# The rain equations

A look at how the Duckworth-Lewis and VJD methods work in different situations

June 8, 2012

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 Which method will work best here? © Associated Press Enlarge

Over the last week, the rain rule used in international cricket has come in for plenty of scrutiny after the Duckworth-Lewis method was recommended by ICC's Cricket Committee as the one that should be used for all international games. The move prompted intense protest from V Jayadevan, an engineer from India, who has devised an alternative method, which was also evaluated by the committee. Jayadevan has claimed he didn't get a fair hearing, while the ICC has refused to get into specifics, only stating: "The committee unanimously agreed that there was no evidence of any significant flaws in the D/L method nor did the committee believe that any improvements could be offered by the VJD method."

Is one of the methods superior to the other? That's difficult to answer without knowing all the facts, but it's also true that there are differences between the two that come to the fore, especially in extreme cases. A study of a few scenarios also brings out a mathematical anomaly with the Duckworth-Lewis method. (Some of these examples have been provided by Jayadevan, but the results have been independently verified.)

Scenario 1: The team batting first in an ODI scores 50 (or 100) in 20 overs when its innings is cut short, and the second team gets 20 overs to chase down the target. Let's examine the target scores by both methods in cases when the team batting first has lost 0, 1 and 2 wickets (see table below).

The aspect that stands out is that by the D/L method the targets, and the projected scores, vary widely. The target for the team batting second is 140 when the team batting first is 50 without loss, and 95 when the team batting first is 50 for 2, a difference of 45 runs. By the VJD method, the difference is only ten, to reflect the fact that losing two wickets after 20 overs doesn't significantly impact a team's ability to score, given that they still have eight wickets in hand for 30 overs.

The other aspect about these targets is the projected score, which is the estimated total that a team would have reached in 50 overs from a given situation. From a score of 50 without loss, D/L estimates that the first team would have ended up scoring 231, which means they'd have scored 181 more in the remaining 30 overs. That score, however, dips to 156 when the 20-over score is 50 for 2, which means they're expected to add only 106 more in the last 30 overs, with eight wickets in hand. Is that a harsh call on the batting abilities of those coming in at No. 4 and lower, or is it fair?

With the rival VJD method, the projected scores are much closer, again suggesting the belief that the loss of two wickets after 20 overs won't significantly alter the scoring ability of a team: with all ten wickets in hand, a score of 50 in 20 overs translates into 191 in 50; with two wickets down, the score drops by only 19 runs, as opposed to 75 in the D/L method.

 Scenario D/L target Projected score VJD target Projected score 50/0 in 20 overs, target in 20 140 231 113 191 50/1 in 20 overs, target in 20 115 188 108 181 50/2 in 20 overs, target in 20 95 156 103 172 100/0 in 20 overs, target in 20 179 344 170 325 100/1 in 20 overs, target in 20 172 320 164 310 100/2 in 20 overs, target in 20 163 288 158 296

Also, the difference in D/L targets between scores of 100 for no loss and 50 for no loss in 20 overs is just 39 (179 minus 140). However, for the same scores but with two wickets down, the difference increases to 68. In the VJD method, this difference stays constant (57, 56 and 55), which intuitively makes more sense.

Scenario 2: Five-over par scores in high-scoring Twenty20 matches

The ICC has ruled that five overs per innings is enough to constitute a complete Twenty20 game, which means any system should be able to work out reasonable results even for such a short game. (It's another matter that the ICC probably needs to rethink this policy - five overs is far too short a period for a complete innings in a cricket match.)

Here's a look at the five-over par-score tables under D/L and VJD for high-scoring Twenty20 matches. The totals here range from 200 to 280, and a look at the five-over par scores shows major differences between the two systems. With D/L, the maths seems to be wrong - the par scores at six and seven wickets down are lower for a target of 281 than for a target of 201. A team chasing 201 has a par score of 95 when they are six down, but the par score actually reduces by one run when the target goes up by 60. From the table below, it's clear that 116 for 7 in six overs is a winning total when the target is 261, but is three runs short of the par score when the target is 201. For scores of over 200, the D/L par scores are sluggish and actually reduce as the targets get higher.

