How robust mathematically and statistically is the D/L system?
1. Introduction The Duckworth Lewis system is completing its 14 years tenure in the international cricket. Undoubtedly it is a much better system than its predecessors; in fact there is no comparison even. However, right from its introduction, the D/L system also has undergone severe criticisms from the various sectors of the cricketing community and it is still on even in the 14th year. Some of these criticisms were baseless and originated due to the lack of understanding of the dynamics of the game whereas some others were (are) quite genuine. In spite of all these criticisms the system survived long 14 years and the basic reason for it is that the architects of the system succeeded in creating an impression in the cricket community that it is developed based on sound mathematical and statistical principles. The inherent fear of people for mathematics helped D/Ls from being questioned beyond a limit. 2. The launch of D/L system Before introducing to the international cricket, the D/L system was first experimented in 1997-98 at English County. In that version, the target in 25 overs for team-2 against team-1’s score of 300 was 207. The par score for team-1 in 25 overs without loss was just 94. If an interruption occurred while team batting first was at 94/0 in 25, the target for team batting second in 25 was 207. Needless to say that it faced severe criticisms. The scientists managed to convince the authorities the first two cases i.e. target of 207 against 300 and par score of 94/0 in 25 overs against 300. However, a target of 207 against a team-1’s score of 94/0 was beyond all tolerance limits and D/Ls had to look out for alternatives. It was there that the novel concept of G50 came in. G50 is a silly concept both statistically and mathematically. But by skilfully presenting it and publishing papers on it, D/Ls have successfully made the whole cricket world believe for an unbelievably long period that it is a highly mathematical concept. When an interruption occurs while team-1 is batting, you need to estimate the probable runs that team-1 would have made in 50 overs in order to compute for team-2 a target. What G50 does is, whatever is the score of team-1 (whether it is 125/8 in 40 overs while the interruption occurs or it is 250/4 in 40 overs when the interruption occurs), for estimating what team-1 would have made in the remaining overs, this G50 which they call an average score in ODI matches of 50 overs and which is equal to 225 would be used. Later on, they revised it to 235. Anyway, G50 served the purpose and the target for team-2 reduced to 174 from 207. Needless to say that at the time of interruption, if team-1 is doing well, i.e. well set to go much beyond 225, they suffer and on the other hand when they are struggling and not likely to make even 200, they benefit by the G50 factor (Setting a target of 153 for South Africa in 32 overs against a score of 114/5 in 32.4 overs by New Zealand is a typical example). They changed the model slightly by 2001, and the new values of target and par score became (in the above example) 200 and 101 instead of 207 and 94. The G50 concept was retained. Subsequently after ICC has approved a completely computerized system in 2004 the CODA version of the system was introduced. Coda-7 series were giving comparatively better results for ODI matches. It was also free from the controversial G50. In the examples under consideration the results were changed to 186 and 115 against the original 207 and 94. The South Africa New Zealand result became 142. It was this version that first compared with the VJD-system in 2005. The reviewer rated both the systems satisfactory but voted in favour of D/L system for its better mathematical and statistical robustness. When T20 matches became internationally popular this version was found wanting in dealing with T20 situations. Also there was a prospect for a more user friendly Window version.
An elegant looking and extremely user-friendly version called Wincoda was released soon and in 2009 the widely discussed version Wincoda 2.0. Most surprisingly this perhaps was the worst of the D/L versions; it was a jar (as Java files are often called) of blunders. The controversial G50 was also back in place. One may visit http://vjdcricket.yolasite.com to read the details of the terrible flaws in Wincoda 2.0. Ironically it was just after the release of this version that the British Government honoured the scientists for their contribution. Let me sight just one example of its blunders. Suppose an interruption takes place after 20 overs when team-1 is batting and has not lost any wicket, the target for team-2 for team-1’s score of 57 is 148, for 62 it is 147 (1 run less) for 67 it is again 148 and for 100 it is 158. During my presentation in front of an ICC expert committee last year at Hong Kong, I pointed out all these errors, but the response I got was that D/Ls have already identified these mistakes and they rectified it in Wincoda 3.0 which they are going to release soon. If D/Ls were aware of these mistakes and still let ICC use it for an event like world cup 2011, it is a serious offence. I am very disappointed that ICC has taken it very lightly. The impression I got after the Hong Kong meeting was that ICC is going to compare VJD system with new Wincoda 3.0 version and use the best one by October 2011. But D/Ls submitted their new version only in September 2011 and hence ICC did not get enough time to check it. Perhaps base on the assurance that all those terrible mistakes are rectified, ICC has decided to employ Wincoda 3.0 from October 2011. 4 Wincoda 3.0 It is true that some of the shocking mistakes of Wincoda 2.0 are rectified in Wincoda 3.0. But there are still a bunch of statistical blunders and some minor mathematical anomalies in Wincoda 3.0, the current system being used to set targets in international cricket. Let us have a closer look at the same. 4.1 A typical illustration for abnormal variations in targets indicating serious flaw in statistical analysis: Let us consider a case where team-1 scores 53/0 in 20 overs when the match gets interrupted and their innings gets terminated. The target for team-2 in 20 overs is 148 runs. 148 runs means it is the 20 over target corresponding to a 50 over score of 244. That means, from a score of 53/0 in 20 overs (2.65 runs/over), the D/L model expects team-1 to add another 191 (6.36 runs/over) in the remaining 30 overs. I think very few people will disagree if I say that 148 is a very high ask. But let us accept it for argument sake. Now let us see what will be the target if team-1’s score at this stage was 53/2. The target would be just 101 runs. A difference of 47 runs from the previous, the projected 50 over score is only 165, a difference of 79 runs. That is, team -1 are likely to add only 112 runs in the remaining 30 overs (3.73 runs per over). 20 overs correspond to 40% of the allotted quota. Losing two wickets at that stage is quite normal. A prediction that a team would score @6.36 runs per over if they have all 10 wickets in hand and they would score @3.73 runs per over if they have only 8 wickets in hand in the remaining 30 overs is just awful. Is not the D/L model insulting the batsmen coming in at No.3 and after? This 53 runs seems to be a critical number in D/L model for 20 overs and no loss of wickets. For every 1 run decrease from 53, the target can be seen reducing by almost 3 runs. For increase of every run from 53, the target is increasing only by less than a run. This is another statistical blunder. If team-1’s score is twice this, that is 106 runs instead of 53, the targets respectively for zero and two wickets are 183 and 167. For an increase of 53 runs, just an increase of 35 runs in the first case (when no wickets are lost) and an increase of 66 runs in the second case (when two wickets are lost)!
