A biology-inspired rain rule for T20s

Despite its unprecedented growth over the past few years, there has been no specialised "rain rule" for Twenty20 cricket. Even in the sport's shortest format, there is a provision for overs to be reduced, primarily due to weather interruptions.

The Duckworth-Lewis (DL) method, now modified and morphed into the Duckworth-Lewis-Stern (DLS) method, is the "rain rule" that has been used to calculate targets in rain-affected games in limited-overs internationals. Originally meant for 50-over matches, the DL method has been utilised for 20-over games as well.

The method has, over the years, been criticised by players and fans. The DLS method is now a computer-based approach that utilises proprietary software and is not accessible to a majority of cricket watchers. This article is a modest attempt at devising a consistent and accurate but spectator-friendly rain rule specifically for T20 cricket.

The construction of this rain rule is based on the analogy between a team's run-scoring propensity and the metabolism rate in animals. The latter is governed by Kleiber's three-quarter power law, which states that the metabolic rate (B) varies with the mass (M ) as B ∼ M3/4. If the resources of a team, which consist of the overs remaining and wickets in hand, are considered analogous to the mass, then the run-scoring ability of the team can be regarded as the analogue of the metabolism rate. Motivated by this analogy, we define the utilisable resource percentage of a team (on which the run-scoring depends on) as,

This specific construction of the resource percentage formula is primarily driven by the requirement that the resources must diminish as the innings progresses and more wickets are lost. For sake of brevity, I will skip the details but merely state that this model does a reasonable job of mimicking the progression of a typical T20 innings, thereby providing the confidence to employ it as a rain rule. It must also be noted that unlike the DLS and VJD methods, this model does not employ data from past matches in its construction, which is a unique aspect of the proposed model.

The determination of revised targets in interrupted games is simple and requires only the calculations of the resources of the two teams. If B1 is the resources utilised by Team 1 (batting first) in scoring S runs and B2 is the resources available for Team 2 (batting second), then the par score (to tie the match) is simply S*(B2/B1) and the target is one added to the par score (both rounded off to the nearest integer). The effective resources for the teams are calculated by subtracting the resources lost, such as those due to multiple interruptions in either innings, from the resources available at the start. Since the Kleiber mathematical rule forms the basis of the resource calculations, the target obtained is referred to as the Kleiber target and the rain rule as the Kleiber rain rule.

The true test of any rain rule is to look at its performance for several scenarios, both in the absolute sense and relative to its predecessors. We can look at utility of the Kleiber rain rule for practical scenarios by considering real-life examples of interrupted T20 games and comparing it with DLS targets to allow for a clear judgement.

Example 1:
The match between Trinidad & Tobago Red Steel and St Kitts and Nevis Patriots in CPL 2015 started as a 20-over game, with Red Steel batting first. Rain stopped play when they were 56 for 1 after nine overs and the match was reduced to 14 overs per side. Red Steel finished at 134 for 5 in 14 overs. What is the 14-over target for the Patriots ?

To demonstrate the application of the Kleiber rain rule, we provide a step-by-step calculation for this example only. For all other examples, similar calculations may be performed either with a scientific calculator or simply using the Spreadsheet calculator. (Click here to download.)

Resources at start for Red Steel = 1 (since O=20 and W=0)

Resources at start of first interruption = 0.6333 (since O=11 and W=1)

Resources at end of first interruption = 0.3506 (since O=5 and W=1)

Resources lost = 0.6333 - 0.3506 = 0.2827
(Only one interruption, hence this is the total resources lost)

Effective resources for Red Steel (B1) = 1 - 0.2827 = 0.7173

Projected Score for Red Steel (PS = S/B1) = 134/0.7173 = 186.82 runs

Resources at start for Patriots = 0.7653 (since O=14 and W=0)

Effective resources for Patriots (B2) = 0.7653 - 0 = 0.7653
(No second innings interruptions, so no resources are lost)

Kleiber target = 186.82×0.7653 + 1 = 143.97 (rounded off as 144 runs)

The DLS target in this case was 142, and the Kleiber target is in a sense a "close" approximation to the DLS target.

Example 2:
England scored 191 in 20 overs against West Indies in the World Twenty20 2010. Chasing 192, West Indies were 30 for 0 after 2.2 overs when rain delay reduced the second innings to six overs. The DL target of 60 was deemed too low and heavily criticised. The DLS would have given a slightly higher target of 66 while the Kleiber target is 71. Given the high-scoring nature of the game and the significant reduction in the second innings, the Kleiber target appears more reasonable.

Example 3:
In the 2015 Natwest T20 Blast, Durham scored 174 in 20 overs and Northamptonshire were 47 for 5 in reply at the end of eight overs when rain caused an interruption. The second innings was rescheduled for 12 overs after inspection and the 12-over DLS target was 131. The 12-over Kleiber target is quite close at 132. However, persistent rain did not allow any further play and the game had to be called off. The DLS par score at that stage was 88, but the Kleiber par score is pegged nine runs higher at 97. It is interesting to note the difference between the DLS and Kleiber targets are stark at the eight-over mark but only one run for 12 overs. The two rain rules appear to differ in the importance they attach to lower-order and lower middle-order wickets, with DLS tending to value them more than the Kleiber rule, particularly for very short second innings. It is debatable which of the two rules is more correct in this scenario, and the lack of sufficient data to validate their underlying mathematical models for such scenarios (that are not very common) is indeed a significant bottleneck in the development and deployment of rain rules.

The article is a concise description of a new proposal to developing a do-it-yourself rain rule targeted exclusively at T20 cricket. Despite its simplicity, the Kleiber rain rule satisfies the seven objectives laid down by the ICC and provides reasonable and fair targets for different scenarios, including those with multiple interruptions. Interested readers are welcome to test the Kleiber rule (using the Spreadsheet calculator or otherwise) for interrupted T20 games (both in the past and likely in future) to arrive at their own judgement, both on the utility of this rule and as a possible alternative to the DLS in the T20 arena.

A comparative study of DLS and Kleiber rain rules for a selected number of T20 matches in the past can be found here. For those readers who wish to delve into the details of the methodology, a more descriptive study may be found here .

While it is true that no mathematical system is 100% perfect when applied to limited-overs cricket, it is a pleasant surprise that one of biology's most well-known scaling relation could hold the key to T20 cricket's rain woes.

Note: I thank Prof. Steven Stern for several useful discussions and insights as well as for providing the DLS targets in the comparative study. The work is dedicated to the cricketing group during my student days, the memories of which provided the initial impetus for this work.