What makes the VJD method the superior rain rule

Over the last month there have been a couple of articles on this site comparing the DLS and VJD methods - S Rajesh's Numbers Game article, which raised some concerns with the DLS method, and Steven Stern's explanations and clarifications on the points raised by Rajesh. I would like to thank both, especially Professor Stern for coming forward to discuss the matter in an open forum.

In his article, Stern has sought to establish that the Duckworth-Lewis System (D/L) is more internally consistent than the VJD system. His comment that the VJD method isn't internally consistent is based on one argument: in the VJD method, the sum of the target in X overs and the par score, for zero wickets, in (50-X) overs isn't equal to the original target in 50 overs.

However, it is a lack of understanding of the VJD system that leads to that conclusion. The VJD method does not assume that every higher over is more productive than the preceding one (for the same number of wickets lost), and hence overlapping of overs does exist, and therefore the sum of the target score in X overs and par score, corresponding to zero wickets, in (50-X) overs need not be equal to the target in 50 overs.

For example, the VJD target in 25 overs for a score of 300 in 50 overs is 192, and the par score in 25 overs is 112. The sum of the two is 304. (Stern says this should be 301.) This happens because the VJD system does not consider all 25 overs in the second half to be more productive than all 25 overs in the first half of an innings. This is sensible when we observe in actual matches several high-scoring overs during the Powerplays, which are mainly in the first half. Stern is referring to this issue in the first three tables as the ambiguity. However, because of the underlying construct of the VJD method, it's clear that there is no ambiguity. If you exclude this last column from the tables that Stern has presented, and look at the same examples, you'll realise that the VJD results are more consistent than the DLS ones.

Since it takes a lot of space to explain the theory in detail, I am not attempting it here and I suggest reading the portion on "theoretical comparison of the two systems" in my paper published in Current Science in 2002 for further learning. Incidentally David Kendix, member of the ICC Cricket Committee, had also raised the same issue during his review way back in 2005, but after listening to my explanations - and perhaps reading the above paper - he was satisfied with my argument.

In DLS calculations, the concept is that the resource utilised plus resource remaining is equal to the total resource that is originally available. Against Team 1's scores of 300 in 50 overs, the DLS par score in 25 overs without losing any wickets is 110. That means, with all wickets remaining, in the next 25 overs they are expected to possess resources to make 190 runs (300-110). Therefore, in the beginning, if Team 2 gets only 25 overs, their target will still be 190 runs, plus 1 of course (191 runs). As per the DLS philosophy, this is a necessary condition for consistency, but under no circumstances can this be regarded as a sufficient condition for overall consistency.

In fact, the main problem with the DLS system is its statistical inconsistency. Fixing a score of 250 and assuming two glaringly different behaviours in scoring patterns before and after this score is not scientific. In the VJD system, six scoring patterns are assumed, one for matches with scores of 100 and below, and one each for scores of 200, 300, 400, 500 and 1800. For any score, for example, say, 263 runs, the software will choose the two patterns in between which the score lies (here 200 and 300) and interpolate and create the most apt table to use for the calculations. So the transition is smooth, without the abrupt changes that are seen in DLS. The right table is chosen from a possible 1701 tables. That is the strength of VJD system.

Here are three more examples that bring out the different outputs of the two methods, and highlight the inconsistency of DLS:

Example 1: If Team 1 is at 57 for 1 after 22 overs at the time of an interruption, the DLS target score for Team 2 in 22 overs is 134 runs. If the score is 70 for 1, i.e. 13 runs more, the target shoots to 164 (30 runs increase, at 2.31 runs per every further run scored by Team 1). For a score of 83 for 1, which is a further 13 runs increase, the target score is 167 runs, just a meagre increase of three runs. (Just 0.23 runs per every further run scored by Team 1). The rate of increase is inconsistent. Also, for 57 for 0 the target is 157, 23 runs more than that for 57 for 1, but for 70 for 0 it is 167, just three runs more than that for 70 for 1. VJD targets in the above cases for the loss of zero wickets are 122, 141 and 158 respectively - increases of 4, 4 and 5 respectively, unlike 23, 2 and 4 in the DLS system.

Example 2: This one is similar to the example in Rajesh's piece, mapping 20-over targets by both methods for various scores by the team batting first, with additional columns for increase in target per run increase in the total. If Team 1 scores 217 runs in 50 overs, the DLS target for Team 2 in 20 overs is 134 runs. If Team 1's score is 233, the target for Team 2 is 144, and for 249 it is 154, a steady increase of 10 runs in target per 16 runs increase in Team 1's score. However, for the target to get to 164, Team 1 needs to score 305, which is 56 more than the total needed for a target of 154. Even for a score of 309 runs (60 runs more), the target is 164. For a score of 260 and 266, the DLS target is the same - 156. The VJD targets are far more responsive to runs scored by the team batting first.

Example 3: Suppose Team 1 is at 106 for 3 in 22 overs when the interruption takes place, the DLS target for Team 2 in 22 overs is 165 runs. If Team 1 makes the same score in 24 overs and Team 2 then gets only 22 overs, the target for Team 2 is again 165. Since 106 for 3 in 22 overs is a better score than 106 for 3 in 24 overs, the target in the second case should be lower by a few runs. If the score were 100 instead of 106, there is six runs difference in targets.

Comparing 22-over targets

Team 1's score	DLS target for Team 2 in 22 overs	VJD target for Team 2 in 22 overs
106/3 in 22 overs	165	152
106/3 in 24 overs	165	148
100/3 in 22 overs	164	148
100/3 in 24 overs	158	142

Ambiguity in DLS targets. VJD targets look more reasonable

As Stern has also written, no statistical model can be perfect or satisfy everybody. The VJD version for ODI matches also starts losing its accuracy once the rescheduled overs go much below 20 overs. That's the reason a different version is introduced for T20 matches. Both these versions are perfectly consistent for both international versions of the game (ODI and T20).

Conclusion
The VJD method is more responsive than all D/L versions, and the current one certainly gives superior outputs compared to the DLS method. It is the obligation of the ICC to seek and find out the best available systems not only for this purpose alone but in all other areas as well. I hope the ICC takes some pragmatic steps in this direction at their annual meeting in Barbados.