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Reinventing the DLS method: Part 3b

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Roar Guru
9th December, 2020

Continuing on from where I left off in Part 3a, I will now complete Part 3 with a simple example of the team batting first having full access to its maximum 50 overs with the chasing side’s innings reduced to 45 overs due to a delay between innings.

Let’s imagine the first side scores 299. The written equation the umpire presents to the two captains, as outlined earlier in Part 3a, is as follows:

Par Target in Runs: 270 VBO: 9 Par Winning Score: 8 for 270 (off 45 overs)

Then the two captains get out their respective handbooks with the complete tables as illustrated earlier in Table 3 (from Part 3a), and together with their respective teams quickly note down the above and below par scores as follows:

Above Par
0 for 252 1 for 254 2 for 256 3 for 258 4 for 260 5 for 262 6 for 264 and 7 for 267.

Below Par
9 for 295

The question to be answered is whether or not that sole below par score is unfair to the chasing side. For example, if the chasing side finished 9-293 off their 45 overs, would it not be reasonable to back their last pair at the crease to scramble seven runs with five overs in which to do it? At face value, maybe.

I could just as easily argue that having spent nearly 40 years watching limited-overs matches on television – both international and domestic – as well as having done 25 and a bit seasons of umpiring warehouse cricket in Brisbane (both summer and winter) at every level, as well as a sporadic amount of QCA in Toowoomba, Bundaberg and Beaudesert thrown in just for good measure, that I do not need to consult any sort of statistics to be able to stake everything I have (as a non-gambler!) on the fact that the number of one-wicket victories being completed in the final five overs of a run chase have been far outnumbered by either occasional ties, or far more commonly, victories to the defending team by small single-digit-run margins – almost always with one of the last pair being dismissed, even when the match reaches the final over.

Far more often than not, a chasing team’s last pair still together in the final overs will crack under the burden of the stifling pressure that has built to that point. However, this is not the sole issue – in fact there is a considerably bigger issue at play.

Josh Hazlewood and Pat Cummins

(Photo by SAEED KHAN/AFP via Getty Images)

Allow me to present the mathematical concept of the virtual (batting) line-up. It works like: in this reduced-overs situation, the first team had 11 men to bat over 50 overs, and therefore had a maximum of ten wickets or partnerships to spread across those 50 overs in order to maximise their final total. When it comes time for the chasing side to bat for their reduced amount of 45 overs, we set their VBO at nine in order that they only get to utilise the same WAF of a wicket every 30 balls, hence the par score after 45 overs of 8-270.

However, we also allow the 11th man in their line-up to bat. But the moment the chasing side’s ninth wicket falls and their 11th player goes into bat within those 45 overs that their innings has been reduced to, it is the mathematical equivalent to allowing a team to bat 12 players over a full 50 overs, and therefore granting them an extra 11th wicket to lose – 11.11 recurring to be mathematically precise, with the virtual equivalent of 12.12 players being allowed to bat.

Obviously, it is physically impossible to play fractions of people or lose fractions of wickets. Batting 12 actual players (with 11 wickets to lose) over 50 overs, chasing 300, would actually shift the par score off 45 overs from the aforementioned 8-270 to 9-273 off 45.3 overs because the WAF over the entire innings would reduce from 30 to 27.27, as would the required run rate of runs per partnership (though the required run rate per over would remain the same).

However, our purpose is for mathematical accuracy (rather than physical/physiological reality) in the context of this example reduced-overs target match situation, hence, the virtual line-up of 12.12 players equating to a virtual 11.11 wickets up the chasing team’s sleeve.

Consider Table 5 below. A theoretical extra batsman – or fraction thereof – amounts to a lower WAF and therefore increased scope to further maximise a team’s final total off the overs they have at their disposal.

Please do not misinterpret the purpose of this table. It is not saying that if the side batting first scores 299 off 50 overs that the chasing team has to score 301 to win should they happen to face 49.5. It is saying that the virtual extra 1.11 (recurring) wicket afforded to a team that gets to have ten partnerships across 45 overs in comparison to ten across 50 can be mathematically applied in a discounted manner all the way down to 49.5 overs. Obviously, if you are chasing 300 to win off 50 overs, and you reach that 300 off 49.2, 49.3, 49.4 or 49.5 then you win the game hands down. Also, a chasing innings will never be reduced to a partial over figure before it begins, or is interrupted along the way (and then resumes), unless the minimum overs required is set at half the maximum overs, then a 35-over match, for example, could have a set minimum of 17.3 in the event.

