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### Reverse Engineering the ICC Test Batting Rankings

TLDR: I tried to replicate the ICC Test Batting Ratings formula from a 30-year-old book and got decently accurate results.Skip to Results for the graphsLink to spreadsheet where I did all my calculations Link to sections of the book that describes the algorithm For a while now I’ve been interested in finding the formula for how the ICC Player Ratings are calculated. I figured that, although it might be quite complex, there would be some complete formula or algorithm specified somewhere online. But alas, after quite a few google searches, I couldn’t find exactly what I was looking for. The most information I could find was from this site, which is either old and has been superseded by the more current site or was never official in the first place. So eventually, I decided it would be fun try to reverse engineer them for myself. Disclaimer: This was really just a proof of concept, the method I used was inexact and often not very scientific. If I wanted to do this properly, I’d probably need use a lot more sophisticated tools and software that I’m unaware of. All of this is to say that this is largely just to get the jist of the formula and I could be talking out my arse at points, but hopefully it is still interesting! The Ancient Sacred TextsIn order for this to be remotely possible I needed data in the right format I needed to know what variables were actually taken into account. I had some idea of that from the aforementioned FAQ but I eventually found myself asking around on the member forums of the ACS (which if you haven’t heard of, I strongly suggest you check it out). They very kindly pointed me to this book, which provided almost all the information I needed to try to replicate the rankings. The final section of the book very handily gives a fairly detailed description of the algorithm used by the Deloitte Ratings, which went on to become the official ICC Ratings. However, it was written all the way back in 1990 and it is very possible that the rankings have changed quite a bit in the past 30 years. As well as this, there are some aspects that are left out that I had to guess/figure out for myself, which we’ll get onto later The DataOf course, I also needed to have all the data, from the description in the book I knew the raw data I needed to calculate the change in rankings after a match were as follows: · The scores of each batsman in each innings · Whether or not the batsman was not out at the end of his innings · The bowling rating of each bowler at the start of the match · The number of overs bowled by each bowler · The batting rating of all batsmen before the match · The winner of the match · The number of innings played by the batsman before the match Most of these things can be taken from the scorecard of a given match. I used CricketArchive because it seemed more consistent and easier to parse than cricinfo scorecards. Thankfully, you can also find the batting and bowling rankings at any given date in the history of Test Cricket online pretty easily here. So after messing around in Power Query for a few days I was able to fumble together a script that could take the scorecard link as input and then combine all this data together for all the batsmen involved in the match and spit it out. My dodgy script only worked completely on about half the matches I gave it and the webpages only show the top 100 at any given time (meaning you had to be in the top 100 batsmen both before and after the match for me to be able to find your rating), so after throwing it around 35 test matches since the start of 2017 I was left with 218 individual match performances as data points with which to experiment. The AlgorithmDeriving the Match ScoreThe ratings are a weighted average of scores given to each individual innings, and the book provides this equation for getting the new rating after an innings https://preview.redd.it/nxnloha7my061.png?width=572&format=png&auto=webp&s=ab24a8304af9aa5dd9ed523c204ef888a91a1fb9 *After looking at the book I tried to confirm the derivation of this formula but kept on ending up with (k * Old Rating * (1-k) instead of (k * Old Rating * (1-k^(n)). However, that through the numbers off so I think what is in the book is correct and not a typo. It would be really appreciated if someone could double check this though, and point to where I’m wrong if I am. Where k is the decay constant that they set at 0.95 (I assumed it hasn’t been change since then) and n is the number of innings played by that batsman before that innings. We only have the ratings before and after each match as that is when they are updated, but we can make an approximation that I will call Derived Match Score (DMS), by manipulating the equation to get https://preview.redd.it/52ktfva9my061.png?width=696&format=png&auto=webp&s=f729efc3b8ab1ce49505087147aecd0d046a81df In theory, DMS should be equal to the weighted average of the first and second innings scores given to the batsman in that match, so I can define Match Run Value (MRV) as follows, and then plot it against DMS to verify my results https://preview.redd.it/cuzvdlsamy061.png?width=479&format=png&auto=webp&s=dc61ca82ff458ba73d8967eb785251e61e00a393 Which leads us on to the meat of the problem… Calculating the Innings ScoresThis