Logarithmic ranking system
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6th December 2011, 17:27 at 5:27 pm #130569
This morning’s roundup got me thinking about the perfect* ranking system for F1.
It should award points to every finisher, so that even the battle for 17th vs 18th makes a difference. But ultimately, a win should still be worth more a lot more than any other finish; and coming 1st & 3rd should be better than coming 2nd twice.
(mathematicians have a snappy word for rankings where “coming 1st & 3rd is better than coming 2nd twice.” They’re called convex.)
At this point I started wondering, how much better? And is 1st & 4th better than two 2nd places?
Then I realised the fairest way of doing it. This season we had 24 starters each weekend (even if they didn’t all make the race). Thus the chance of winning was 24:1. The chance of coming in the top 2 is 24:2, or 12:1. The chance of coming in the top three is 24:3, or 8:1. And there are your points.
But! We wouldn’t just add these points up. You see the chance of winning two races in a row is 24:1 squared, or 576:1.
Actually, this is a bit of a problem: nobody wants to do long multiplication to find out people’s scores. And Vettel’s score this season would be over 18 billion trillion!
No problem – mathematicians have a trick for that too. They’re called logarithms. Make the score for nth place be log(nth/24), and then we can just add the scores up at the end of the season.
Here’s the table I ended up with:
{Position} {Points} {Equivalent to} {1} {3.18} {25} {2} {2.48} {20} {3} {2.08} {16} {4} {1.79} {14} {5} {1.57} {12} {6} {1.39} {11} {7} {1.23} {10} {8} {1.10} {9} {9} {0.98} {8} {10} {0.88} {7} {11} {0.78} {6} {12} {0.69} {5} {13} {0.61} {5} {14} {0.54} {4} {15} {0.47} {4} {16} {0.41} {4} {17} {0.34} {3} {18} {0.29} {2} {19} {0.23} {2} {20} {0.18} {1} {21} {0.13} {1} {22} {0.09} {1} {23} {0.04} {0} {24} {0} {0} {DNF} {0} {0} {Position} {Driver} {Points} {1} {Sebastian Vettel} {51} {2} {Jenson Button} {38} {3} {Fernando Alonso} {37} {4} {Mark Webber} {37} {5} {Lewis Hamilton} {33} {6} {Felipe Massa} {22} {7} {Nico Rosberg} {20} {8} {Michale Schumacher} {16} {9} {Adrian Sutil} {16} {10} {Paul di Resta} {15} {11} {Vitaly Petrov} {14} {12} {Kamui Kobayashi} {13} {13} {Jaime Alguersuari} {12} {14} {Sebastian Buemi} {11} {15} {Sergio Perez} {10} {16} {Rubens Barrichello} {10} {17} {Nick Heidfeld} {9} {18} {Pastor Maldonado} {7} {19} {Heikki Kovalainen} {6} {20} {Jarno Trulli} {5} {21} {Jerome d’Ambrosio} {5} {22} {Bruno Senna} {5} {23} {Timo Glock} {4} {24} {Viantonio Liuzzi} {2} {25} {Daniel Ricciardo} {2} {26} {Narain Karthikeyan} {1} {27} {Pedro de la Rosa} {1} {28} {Karun Chandhok} {0} So, what do you think? Do you think the rankings shown here are fairer than the ones we have at the moment? A lot of drivers have changed rankings, and I imagine so have the teams. Does this give a fairer picture of the performance of the backmarkers?
*sadly, this isn’t it. It’s a little bit complicated.
6th December 2011, 17:43 at 5:43 pm #186553I like the concept, but how about scaling the points up?
If you multiplied the points by 8, not only would you they be on the same scale as the current system, but you could do away with the decimal point and leave it as an integer.
So: 25, 20, 17, 14, 13, 11, 10, 9, 8, 7, 6, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1, 1, 0, 0
True, when you get near the bottom you start getting the same number of points for two different positions. BUT, it’s still better (as you point out) than having 0 points for positions 1124.