While a 50-over target could be reduced beforehand to 49, an innings could also be prematurely terminated by rain or some other cause at 49.1 overs, in which case, according to the mathematical calculations in the table, with nine wickets down, 299 would not be enough to be awarded the game in a run chase of 300 off a maximum overs of 50.


Table 5
(The bracketed numbers in the middle column represent the runs per partnership in the imaginary 12.12 line-up with 11.11 wickets to lose and the runs per over)

Reduced Overs (from 50) Discounted Virtual Partnerships Real Score Virtual Target with Virtual Partnerships Virtual RPO Needed in Virtual Partnerships Context
45 11.11 295 (29.5) (6.6) 328 6.6
46 10.87 296 (29.6) (6.4) 322 6.5
47 10.64 297 (29.7) (6.3) 316 6.3
48 10.42 298 (29.7) (6.2) 310 6
49 10.2 299 (29.8) (6.1) 305 6 (6 runs/6 balls)
49.1 10.17 299 (29.9) (6.08) 304 6 (5 runs/5 balls)
49.2 10.14 300 (30) (6.08) 304 6 (4 runs/4 balls)
49.3 10.1 300 (30) (6.06) 303 6 (3 runs/3 balls)
49.4 10.07 300 (30) (6.04) 302 6 (2 runs/2balls)
49.5 10.03 300 (30) (6.02) 301 6 (1 run/1ball)
50 10 300 (30) (6.00) 300 n/a

To further highlight the mathematical logic in Table 5, Table 6 below represents a revised par diagonal of equilibrium with a 50-over target of 330 using an imaginary line-up of 12 batsmen, and therefore 11 wickets to fall. Assume all else as equal, for example, the opposition have only the usual 11 players in the field. I have used 330 instead of the 328 in Table 5 for two reasons. Firstly, for simplicity I wanted to keep it in whole multiples of 11, and secondly, the point of last par wicket, now nine with an extra 11th once the tenth falls, now shifts to 45.3 overs because the WAF across the 50 overs is reduced to 27 balls, down from 30, on account of that imaginary extra wicket afforded.

The decimal fraction at that nine wickets down par point of 45.3 overs is 1.1 (precisely) as opposed to 1.1111 (recurring) at the eight wickets down 45-over point in a standard 11-batsman line-up with only the usual ten wickets to utilise. Being 9-295 after 45.3 overs projects to a 50-over score of 325, which is three less runs compared to Table 5 but also off three less balls.

With eight wickets now being a wicket above par, the decimal fraction at 45.3 overs with three wickets standing is 1.1074. Multiply this by 270 (par score with eight wickets at 45 overs in previous Table 2) and we get a 50-over projection of 299, which is only one run less than in the proper Table 2 shown earlier, but off three less deliveries, as eight down is now, as stated in the previous sentence, one wicket above par rather than par itself.

Table 6

50max 11 10 9 8 7 6 5 4 3 2 1
18.3 73 89 105 120 145 170 195 220 245 270 295
23 101 114 126 138 150 175 200 225 250 275 300
27.3 132 142 152 162 171 180 205 230 255 280 305
32 165 173 181 189 196 203 210 235 260 285 310
36.3 200 206 212 218 224 230 235 240 265 290 315
41 237 242 246 250 254 258 262 266 270 295 320
45.3 277 280 283 286 289 292 294 296 298 300 325
50 330 330 330 330 330 330 330 330 330 330 330

Table 7, which is a remake of Table 4 in the previous Part 3a in this series, shows the steadily decreasing WAF as well as runs per partnership requirement to fit the previously outlined imaginary line-up of 12 batsmen with 11 wickets to lose, with all else being equal, i.e. still only 11 opposition fielders and still the maximum overs allowed for any bowler of ten.