is the actual formula that gives a score to each innings, the book denotes this as Runs Value (RV) and the crux of the formula is as follows https://preview.redd.it/esjmnvzbmy061.png?width=544&format=png&auto=webp&s=6be7944fe125c654d7287f3cb5399c8ae1711d4f So what are all these variables? Runs is simply the number of runs scored in the innings. Average is the average runs per wicket over all of test cricket (the book states this as “approximately 31”, however I used 30.5 as it is closer to that now) MPF, IPF and Quality require a bit more explaining. MPF, or Match Pitch Factor can be thought of as the average runs per wicket during the match, however there is some nuances that I will get to later. Similarly, IPF is Innings Pitch Factor and can be thought of as the average runs per wicket of that innings (with the same caveats as MPF). Quality is a sort of expected average runs per wicket, which is derived as some function of the weighted average of the bowling ratings of the opposition bowlers (weighted by the number of overs each bowler bowled in that innings). You can sort of think of this formula as taking the runs scored by a batsman, making an adjustment for how difficult it was for the average batsman in that match, making a smaller adjustment for how difficult it was for the average batsman in that specific innings, and making a much bigger adjustment for the quality of opposition bowling. Also note that these adjustments are multiplicative, and that we’re still ending up with a score on the scale of runs. A batsman up against a perfectly average attack, in a perfectly average innings in a perfectly match will have the same Runs Value as the runs he made in that innings. Innings Pitch Factor and Match Pitch FactorThis is the first place where there is a major lack of information in the book. Regarding the ratio of runs to wickets in a match, it states: “Incomplete innings have to be adjusted first, as 180 for 2 would very rarely be equivalent to 900 all out. A separate formula thus transforms the simple ratio of runs per wicket to the much more important sounding ‘match pitch factor’ (although, it should be stressed, the actual pitch is not being assessed in any way)”The only problem is that they don’t give any formula for this, so I was stuck. Ultimately, with no information on the functional form of said formula, the only way I could treat this was to guess a reasonable function and continue from there. I decided the most reasonable assumption to make was that MPF was simply the average of the IPF for each innings, and that I would calculate “my” IPF as follows. Consider the average percentage of innings runs scored by the fall of the nth wicket, and denote it as C(n). I found data for partnerships in this paper, and used it as a proxy (I know that adding all the means and finding the cumulative percentage is not necessarily the same thing, but I figured it was a good enough approximation for my purposes).
https://preview.redd.it/puv3nkezeu061.png?width=162&format=png&auto=webp&s=2af97a43c02011d1faf8edd9ea7da1fd5adc3a88 This IPF isn’t perfect, but it made a slight increase to the accuracy of the results QualityAfter sorting out the IPF and MPF I still had to figure out how to calculate the Quality variable. As with the other 2, the book doesn’t give a formula or really any hints towards it other than it uses the weighted average of bowler’s ratings. So I made the assumption that it could be approximated by the basic formula https://preview.redd.it/jfojiclemy061.png?width=407&format=png&auto=webp&s=c4023f84034d8a5c4092f4ed3d362d73f1d8d7b7 Where a and b were parameters to be estimated. I thought I could use a simple linear regression on this with the data I had, but I couldn’t easily extract the quality rating from the derived match score (for reasons I’ll get too soon). I considered trying to make this estimation based on a regression predicting the actual innings totals in the matches from the bowler’s ratings - that is what the Quality variable is supposed to account for – but the data for that would be too noisy to do it properly. So I ended up to resorting to the, not very scientific, method of using Excel's solver to find values that best fit the data, then rounding them to correct significant figures. I was left with a = 1800 and b = 30. AdjustmentsThe book then describes adjustments made taking into account the result of the match. I won't cover them in detail here because this post is already massively long and they are in the pages of the book I linked to above if you are interested. Basically, batsmen with high scores in winning games have their score for that innings increased proportionally to how well they did, whilst low scores in losing efforts get quite severely punished. It was all described completely which was nice as it meant I didn't have to do any guesswork but the fact the adjustments were there meant that it wasn't simple to directly work out Quality as a function of the oppositions bowling ratings. There are also adjustments made for if a batsman finishes not out but they aren't described at all beyond a brief mention so I decided to omit them from this. Dampening First InningsIn order that a player doesn't reach the top of the rankings immediately if they have a particularly good debut. The book puts it like this: "The system works for all but the