With VJD, on the other hand, the par scores move up with the targets, which is as it should be, but those pars are also much higher than the D/L ones. For a target of 200, for example, the par score at six down is 126 in five overs, which means the last four wickets need to score 75 in 15 overs with four wickets in hand. Is that too low an asking rate, given that the rate at the beginning of the innings was ten an over, or is it justified given that six top-order wickets have already fallen?

 Team 1 total D/L par-6 down 7 down VJD par-6 down 7 down 200 95 119 126 145 220 97 121 139 160 240 96 119 151 174 250 95 118 158 182 260 94 115 164 189 280 92 111 177 203

Let's look at a couple more situations where the par scores don't quite conform to cricketing logic. In a chase of 200, a score of 104 for 1 after 11 overs is below par - ie, the team batting second would have lost with that total - but a score of 105 for 4 after ten, or 104 for 5 after nine, is a winning total according to D/L. You'd expect the loss of four or five wickets to have a more adverse impact on par scores, but it doesn't.

The par scores in the VJD method, meanwhile, reacts sluggishly to the fall of the first couple of wickets, but springs into action thereafter. So, the par scores after ten overs in this method are 91 for 0, 92 for 1, 92 for 2, and 93 for 3, but jumps to 107 for 4 and 122 for 5 - the VJD logic is that, with only ten overs to go, it doesn't matter much if a team has ten wickets in hand or eight. Comparing with the situations given above in the D/L method, the 11-over par is 101 for 1, the ten-over par is 107 for 4, and the nine-over par is 119 for 5.

Thus, while 104 for 5 after nine is a winning total under D/L, a team would have to score 120 for 5 under the VJD method at the same stage to win.

Scenario 3: Internal consistency in Twenty20 targets

Here's another look at internal consistency of the two methods, but in Twenty20 matches. Like in the first case, this looks at the targets set by each method when teams have made two sets of scores (35 and 50), losing 0, 1 and 2 wickets. In the VJD method, there's little difference in the targets regardless of the wickets lost; in D/L, the targets don't change much when the score is 50 - 50 for no loss gives a target of 63 and 50 for 2 throws up 60 - but the wickets influence the target much more when the score is 35 - the difference there is ten runs. What's interesting is that there's a difference of only six runs in the D/L target between scores of 35 without loss and 50 without loss; the target difference between 35 for 2 and 50 for 2 goes up to 13. In the VJD method, the difference remains constant at 12.

 Scenario D/L target in 6 VJD target in 6 35/0 57 53 35/1 52 52 35/2 47 51 50/0 63 65 50/1 62 64 50/2 60 63

These examples don't provide an exhaustive list of differences between the methods, but they offer a glimpse into some of the salient ones. The ICC has so far refused to get into specific advantages and disadvantages of the two - and Duckworth-Lewis haven't said much either - which has allowed the debate to degenerate along regional lines in the media. Are there areas where the D/L method outperformed VJD? Are there examples to illustrate D/L superiority? Are there mathematical flaws in the VJD method leading to erroneous outputs in certain situations, which made the cricket committee choose the D/L method? The ICC could start by offering answers to some of these questions.

Feeds: S Rajesh

Posted by   on (June 9, 2012, 14:35 GMT)

Since these days all the matches are administered by neutral officials, my suggestion is that all five (two on field umpires, third umpire, reserve umpire and the match referee) should form a jury and vote or allot marks to both the teams in such conditions and best of five should win. Alternatively, highest & lowest from the marks given by the officials be dropped and the team getting maximum from the rest three should win. Result should be a tie, in case both get equal marks. believe this is a widely used practice in games judged by a jury. ICC can provide a guide line to officials for awarding marks in such conditions.

