Is Jayadevan's proposed method better than the Duckworth/Lewis method?

It is going to be difficult to answer this question. My gut feeling is that Jayadevan's proposed method to reset targets in interrupted limited over international (LOI) cricket matches could indeed challenge the Duckworth/Lewis method. But we will have to do a lot of work before we can reach a decisive verdict.

One way to reach this verdict would be to determine objective criteria to compare the two methods, look at a large number of examples and, finally, simply argue. This should certainly get us somewhere.

The "objective" criteria

I can think of five criteria, perhaps in that order:

• the target set should be fair and reasonable
• calculations must be easy and understandable
• the method must handle all kinds of interruptions
• the underlying mathematical model should be theoretically valid, internally consistent and, hopefully, even elegant
• the method must be able to handle special or unusual situations sufficiently well.

In the following discussion we will consider each of these five criteria. In doing so, I will draw heavily from my earlier articles on Rediff introducing both the Duckworth/Lewis (D/L) and Jayadevan (J) methods. I will also quote extensively from private communications which I have received both from Duckworth and Lewis and Jayadevan.

"Fair and reasonable"

There have been over 200 applications of the D/L method in the last four years, and I can recall only a handful of cases where the target has been seriously questioned. This is excellent by any yardstick, but we could still argue! For example, if the 1999 India vs Sri Lanka World Cup match had terminated with Sri Lanka at 117 for no loss in 25 overs, in reply to India's massive 371, Sri Lanka would have won according to the D/L method!

I cite this instance to contend that although D/L has performed remarkably, it might also be lucky that no really "bad", albeit very improbable, result has so far been actually encountered. But as cricket match preparation gets more scientific, strategies could evolve to exploit such weaknesses. If I had been advising SL in that match, I'd have said: "Look, you can never win chasing 371. So pretend that this is a test match and try to reach 117/0 in 25 overs. Then pray hard for rain!". [The J method sets a winning target of 150/0 in 25 overs; a revised D/L model, which requires the use of computers, sets a target of 152/0 in 25].

We'll return to discuss more examples of "bad" targets, using both the J and D/L methods, a little later. But let's first look at the J and D/L targets in more "normal" situations. Let's look at two tables comparing the J and D/L targets. The first table looks at likely, but hypothetical, match situations. The second table looks at real match situations.

These 14 examples suggest that while the D/L and J targets are never too far from each other; there are six cases (*) out of these 14 when they differ by more than 10 runs.

Let's see how the comparison involving real LOI matches looks like. In some of these matches the D/L method was not actually applied.

In the next set of 14 matches, the D/L and J targets are in much better agreement, with only one case (*) differing by more than 10 runs. This is re-assuring; it must seem that real matches "behave" better than hypothetical ones! More seriously, this really means that the D/L and J targets match very well on the average. The disagreements arise only in more unusual or extreme situations.

It is therefore important to look at these more unusual situations. It is my personal view that D/L targets are questionable in two situations. The first, and this is well-documented, is when Team 2 is chasing a "well above average" total and holding on to its wickets. In such situations, D/L sets "well below average" winning targets for Team 2. We talked of the extreme situation in the India vs SL match. But this anomaly persists even in less extreme situations: With 94/0 in 25 overs, or even 115/2 in 25, Team 2 can defeat Team 1 who have scored 300 in 50 overs.

The second questionable situation can occur in low scoring (150 or below runs) matches where Team 1's innings has been severely curtailed. For example, in a recent (October 2000) match between New Zealand (NZ) and South Africa (SA), after a one-over interruption at 81/5, NZ recovered to 114/5 in 32.4 overs in difficult batting conditions when their innings was terminated. In reply SA were asked to score 153 in 32 overs to win, i.e. almost 40 more runs in the same 32 overs. It required the genius of Lance Klusener to lead SA through, but did they deserve to lose if they had scored 150/5 in 32 overs?

When I discussed these two examples with Duckworth and Lewis, they acknowledged that the first situation does reveal a basic weakness in the D/L model (and indicated that their revised computer-dependent model would correct this anomaly). They however defended the D/L target in the second situation stating that extensive analysis indicated that, on the average, teams in situations similar to NZ's would reach around 190 in 49 overs. So, given that SA knew their target from the start, 153 was quite a realistic target.

