Performance Evaluation and Ranking of Drivers in Multiple Motorsports Using Massey’s Method

Presented in Mathsport International 2025 (https://math.uni.lu/midas/events/mathsports2025/)

Abstract:
In four-wheeled motorsports, various championships, such as Formula 1 (F1), the World Endurance Championship (WEC), and Super GT, are organized globally. While these championships have significant differences in vehicles used and the regulations applied, they share the common characteristic of employing four-wheeled cars. These characteristics allow drivers to compete within the same championship, participate in multiple series simultaneously, or transfer to entirely different championships. Furthermore, participation in F1 requires a Super License, and some championships outside of F1 award Super License points to drivers for obtaining this license. Despite these connections and hierarchical relationships between championships -such as drivers participating across different championships and the Super License system itself- there is no official ranking system for drivers across multiple championships. In addition, the authors' investigation found no existing studies that use validated rating methods across various championships to evaluate the performance of drivers.

This study aims to develop a method for quantitatively evaluating the achievements of all drivers who have participated in multiple championships, using Massey's rating method, a well-known quantitative performance evaluation approach. As a result, the study tries to establish a unified ranking system of drivers in a wide range of four-wheeled motorsports championships.

In order to assess the performance of each driver, it is necessary to compare the results of individual drivers and compute evaluation values by using drivers who have participated in multiple championships as a reference point. For this purpose, the evaluation methodology is based on Massey's rating method. This rating method is commonly applied in sports where two competitors compete against each other for scores. It assumes that the difference in the rating values between players explains the score difference of one match, and then estimates the rating value of every player based on the match results using the least squares method.

In this study, Massey’s method is extended to ranking-based race competitions by replacing the score difference with the logarithmic difference in race positions. The resulting ratings can be interpreted as performance evaluation values for the drivers.

Data has been collected from championships involving formula cars in Europe, Japan, and the United States to calculate the performance evaluation values. Eight series were analyzed: Formula 1, Formula 2, Formula 3, Formula E, Super Formula, Super Formula Lights, IndyCar, and IndyCar Lights. The data collection spanned three years, from 2021 to 2023, encompassing 275 drivers.

Using this method, we calculated the performance evaluation values of drivers participating in these championships. The results revealed performance evaluation values that reflect hierarchical relationships between championships. Additionally, in championships such as F1, where competition in vehicle development plays a significant role and dominance tends to persist, the drivers with consecutive victories were to have exceptionally high-performance evaluation values.
Future work is focusing on analyzing the transitions in performance evaluation values for individual drivers to validate the predictive accuracy.