6th December 2011, 17:46 at 5:46 pm #186554Um – apologies – I seem unable to either add a table properly, or edit my posts. That makes this all rather illegible.
I’ve put a copy of my spreadsheet onto Google Docs, in case anyone cares to see what I was talking about:
https://docs.google.com/spreadsheet/ccc?key=0AorIV213RTSEdFRiMDdTU0pSNF95ai1Td282eVVsa3c
6th December 2011, 18:27 at 6:27 pm #186555That’s definitely interesting, but (if I get it correctly) it awards a lot of points for finishing in the same position many times? Di Resta in front of Petrov doesn’t make sense.
6th December 2011, 18:34 at 6:34 pm #186556Honestly, I think the scoring system is fine as it is.
6th December 2011, 18:49 at 6:49 pm #186557Comparing Di Resta and Petrov, down to their 13th place finishes:
Petrov is up 3rd, one 5th, and two 9ths vs Di Resta’s 6th, 7th, 8th, 10th, 12th and 13th.
My system has said the chance of coming in the top 3, 5, 9 and 9 is (3/24)*(5/24)*(9/24)*(9/24) = 1 in 273.
The chance of coming in the top 6, 7, 8, 10, 12 & 13 is (6/24)*…*(13/24) = 1 in 365
Thus it thinks Di Resta’s achievement is more impressive.The question is, do we agree? This was a good spot because it’s probably one of the more contentious ones (whereas most people would probably be happy that Alonso is ahead of Webber, and Kovalainen is ahead of Trulli).
I think the system might be wrong here. It’s basically saying that coming twelth twice is the same as coming sixth once: (12/24)*(12/24) = (1/2)*(1/2) = (1/4) = (6/24). In retrospect that’s a poor weighting, but I’m not sure what to do about it.
Any suggestions? With the spreadsheet I’ve got set up, it would be easy enough to add another row of scores, and another column of total points. So we can compare as many systems as we want, here.
6th December 2011, 18:54 at 6:54 pm #186558
AnonymousHonestly, I agree with @magnificentgeoffrey . It hurt my head reading that. :P
6th December 2011, 19:01 at 7:01 pm #186559This isn’t a crusade to change the rules – I just find it interesting to think about how they could be different. If you don’t find it interesting, that’s fine too!
And besides, this does give us a better way of comparing the midfield and lower runners – say, from Perez downwards (which is half the drivers!). So there’s interest there, in seeing how the new teams change over time.
6th December 2011, 20:53 at 8:53 pm #186560Or:
With a DNF = 0
24th = 1
every place up the order is multiplied by 2
so 23rd = 2, 22nd = 4 etc so the winner gets 2^23 (8388608 points!)
Make every place worth doubleOr alternatively, DNF = 0
24th =1^2
23rd = 2^2
so 1st gets 24^2 = 576 pts
Seen as the current system makes comparing points ridiculous now, why not make a real incentive for more points!6th December 2011, 21:17 at 9:17 pm #186561Nice bit of work and a good way of comparing the drivers a bit better than the current system. I also like the idea of evaluating every battle for position, not just in the event of a tiebreak – e.g. Glock v Liuzzi is much fairer this way.