newest Test players, who for the first few games of their career have their ratings damped by gradually decreasing percentages to stop them rising too high and too quickly.It is unclear here whether or not this means that their real rating is kept and used to calculate new ratings, which then reduced by a different percentage after each match, or if a player's first innings simply gets counted for less forever. As it was simpler to implement, I chose the later. So now a player only ever receives a given percentage -p- of points for his first inning, and the percentage of points he receives for his second and third innings, and so on, are increased linearly until his -n-th inning, at which point all innings are worth full points in the ratings. So we have parameters p and n to consider Using the same method as that used to estimate the a and b parameters for Quality, I determined that p = 50% and n = 10. In other words, a players first inning is worth 50%, and this increases until his 10th Inning which is worth 100%. ResultsSo how does my hacked together approximation of the ratings compare? As mentioned, the MRV should be equivalent to DMS (up to a transformation). If we plot them together we see that they agree pretty well with each other. In fact MRV can explain roughly 90% of the variation in DMS https://preview.redd.it/9m0k8fmlmy061.png?width=500&format=png&auto=webp&s=ce4a5a3dc88509129fcc7227b800f81d4dc27454 You may wonder why this isn't a trendline with equation y = x, but rather y = 22.2x +79.9. This was to be expected as the ratings (and therefore DMS) are all based on a scale of 0 to 1000 whereas Innings Scores (and therefore MRV) are still always on the scale of runs. But we can use the information from this graph to convert each Innings Score into the correct scale. Then we can use the first equation of this post to work out the rating after the first innings, given the rating before the match and the newly converted innings score for a batsman's first inning. We can then predict what the rating should've been after the match using the calculated rating after the first innings and the second innings score. This gives us a set of ratings that we calculated using our algorithm, along with the actual ratings calculated by the ICC after the match. Plotting them together looks like this https://preview.redd.it/r99idlynmy061.png?width=453&format=png&auto=webp&s=7994b26a8a98377ec7dcdda91c7db765ce034a75 That's an incredibly close fit, but can be a bit misleading, as ratings after a match would be close to the rating before the match, which we use in our calculations anyway. It would be more informative to take a look at the change in the ratings compared to the predicted change in the ratings. https://preview.redd.it/ls3xc45qmy061.png?width=487&format=png&auto=webp&s=4b7f8e23822c46227fecc40ef8209b92edec76b7 So this is still a good fit. In fact, this algorithm can explain nearly 92% of the variance in the change in official ratings after a test match. Is that good? I'll leave that for you to decide. In theory it should be possible to get it pretty close to 100% as we're trying to predict a process which is itself driven by an algorithm and completely non-random. Still I think this shows we have an algorithm who's results tend to line-up pretty well with those of the official ratings, and I think it was not too bad for a first try. Where do the uncertainties lie?I think the biggest uncertainties are in that we don't really know what sort of function the Quality, MPF and IPF variables follow, and it seems impossible to ever know that with certainty. Similarly, there are a lot of parameters to be determined. There were at least 4 that were determined here and hey are all linked together in complicated ways its impossible to take one in isolation and determine its value. Even more parameters were taken as given and could've been changed since the book came out. The nonlinear weights for each factor as well as the decay constant were examples. If I had not considered them fixed I don't think I would've had enough data to confidently determine every parameter. So next time more data and more sophisticated parameter estimation techniques would be required. What next?The first thing I wanna do with this is to forecast the changes in ratings after each test in India's tour of Australia. That way I can test if it actually works on new data it hasn't seen before, or if its complete junk. Also, now that we have a similar process for determining rankings as that used in test. We could use it to make our own batting rankings for first class competitions. I think that would be really cool and interesting, if say we had a complete rankings table for the County Championship The obvious next step is to work out the bowlers ratings, but they are even more hideous than this algorithm, so I'll leave it a bit for now. Would be interesting to come back to some time in the future though. If someone who actually knows what they're can pick this apart or point out a flaw in what I've done, I'd love to hear from you. I'm genuinely curious as to how someone would go about doing this sort of thing, and I'd love to learn more (even if it necessitates telling me this is complete garbage)! If you made it this far thanks for taking the time to read this! |

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