Numbers Game

# The rain equations

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

S Rajesh
08-Jun-2012
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.