I am still not fully convinced. Those who know the D/L method sufficiently well will recognise that, in obtaining the target of 153, D/L uses the G50 = 225 rule which assumes that, on the average, Team 2 are expected to score 225 in an innings. In low scoring matches (usually due to difficult batting conditions), if Team 2 appear capable of only reaching about 150, the assumption that they will get to 225 is significantly off the mark. It is this assumption which pushes the target up abnormally. Put another way, if a supremely talented Klusener is required to do his best to reach the target, the target is, generally speaking, very unfair.

How does the J method perform in these two questionable situations? If Team 1 score 300 in 50, Team 2 can win if they reach 122/0 in 25 (as against D/L's 94/0) or 130/2 in 25 overs (D/L: 115/2). These targets certainly appear more reasonable. In the second situation, the J target for SA would be 128 in 32 overs, which, again, appears more reasonable.

When would I want to question the J targets? I am just a little concerned about how the J model responds to the fall of wickets. In the example above, I find it hard to accept that 122/0 and 130/2 have the same "parity" at the half-way stage; 122/0 is distinctly more comfortable!

To study how J targets vary with the fall of wickets, let's consider the situation where Team 1 score 125 in 25 overs when a heavy downpour causes the match to be curtailed to 25 overs per side. We'll tabulate Team 2's target in 25 overs for all possible "loss of wicket" situations.

Table 3
Team 1's end score	D/L target	J target
125/0	210	197
125/1	203	191
125/2	194	185
125/3	184	179
125/4	170	153
125/5	153	133
125/6	134	126
125/7	116	126
125/8	103	126
125/9	93	126

Table 3 indicates that while the D/L target reduces after the fall of every wicket (and very steadily), the variation of J's targets with the fall of wickets is less smooth and a trifle "sluggish". Jayadevan doesn't agree with this reaction (and counters that D/L is "over-responsive" to the fall of wickets), but I still consider D/L's fall-of-wickets "distribution" to be superior especially when the last few wickets are batting.

Notice also how the D/L target falls below 125 from 125/7 onwards. This lower target for Team 2, in the same number of overs, initially surprised me, but it is entirely consistent with the D/L philosophy of compensating the team that has fewer resources. J, on the other hand, does not allow a smaller target for Team 2 in the same number of overs partly because he believes that the cricket fan will not accept this situation.

"Easy and understandable calculations"

Jayadevan told me recently that many Indian umpires couldn't set a D/L target without some help. This is surprising because the D/L method is really quite simple. If this is indeed the case, umpires will also have trouble with the J method, especially in situations involving multiple interruptions.

It is my considered view that a target resetting method need not be unnecessarily simple, especially if this requirement weakens the method. But every cricket person I know talks of that "perfect method" which is both very easy to understand and work out, and also sets impeccable targets in all situations. This is a pipe dream. Both the D/L and J methods suffer marginally because of this forced requirement of being "easy to calculate". I personally advocate the use of the most sophisticated available method, even if it needs a computer. But this isn't a very popular prescription! My friend Vidyadhar Mudkavi has an even better suggestion: build cheap, attractive and portable D/L or J calculators in which the target resetting computer programs are "hard coded". This would solve everyone's problem!

To return to our D/L vs J comparison, a calculation using either method is really no big deal. It essentially involve looking up tables (see typical extracts in Tables 4,5), noting down a few values "on the back of the envelope" and then doing some simple arithmetic.

The tricky part is to decide what to look up from the table. My experience, after a few hundred calculations using the D/L and J methods, is that the J method is marginally simpler for situations involving only a single interruption. For multiple interruptions, however, D/L is significantly simpler.

I have also found it just a shade difficult to explain the D/L method to the layman, chiefly because it takes a little while to explain the abstract idea of a "resource percentage" (RP). D/L calculations also involve a moment's confusion between "RP available" and "RP used". With the J method, there is the overhead of converting balls remaining in an over into a percentage. Since J currently doesn't have a ball-by-ball percentage table for overs% and runs%, we are also required to do some trivial interpolation.

Both the D/L and J methods are capable of calculating the winning target on a ball-to-ball basis when Team 2 is chasing its victory target; but the D/L method is better suited for such a calculation. In a recent LOI match (see last row of Table 2) between Sri Lanka (SL) and New Zealand (NZ), SL, chasing 183 in 35 overs, had reached 155/5 in 31 overs when the rain returned to end the match. For a moment, no one knew who had won! A D/L calculation eventually declared SL winners by 3 runs. The confusion arose because neither team kept track of the winning score on a ball-to-ball basis.

