THE CHAMPIONSHIP POINTS SYSTEM A mathematical analysis with some surprising results.Between 1935 and 1939 the Association Internationale des Automobile Clubs Reconnus (AIACR) organized a European Drivers' championship. No one, as far as I know, has ever claimed that the results of the first four years would in any way be considered unfair, Bernd Rosemeyer being the 1936 Champion while "Rudi" Caracciola claming the other three titles. However the 1939 championship remains controversial because if using "common sense" Lang was the champion with his two victories while by following the 19351938 rules the rightful champion was in fact Müller. We will take a short look at the points system trying to explain why this happened. When the championship started in 1935 AIACR decided on a variant of something that could be called a "placing point" system. That was a curious decision to say the least and we can only assume that the system had been invented by an AIACR "bigwig" who had a "bright idea" how to do the rules, probably against the better judgement of those who had a better understanding of the troubles it would create. There are of course the terrible problems of keeping the points tables up to date as one has to know the full results for each driver including on which lap they retired on, where today you only need to know the top six positions for a race. But there are other hidden problems with the rules, far harder to recognize at a fast glance. Now, the "placing point" system where a competitor gets points equal to his position at the finish is seldom used in sports. In the Olympics for example the only competitions I know using it were the now discontinued "pentathlon" during 19121924, sailing from 1996 onwards and "modern pentathlon" during 19121952. (Note 1) In the AIACR's variant of the placing point system the drivers scored as follows:
2 points to 2nd position 3 points to 3rd position 4 points to those that completed 75 % of the race 5 points to those who completed 50% of the race 6 points to those who completed 25% of the race 7 points to those who did not complete 25% of the race 8 points to those who did not start. A first look seems to indicate that that AIACR's system especially rewards the winner. After all he scores "twice better" that of the 2nd positioned man and "three times" that of the 3rd positioned man. That would compare to something like 1264 points for the top three against the current 1086 with an extra two bonus points for the winner, right?
This is how the transformation should be done correctly: We want to change the AIACR's points table to something we are more used to, where maximum points, not minimum points, give the winner. Then let's change the rules a bit and use negative points instead!
 2 points to 2nd position  3 points to 3rd position and so on 1935: Caracciola 1 3 1 5 1 = 11 Fagioli 2 4 2 7 2 = 17 Stuck 8 2 4 1 6 = 21 . . . .Now the new table would look exactly as AIACR's tables but with a minus sign in front of each of the drivers' points. The main difference is that now the champion is the person with MOST points even if all are LESS than zero. (In 1935 Caracciola's minus 11 points is "more" points than Fagioli's minus 17 points and so on.) However, the new table is surely a bit troublesome to use for most of us. So let us make a simple mathematical trick and give an additional 8 points for EVERY single entry at EVERY single race. Naturally that would not change any order in the table as every driver gets the equal amount of bonus points. (In the 1935 season that is 5 races x 8 points = 40 bonus points for each driver.) Now this is what we get:
 2 + 8 = 6 points to 2nd position  3 + 8 = 5 points to 3rd position  4 + 8 = 4 points to those that completed 75 % of the race  5 + 8 = 3 points to those who completed 50% of the race  6 + 8 = 2 points to those who completed 25% of the race  7 + 8 = 1 points to those who did not complete 25% of the race  8 + 8 = 0 points to those who did not start The prewar rules were equivalent to:
6 points to 2nd position 5 points to 3rd position 4 points to those that completed 75 % of the race 3 points to those who completed 50% of the race 2 points to those who completed 25% of the race 1 points to those who did not complete 25% of the race EXAMPLE: 1935: Caracciola 7 5 7 3 7 = 29 (= 40 11) Fagioli 6 4 6 1 6 = 23 (= 40 17) Stuck 0 6 4 7 2 = 19 (= 40 21) . . . .Now our new way of writing the rules looks both better and worse than the old way. By adding 8 points to each score we have managed to eliminate the silly "points to non starters" from our rules and the winner is the one with most points. But the wonderful 1264 score some may have expected for the top three has suddenly changed to a miserable 765 ! But it will get even worse as we look further. One of the worst failures in the rules are the 4 points given to all those that completed 75 % of the race . Let's look at the actual results from the era. In the 22 races included in AIACR Championships between 1935 and 1939 the points were given as follows: 1 point 22 times 2 points 22 times 3 points 22 times 4 points 137 times 5 points 33 times 6 points 39 times 7 points 77 times 8 points 393 timesOr by dividing with 22 we get: 1 point was given once per race 2 points once per race 3 points once per race 4 points 6.23 times per race 5 points 1.50 times per race 6 points 1.77 times per race 7 points 3.50 times per race 8 points 17.86 times per raceTransforming those results to our new way of writing the rules we get the following table: 7 points once per race 6 points once per race 5 points once per race 4 points 6.23 times per race 3 points 1.50 times per race 2 points 1.77 times per race 1 point 3.50 times per raceand by rounding off we now get something like the following points table to use for each race: (This one giving 55 points per race against the actual 54.46 per race) 7  6  5  4  4  4  4  4  4  3  3  2  2  1  1  1 That is how the prewar point scoring system would look like in modern terms! Compare that to FIA's : 10  8  6  5  4  3  2  1 or the old 10  6  4  3  2  1 or 20  15  12  10  8  6  4  3  2  1 or to CART's 20  16  14  12  10  8  6  5  4  3  2  1 and we can start to see the real problems with the prewar championship rules. They are closely following the old rule "to finish first, first you have to finish". Victories aren't necessary decisive but a DNS or an early retirement could be devastating to a driver. On the other hand rules are rules and a clever driver would always adapt his driving to take the best advantage of them. That's why it is silly when sore losers clam Prost was some kind of "moral winner" of the 1988 championship just because he scored more "brutto" points (Note 3). Well, the old rules said points for "half of the events plus three rounded upwards" not all races and that's what Senna was adapting his driving to. Sadly the 1988 season results were enough reason for FIA to give in to a journalist mob that couldn't count 2+2 and change the rules with fatal consequences to Formula 1. The 1/2 +3 rules was a quite ingenious "safety valve" and it is no coincident it is about this time things like controversial crashes and controversial FIA disqualifications or nondisqualifications started to appear. Go to any F1 forum and you can read what kind of paranoid atmosphere it ultimately has led to. Also don't believe for a second it was just bad engineering that made cars to retire more often earlier. Cars were simply optimized to retire 5 times out of 16 in one way or another (one can argue if this was an interesting variant of Darwinism or deliberate engineering) meaning more interesting results and better racing as it also encouraged the drivers to take chances. A further sad downstep towards the prewar rules was the 2003 points inflation from 26 points per race to 39 with winner's part falling from 38.5 % to 25.6%, a natural consequense of binding points to economic interests while the cars of the top teams were optimized to finish each race and the jelously that followed that. 
FOOTNOTES:
Moder Pentathlon includs: Riding, Fencing, Shooting, Swimming and Running. 2 Including a writer of an article for Autosport in the mid 80s. 3 The 1900 Olympic Games in Paris included "fishing" as an exhibition event. The foreign teams believed they should catch the biggest fish and threw the small ones back into the sea. Result: 17 big fishes in four days while the three French teams had a triple win with their 604+264+104 small fishes. Which simply means, be sure of the rules before you start. (Source: L. Brunnhage  Olympiaboken 1968, S. Lindhagen  Olympiaboken (1961) etc.)
