rd wicket kept India close to the Target score while the 4th wicket saw them surge ahead after 30 overs. Through regular breakthroughs, which can be visualised through ‘the steps’ in target curve, Sri Lanka surged ahead. Dhoni remained calm despite falling significantly behind the curve with 6 balls remaining but won the match by virtue of a 6, 4 & a 6!

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