"Handle all interruptions"

An essential requirement of LOI target resetting methods is the ability to handle all kinds of match interruptions. We could divide interruptions into five types: (a) interruption between Team 1 and Team 2's innings, (b) single interruption while Team 2 are batting, (c) single interruption while Team 1 are batting, (d) more than one interruption while Team 2 are batting and (e) more than one interruption while Team 1 are batting.

Before the D/L method came along in 1998, only the first two types -- type (a) and type (b) -- of interruptions could be handled sufficiently well. Interruptions while Team 1 are batting -- type (c) and type (e) -- were not even "recognised"; the accepted practice was that if Team 1 got to play 30 overs, after one or more stoppages, and scored 120 runs, Team 2 would need to get 121 in 30 overs. This severely disadvantaged Team 1; and so encouraged the popular practice of always inviting the opposition to bat first on a wet day.

Duckworth and Lewis changed all that and completely solved the problem of handling multiple interruptions. I remember writing about it in 1999, and I still consider the greatest merit of D/L to be this ability to handle multiple interruptions. No method can possibly improve D/L's procedure for dealing with multiple interruptions. In fact the D/L method tends to treat single interruptions as a special case of its multiple interruption rule.

The J method must therefore pass the difficult test of first demonstrating its capability of handling all the five types of interruptions, and then showing that it can match D/L's acknowledged prowess in handling interruptions.

The J method, especially after Jayadevan's recent revision, can now handle all the five interruption types. His method performs well, and is certainly on a par with D/L, while handling single interruptions. J's procedures for multiple interruptions work equally well; but they are not as "natural" as D/L.

"Good mathematical model"

I remember receiving a mail from Duckworth and Lewis asking if the J method was based on firm mathematical principles. It is, although the J and D/L equations are very different. Both methods use average scoring patterns; D/L derives its target from an exponential relationship while J uses a regression fit.

One major difference between D/L and J lies in the way the methods respond to the fall of wickets. Generally speaking, D/L resets targets, after the fall of each wicket, exceedingly well, although its targets tend to be low till the first couple of wickets fall. J, as we have remarked, is much less responsive to the fall of wickets.

After a first tutorial on the D/L method, the most frequently asked question is why D/L has different target resetting rules for the "R1 > R2" and "R2 > R1" situations. If the ratio of the resources can be used to "scale down" targets, why not to "scale up" targets? One of the merits of the J method is that it can both "scale up" and "scale down" using the ratio of the same variables. This offers J the additional advantage of ensuring that Team 2's target is always proportional to Team 1's score.

In its early versions, the J method contained several technical anomalies, most of them identified by D and L. After his recent revision, Jayadevan is now confident that all anomalous situations have been corrected.

"Handle special situations"

LOI matches, especially those played in India, Pakistan, SL and Bangladesh, now have two very high scoring phases: the first 15 overs -- to exploit field restrictions -- and the last 5-10 overs. While the D/L model is "well-prepared" for the end innings slog, people have often asked if D/L is equally well prepared for the first 15-over slog. D and L maintain that all available statistical evidence suggests that the higher scoring is adequately compensated by the more probable loss of (more valuable) wickets. The J model, on the other hand, recognises and accommodates the expected early slog.

In matches where a penalty is imposed for a poor over rate, the D/L method has to make an awkward correction in the RP available value. The J method, which is based on the overs percentage, has no such problem.

Winding up

It is now time to take stock and see if we can reach a verdict on D/L vs J. Just to make things simpler for all of us, let me put down the principal arguments for each of the five criteria in Table 6.

I think that Table 6 is quite useful. Each one of us can make our own comparative assessment based on the importance that we would like to attach to each criterion, and the related arguments. Individual assessments could point either to D/L or J; this won't surprise me, because both are worthy methods.

Who would I vote for? I have been trying to put off this question to the very end, but, if I am asked to judge, I would raise my finger for Jayadevan. If this looks like an Indian umpire upholding an Indian bowler's appeal, then so be it!

Acknowledgements

It gives me very great pleasure to thank Frank Duckworth, Tony Lewis and V Jayadevan for being so very helpful in helping me prepare this article. We have exchanged dozens of e-mails between ourselves over the last 6-8 weeks. I thought it was absolutely wonderful of D and L to share so much information with us, although they were well aware that Jayadevan could go on to challenge their own method. I must also thank Prem Panicker and Rediff once again. It is satisfying to note that all my three articles now appear on what I consider to be India's best website.

Jayadevan's rain rules for LOI cricket

Also read: The dummy's guide to Duckworth-Lewis

Srinivas Bhogle is Scientist and Head, Information Management Division, National Aerospace Laboratories, Bangalore, India.