Please carefully go through the portions highlighted in table. Here, par scores corresponding to higher scores are less than the same for lower scores. For example, if team-2 makes 95/6 at this stage, and the match gets interrupted, they win if team-1’s score is 260 or 280, but they lose if it is 220 or 240 and will be a tie if it is 200. One might argue that these are rare situations, but the influence of this serious mathematical and statistical anomaly will propagate to the normal situations also but may not be so glaring. For example, against team-1’s score of 190 in 20 overs, 136/7 in 12 overs is a winning score, which is clearly quite low. 4.3 Comparison of behaviour of par-scores in ODI and T20 A score of 220 runs now is not an uncommon score either on ODI or in T20. Let us now compare the par scores corresponding to 5, 6, 7 wickets after 5 and 10 overs in ODI as well as T20.
In ODI, par score when you lose 6 wickets is 147 runs in 5 overs and 148 in 10 overs which look somewhat reasonable. Though you have 45 and 40 overs remaining respectively, you have lost most of your batting strength and hence it is not the overs remaining that mainly matters. But see the T20 par-scores. Here you have 15 and 10 overs respectively are remaining and your par scores are 97 and 128 for 6 wickets. This is nothing but rubbish. You have lost most of your batting strength and still the model expects you to make 123 runs in 15 overs and 92 runs in 10 overs with the remaining 4 wickets. Normally, the par scores in T20 are expected to be higher as you have much less number of overs remaining. But here, just reverse is happening, that too with huge difference. This is a clear evidence to prove that D/L system is not statistically sound at all. 4.4 Some mathematical anomalies in D/L system I was rather surprised to see that in spite of being in use for the past 14 years in international cricket there are still a few mathematical anomalies too in D/L system. However, the major mathematical anomalies of Wincoda 2.0 have been successfully rectified in Wincoda 3.0. So I do not say that these anomalies can create very serious problems in any match. However, I need to point out them to prove that D/L system is not mathematically robust system as averred by the D/L fans. The general dynamics of the match is that, when team-2 is chasing a target, the later the interruption occurs tougher it will be for the chasing team. To compensate this, in a scientific method, the target will be reduced accordingly (when there is no additional loss of wickets between the stages under consideration). To explain it clearly let us consider an example: Team-1 makes X runs in 50 overs. Team-2 are X1 runs for 2 wickets after 10 overs when an interruption took 5 overs away. Suppose the fixed target is T1. Now suppose the interruption had actually taken place after 20 overs (not at 10 as assumed just before) when team-2 is at X2/2 and the target computed is T2. T2 will be, has to be, less than T1. If the interruption occurs after 30 overs when team-2 is at X3/2, the target T3 should be further less. Here we have considered a difference of 10 overs. In the case of differences of 1 or 2 overs this reduction can be marginal, can be zero also. But under no circumstance the target can increase. In Wincoda 3.0, you may take any score; you can find situations where the target is increasing for interruptions taking place at higher order overs. This is not mathematically correct.
1. In a T20 match, team-1 makes 125. Team-2, when X/2 after 5 overs an interruption takes away 2 overs. D/L target for team-2 in 18 is equal to 116. If the interruption had occurred after 7 overs instead of 5, the target in 18 is 117. You can see that between overs 5 & 7, for each ball the target is hopping between 116 and 117. 2. In an ODI, team-1 make 280, in reply when team-2 are X/3 in 19.4 overs an interruption cuts two overs. D/L target for team-2 in 48 overs is 275 runs. If the interruption had occurred after 22.2 overs (16 balls later), the target is increased to 276. 3. Team-1, 320. Team-2 when X/2 loses two overs. If it happens at 15.2 overs the target is 313, if it happens in 21.2 overs then also the target is 313. When it happens in 17.2 overs the target is 314 and when it happens in 19.2 overs the target is 312. 5. Conclusion The D/L system is not a statistically or mathematically robust system as it is emphatically established here. It is just a mediocre system. ICC is considering VJD system as an alternative to D/L system in their Cricket Committee meeting on 30th May, 2012. In spite of severe criticism from various sectors, D/Ls are riding on a false image, an Illusion so far. The lack of commitment from the part of the creators to provide a good system to the cricket community is very clear from the childish mistakes they have in their model even in the 14th year. In spite of being aware of the serious anomalies in Wincoda 2.0, D/Ls let ICC use it in a major event like world-cup-2011, which is an unpardonable offence. Still, if ICC still decides to continue with this faulty system, it will be highly unfair in general to the whole cricket community who are fed up with D/L system and badly looking for a change, and in particular to me who invested a major part of my life to develop the best method currently available in the world and still remain without getting any appreciation or recognition. [1] The author is one who developed the vjd-system, the focal challenger of the D/L system. vjdcricket@yahoo.co.in | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