Table 7

Overs Bowled Wickets Standing Progressive WAF Runs Still to Get: Target 330 Runs Per P’ship Needed: Target 330
0 All (Virtual Line up 12) 27.3 330 30
4.4 11 24.7 300 27.3
9.2 10 24.4 270 27
14 9 24 240 26.7
18.3 8 23.6 210 26.3
23 7 23.1 180 25.7
27.3 6 22.5 150 25
32 5 21.6 120 24
36.3 4 20.3 90 22.5
41 3 18 60 20
45.3 2 13.5 30 15
50 1 n/a 0 Target Achieved on Par = 10 for 330 (50 overs)

To finish off this Part 3b in this series, let’s have a look at two games from the 1992 World Cup, including the infamous England versus South Africa semi-final. What would McWarehouse do in each situation, and what would the Duckworth-Lewis system have done?

That semi-final
England batted first and reached 6-252 off 45 overs. The thing to be clear about here is that it did not start out as a 45-over game, but rather the standard 50 maximum overs. South Africa engaged in time wasting, which was to England’s detriment, as at the point of compulsory closure, time wise, they were still three wickets in hand of VBO in the hypothetical case of an innings reduced to 45 overs.

The first thing McWarehouse does is project England’s 6-252 off 45 into a 50-over total using the decimal figure in the tables that are based entirely around the par diagonal of equilibrium. That decimal figure in the row for 45 overs in the 50 max charts and in the column for four wickets standing is 1.1364. Multiplied by 252 gives 286, which gives England a virtual run benefit (VRB) of six runs. More wickets in hand with more overs remaining would obviously give a higher VRB proportionately with whatever total they had on the board. A score of 1-200 off 40 in a 50-over match, were the innings terminated at that point, would give a VRB of 24, i.e. the 50-over total would be projected to 274 rather than 250. In the case of this 1992 semi-final, the projected total of 286 is instead merely 280.

Then the umpire follows the earlier outlined procedure of presenting the equation to the captains of both teams. The decimal figure in the bold printed (par) two wickets standing column and the 45-over row is 1.1111 and dividing this into the 287 gives us 258. Therefore:

(Guide) Target: 258 VBO: 9 Par Target: 8 for 258 (off 45 overs).

Above par targets, the prerogative of either captain to calculate for their own potential benefit, would be:


0 for 241 1 for 243 2 for 245 3 for 247 4 for 249 5 for 251 6 for 253 and 7 for 255

The sole below par target would be 9-282 (remember the virtual extra 1.11 recurring wickets benefit from Tables 5 and 6).

So, when the rain came the equation would have been 27 off 13 balls with no more than two more wickets lost in those 13 balls. If the current pair were able to remain together, then the 22 the team required at the actual time of events transpiring would have sufficed. Lose just one wicket and 24 are needed. Lose three wickets in those 13 balls, then off the other ten balls they would have a virtually impossible task of scoring 51 more runs.

These would have been the equations at the point of the decision being taken to cut one over off the innings: the VBO would be reduced to eight, so the new par target would be 7-250, i.e. 19 runs off seven balls losing no more than one wicket. The current pair remaining together would need to add another 16 off those seven balls. Lose two wickets in those seven balls then another 26 runs are needed, while three wickets falling in those seven balls would require 50 from the other four balls remaining.

Had they not got on, then the required score for six wickets down – which South Africa were at that 42.5-over point – would be 6-241, which would have seen them ten runs adrift. When they got back on, the South Africans would have needed 11 off that final solitary ball to be bowled.

So, they would have had a shot if England had delivered a no-ball. However, both the no-ball and the extra delivery would have needed to be hit for six, as this was still in the days when the batting side forfeited the one-run penalty for a no-ball that was scored off, while retaining the extra delivery, as well as not being able to be out bowled, caught, LBW, stumped or hit wicket.

That peculiar law changed for wides in the mid-1980s if I am not mistaken, so a wide that the keeper missed and consequently went to the boundary would have left the chasing team needing to score a maximum off the extra final delivery.

Cricket generic

(Mark Kolbe/Getty Images)


The original standard Duckworth-Lewis, first introduced a few short years later, would have required a blanket target of 275 encompassing the whole line-up off the original 45 overs. As noted in the previous Section 2 of this series, blanket targets in reduced-overs situations are mathematically illogical, unattainable from the perspective of any sense of cricketing justice and completely untenable. It should always go on wickets, irrespective of the specific type of interruption that occurs on any particular day.

Come the interruption at 42.5 overs, and the initial reduction by one over to 44, and the standard D\L target becomes 266 (35 needed off seven balls). That further over reduced it to 256, so 25 off one ball, rather than 22. The required score for six down had they not got back on at all would have been 253, which is what they were chasing off 45 overs without any adjusted 50-over projection for England’s innings beforehand. If the interruption had originally come one ball later at 43 overs and they hadn’t got back on, the chasing team would have needed to be 254 at that point with six wickets down at the time.