6th December 2011, 22:27 at 10:27 pm #186562What about no points at all and just highest finishing position, the same thing that decides which of the new teams is ahead? Thanks to your spreadsheet, here’s what I get:
1. Vettel 11 wins
2. Button 3 wins, 4 2nds
3. Hamilton 3 wins, 3 2nds
4. Alonso 1 win, 5 2nds
5. Webber 1 win, 2 2nds
6. Petrov 1 3rd, 1 5ths
7. Heidfeld 1 3rd
8. Schumacher 1 4th
9. Massa 6 5ths, 5 6ths
10. Rosberg 2 5ths, 4 6ths
11. Kobayashi 1 5th
12. Sutil 2 6ths
13. Di Resta 1 6th
14. Alguersuari 2 7ths
15. Perez 1 7th
16. Buemi 2 8ths
17. Barrichello 2 9ths
18. Senna 1 9th
19. Maldonado 1 10th
20. De la Rosa 1 12th
21. Trulli 2 13ths
22. Kovalainen 1 13th, 3 14ths
23. Liuzzi 1 13th
24. D’Ambrosio 2 14ths
25. Glock 2 15ths
26. Karthikeyan 3 17ths
27. Ricciardo 2 18ths
28. Chandhok 1 20th1. Red Bull 12 wins
2. McLaren 6 wins
3. Ferrari 1 win
4. Renault 2 3rds
5. Mercedes 1 4th
6. Sauber 1 5th
7. Force India 3 6ths
8. Toro Rosso 2 7ths
9. Williams 2 9ths
10. Lotus 3 13ths
11. HRT 1 13th
12. Virgin 2 14thsI find it quite interesting. Some finds:
Vettel is of course 1st regardless of the system. Hamilton gets in front of Webber and Alonso because of his 3 wins, as consistency doesn’t matter here. Alonso is also in front of Webber thanks to more 2nd places.
Petrov is up to 6th from 10th, and Heidfeld 7th from 11th, which is because of their earlier podiums. Rosberg drops to 10th as he hasn’t finished a race in the top 4, Massa to 9th for the same reason. Same goes for Sutil as well.
Renault is in front of Mercedes in the constructors’ standings, and Sauber in front of Force India.
6th December 2011, 22:52 at 10:52 pm #186563Basically any scoring system is a comprimise between “what was your best result?” and “how consistent were you?”
At one end is the system you just described, and the other is to simply take average finishing position over the season (with any retirements counting as 24th/25th) – the system Ajokay set out in his comment.
Both systems ‘feel’ wrong, in that Petrov doesn’t seem to ‘deserve’ to be ahead of the Mercedes cars (as in ‘best finish’), but the average system doesn’t do enough to reward risks.
The question then is psychological: what are these feelings based on? Can we make something which ‘feels’ right? And can we really lay down a formula which says, e.g. ‘one win is worth the same as two 3rd places?’
I also wonder how much this shapes our opinions of the drivers. If the FIA used this system, would we be talking about how Schumacher had ‘made good’ on his comeback, and beaten his teammate by two places? Would we be saying that Sutil was deadweight and essentially tied with Di Resta? Or that Petrov was one of the best drivers around, hauling his Renault above both Mercedes and a Ferrari?
I do wonder just how much of our opinions about the drivers is influenced by that little number next to their names in the standings. That’s one of the reasons I like looking more closely at the numbers: it reminds me of the stories behind them, and how different things could look if we approached from a different angle.
7th December 2011, 0:40 at 12:40 am #186564
Anonymous@alex I wasn’t trying to be derogatory towards you in my previous comment there. It is quite an interesting idea and it would give some different results. But can you imagine if this *was* the scoring system, hypothetically speaking course. F1 is can be complicated enough as it is. (I’ve been watching since ’94 and still don’t get a lot of stuff that goes on, to be quite honest!). A newcomer to the sport could see this scoring system and could be put off by it, let alone all the other complicated regulations that F1 has.
Scoring should be kept simple. It’s worked for years with no problem.
7th December 2011, 3:39 at 3:39 am #186565Good Lord. Fantastic idea. Absolutely fantastic!
7th December 2011, 5:23 at 5:23 am #186566Good idea. The current points system is definitely too linear and blocky. There are big chunks of points differences between various positions that don’t make sense.
Not only should the point system be modified, but one should also gain points from qualifying.
In addition, some bonus points could be awarded during the race. For example, on track passes of leading cars that are racing for position should give a driver some bonus points (say 1 or 2 for each pass). This gives all drivers a chance to gain points. Positions gained when drivers crash out (or competitors’ car fails) give no points. Fastest pit stop of the race should give the “team” some bonus constructor points. Fastest lap of the race could give driver bonus points. So there should be several different point structures that when combined gives drivers and teams the total points gained during the overall grand prix.

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