Posted by keecha on (June 9, 2012, 6:50 GMT)

Am not sure if either methods consider the economy rates of the bowlers, strike rate of the not out batsman when arriving at the numbers. These are important figures. If it looks a bit overcast, the captain bowling first might employ his best bowler fully before rain intervenes and he might complete 10 overs for just 30 runs by the 35th over and lets say rain arrives then and reduces it to a 37 overs per side match. The rule should consider the worst economy rate of the bowlers that have bowled in first innings for the number of overs the best bowler bowled extra than the max number of overs allocated to the team bowling second. Also if the chasing team loses a wicket at 50 runs in the 6th over and the batsman that got out had contributed only 10 % of the score with a strike rate of 30, then there should not be a significant rise in the target for the loss of 1 wicket beca the one who has done all the handwork is still out there and could pull it off. These things should be factored in

Posted by keecha on (June 9, 2012, 6:38 GMT)

Everybody unanimously agree that D/L method is the ultimate scale? Think again. Why go mathematically? don't we see that the D/L method is now biased to the team that chases in a rain affected match? as a player, as a captain, as a cricket follower, everyone knows that if it is a rain affected match, the side batting second has an edge. so why not change? and is the DL method factoring in the changing powerplay rules??

Posted by 07sanjeewakaru on (June 9, 2012, 5:28 GMT)

It's ridiculous to see D/L targets when 200 to 280 reduce when 6 down 95 to 92.So, It's massive fail.How about is this happen in T20 WC final or semi like 92 WC semi?I don't think there are many differences in two methods.Think Engineer who built VJD improved the DL.This should be considered and evaluated.It should be stop happening things like SCG in1992.

Posted by Rao_Guru on (June 9, 2012, 4:07 GMT)

Several people have suggested replaying the match on a reserve day as the fairer alternative to adjusted scores on a rain-shortened day. In principle, I agree with this. However, I think that instead of replaying the match from scratch on the reserve day, it is better to continue the match from where it is suspended on the first day. This way, there is a better chance of finishing the game. After all, if it rains on one day, it is also possible to get a similar amount of rain the next day as well. Better use of the reserve day in my opinion.

Posted by Ennarkay on (June 9, 2012, 3:15 GMT)

Rajesh, Can you tell us what the VJD method says about THE famous SA Vs Eng match from 1992 world cup? If it comes out with similar target as D/L, no point discussing it. http://www.espncricinfo.com/ci/engine/match/65155.html

Posted by   on (June 9, 2012, 0:15 GMT)

always hated D/L; why not resume the match next day; loved that option in 99 world cup

Posted by   on (June 8, 2012, 23:58 GMT)

Rajesh, look at the more likely ODI scenario of 100 runs in 20 overs. VJD and D/L targets are less than 5% of each other. What's the fuss? As we say here in the states: "Don't fix it if it ain't broke."

Posted by Steggz on (June 8, 2012, 23:39 GMT)

Big issue for both systems re: T20 targets is that they still need more games for the equations to work.

Posted by   on (June 8, 2012, 22:16 GMT)

@declaration - well said. It's better to accept 'no result' or have a reserve day and replay the match wherever possible. Both these methods are based on statistics/historical performances (that are supposed to work well for most scenarios) and one can always find or create examples to prove or counter-prove a point. Picking only extreme examples to discredit DL method also seems a little unfair. Maybe the DL method needs some tweaking in this new age of high scoring T20s. But I still think at the end of this all, we shouldn't even be trying to predict/extrapolate scores. It's a futile exercise. The beauty of the game is you never know when And planning to use such methods in World cup matches is just outrageousl.

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S Rajesh
Stats editor Every week the Numbers Game takes a look at the story behind the stats, with an original slant on facts and figures. The column is edited by S Rajesh, ESPNcricinfo's stats editor in Bangalore. He did an MBA in marketing, and then worked for a year in advertising, before deciding to chuck it in favour of a job which would combine the pleasures of watching cricket and writing about it. The intense office cricket matches were an added bonus.

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