That semi-final was actually merely the tip of the iceberg for the moronic highest scoring overs method. There were about half a dozen occasions earlier in the tournament where the warning signs were well and truly there that something was simply not right. One of these was the Australia versus India game in Brisbane, a match that I actually attended as a 19-year-old with my then-16-year-old brother.

Australia batted first and totalled 237 off their completed 50 overs. A brief light shower after 17 overs with India 1-45 saw the players leave the field for the equivalent time of three overs, so India’s innings was reduced by three overs. Unfortunately, they had bowled a maiden as well as two other overs of just a run each, so Australia’s highest scoring overs totalled 235 plus one more to win for a target of 236, just one solitary run less than Australia had scored in three more overs. The original standard Duckworth-Lewis would have set a blanket target of 228, without any obvious mathematical principle to justify it.

McWarehouse on the other hand would have done this:

(Guide) Target: 217 VBO: 9 Par Score (after 47 overs): 8 for 217.

Above par scores would be as follows:

1 for 214 2 for 214 3 for 215 4 for 215 5 for 216 6 for 216 and 7 for 217

The sole below-par score would be 9-236 the same total as India were required to score on the day even had they not lost any subsequent wickets after the interruption (when they were only one down).

It’s interesting to note that India reached the final over to be bowled, the 47th, on 7-223, requiring 13 off that last over. Part-time medium pacer Tom Moody bowled it and Indian keeper Kiran More flicked each of the first two deliveries through midwicket for four.

Attempting to repeat the shot a third time in succession, he was bowled, leaving five to get off three deliveries, with two wickets remaining. New batsman, number ten Manoj Prabhakar, scored a single off the fourth. The equation was then four off two.

The same batsman was then run out at the non-striker’s end off the following fifth delivery, setting off for a run and being sent back by the striker, number nine Javagal Srinath. So a boundary was needed off the last delivery and the rest is history.

Srinath swung it high into the air where he was dropped on the boundary by Steve Waugh, who kept his cool and picked up the ball and fired it in to keeper David Boon, filling in for the injured Ian Healy, who ran out the number 11 Venkatapathy Raju, going for the third run. So India finished two short of the extremely dodgy blanket target to give Australia victory.

With India six runs ahead and one wicket in hand of the par McWarehouse score of 8-217 going into that final over, playing out a maiden, or even a wicket maiden would still have seen India victorious. Losing two wickets would have seen the last pair required to reach that same score of 236, so there is an outside chance that the final over could still have played out in exactly the same way it did, although there would have been absolutely no need whatsoever for More to take the genuinely considerable risks that he did on the first three deliveries of that final over. At the end of the day, India were far more shafted that day by the moronic system in place than South Africa actually were in the semi-final when closer inspection is undertaken.

The posthumous McWarehouse calculations for the Indian run chase that day actually depart from the handbooks with the decimal fractions based on the par diagonal of equilibrium because the situation is a slightly different one. When a team sets out chasing the original target from the maximum overs, and are then interrupted along the way, consequently having their innings reduced, then we need to ensure that the equation remains as similar as possible after the interruption as beforehand in respect of the required run rate per over together with the chasing side’s WAF (wicket affordability factor).

How McWarehouse approaches this task will be the main topic of Part 4. DLS claims to deal with it in a mathematically sound manner, but they don’t do so very well, and the method behind the madness in the ridiculous blanket targets they come up with is anything but transparent. Those less than sound blanket targets also do not lend themselves to any genuine semblance of cricketing justice.

Had India’s run chase been reduced to 47 overs before it began, the equation would have been this, using the standard McWarehouse calculation tables:

(Guide) Target: 223 VBO: 9 Par Score (after 47 overs): 8 for 223.

This would have also required India to merely play out a maiden for victory losing no more than one wicket, going into the final over on the aforementioned 7-223.

The sole below par score would have been the same at 9 for 236, while the above par scores would have been as follows:

0 for 214 1 for 215 2 for 216 3 for 217 4 for 218 5 for 219 6 for 220 7 for 222

D\L would have set a blanket target of 231, only five runs less than the highest scoring overs method in place actually set the unfortunate Indians that day.