Transcript

1. Performance Evaluation and Ranking of Drivers in Multiple Motorsports Using Massey’s Method RYOGA YAMAGUCHI*, EIJI KONAKA (MEIJO UNIVERSITY, JAPAN) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
2. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
3. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
4. Motorsport: Classification by rules ⚫Three types of rules for motorsports MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. B) Ref. C) Formula 1 Mass start, Ranking Rally Separate start, Time Drift Single or dual, Appeal to judge
5. Motorsport: Classification by vehicles ⚫Types of vehicle MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Formula Car Ref. D) Ref. E) Ref. F) Prototype Car Touring Car
6. Research target of this study ⚫Championships of Formula cars ⚫Mass start ⚫Compete for the top spot in the race ⚫Objective: Performance evaluation of drivers ⚫Difficulties: Different cars, circuits, regulations MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
7. Championships around the world ⚫Championships using formula cars are held all over the world. ⚫Research target: 8 championships around the world ⚫Region: Worldwide (mainly in Europe), Japan, and USA ⚫Feeder series structure: e.g.: Formula 1-2-3 MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Formula E Formula E Super Formula Super Formula Super Formula Lights Super Formula Lights Indy Car Indy Car Indy Lights Indy Lights Formula 1 Formula 1 Formula 2 Formula 2 Formula 3 Formula 3 Europe (Worldwide) Japan USA
8. Structure of championships and driver move ⚫“Feeder series” structure ⚫Formula1-2-3 ⚫Higher-ranked driver “steps up” to the upper championship ⚫Super License (next slide) ⚫Driver move crossing the sea ⚫E.g., Formula 2->Super Formula MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Formula 1 Formula 1 Formula 2 Formula 2 Formula 3 Formula 3 Super Formula Super Formula Super Formula Lights Super Formula Lights
9. Super License to Formula1 ⚫Super License is required to participate in Formula 1 championship ⚫Drivers can obtain Super License Points in the championships (except Formula 1) ⚫Point Attribution table (right) ⚫The thoughts of FIA might be reflected. ⚫(F3 is more competitive than SF, F2 is more competitive than IndyCar, …) 1st 2nd 3rd … Formula 2 40 40 40 Formula 3 30 25 20 Formula E 30 25 20 Super Formula 25 20 15 IndyCar 40 30 20 MATHSPORT INTERNATIONAL 2025@LUXEMBOURG * FIA (Fédération Internationale de l'Automobile)
10. Driver moves crossing the sea ⚫Driver moves between championships ⚫2021-2023(Left figure) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG US EU JP
11. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
12. Driver ranking for enthusiasts, team owners, and drivers themselves HOW MUCH SUCCESS DID HE ACHIEVE? ⚫It is hard to quantitatively evaluate each driver’s performance ⚫Different rules, vehicles, circuits, … ⚫Each driver experiences different pathway ⚫No official unified criteria MAIN OBJECTIVE: UNIFIED RANKING SYSTEM ⚫ It will be useful for enthusiasts, team owners, and drivers themselves. MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Propose an unified ranking system among multiple motorsport championships
13. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
14. Basic idea of the proposed method ⚫How to compare two different championships ⚫Drivers moved ⚫Adjust drivers based on the result of the moved driver ⚫Make unified ranking ⚫Proposed method: an unified ranking for multiple seasons MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
15. Mathematical background: Massey’s rating WHAT IS MASSEY’S RATING? ⚫A simple rating method ⚫Paired comparison model ⚫Rating difference explains score difference ⚫Equation: DEFINITION ⚫𝑟𝑖 , 𝑟 𝑗 : Rating value of players 𝑖, 𝑗 ⚫𝑠𝑘 : Score difference of a match 𝑘 ⚫𝜖𝑘 : Estimation error MATHSPORT INTERNATIONAL 2025@LUXEMBOURG 𝑟𝑖 − 𝑟𝑗 + 𝜖𝑘 = 𝑠𝑘
16. Massey’s rating: numerical example(1) DEFINITION ⚫𝑟𝑖 , 𝑟 𝑗 : Rating value of players 𝑖, 𝑗 ⚫𝑠𝑘 : Score difference of a match 𝑘 ⚫𝜖𝑘 : Estimation error EXAMPLE 𝒊 𝒋 𝒔𝒊 𝒔𝒋 𝒔𝒌 1 2 5 0 5 1 2 4 1 3 2 3 3 2 1 2 3 1 2 -1 3 1 0 1 -1 3 1 4 2 2 MATHSPORT INTERNATIONAL 2025@LUXEMBOURG 𝑟𝑖 − 𝑟𝑗 + 𝜖𝑘 = 𝑠𝑘
17. 𝑟1 − 𝑟2 = 5 − 𝜖1 𝑟1 − 𝑟2 = 3 − 𝜖2 𝑟2 − 𝑟3 = 1 − 𝜖3 𝑟2 − 𝑟3 = −1 − 𝜖4 𝑟3 − 𝑟1 = −1 − 𝜖5 𝑟3 − 𝑟1 = 2 − 𝜖2 Massey’s rating: numerical example(2) 𝒊 𝒋 𝒔𝒊 𝒔𝒋 𝒔𝒌 1 2 5 0 5 1 2 4 1 3 2 3 3 2 1 2 3 1 2 -1 3 1 0 1 -1 3 1 4 2 2 MATHSPORT INTERNATIONAL 2025@LUXEMBOURG 𝑟𝑖 − 𝑟𝑗 = 𝑠𝑘 −𝜖𝑘 simultaneous linear equations
18. 1 −1 0 1 −1 0 0 1 −1 0 1 −1 −1 0 1 −1 0 1 𝑟1 𝑟2 𝑟3 = 5 3 1 −1 −1 2 − 𝜖1 𝜖2 𝜖3 𝜖4 𝜖5 𝜖6 𝑟1 − 𝑟2 = 5 − 𝜖1 𝑟1 − 𝑟2 = 3 − 𝜖2 𝑟2 − 𝑟3 = 1 − 𝜖3 𝑟2 − 𝑟3 = −1 − 𝜖4 𝑟3 − 𝑟1 = −1 − 𝜖5 𝑟3 − 𝑟1 = 2 − 𝜖2 Massey’s rating: numerical example(3) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG simultaneous linear equations Matrix-vector form
19. 1 −1 0 1 −1 0 0 1 −1 0 1 −1 −1 0 1 −1 0 1 𝑟1 𝑟2 𝑟3 = 5 3 1 −1 −1 2 − 𝜖1 𝜖2 𝜖3 𝜖4 𝜖5 𝜖6 Massey’s rating: numerical example(4) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Matrix-vector form 𝑿𝒓 = 𝒔 − 𝝐 Find 𝒓 That Minimizes 𝝐𝑇𝝐 𝒓 = 𝑿𝑇𝑿 −1𝑿𝒔 Least Mean-Square Method
20. Expansion of Massey’s method EXPANSION OF MASSEY’S METHOD Massey’s method Proposed Match One-on- one Mass start Difference Score difference (Modified) Ranking difference HOW TO EVALUATE RANKING DIFFERENCE? MATHSPORT INTERNATIONAL 2025@LUXEMBOURG ⚫𝑅𝑖 , 𝑅𝑗 : Ranking of drivers 𝑖, 𝑗. ⚫Weight more for top rankings ⚫− log 1 2 = 0.69, − log 19 20 = 0.05 𝑠𝑘 = − log𝑒 𝑅𝑖 𝑅𝑗
21. Proposed rating: numerical example Rank Driver 1 B 2 A 3 D 4 C MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Expand all one-on-one pair 𝒊 𝒋 𝒔𝒊 𝒔𝒋 𝒔𝒌 B A 0 0.69 −0.69 B D 0 1.10 −1.10 B C 0 1.39 −1.39 A D 0.69 1.10 −0.41 A C 0.69 1.39 −0.69 D C 1.10 1.39 −0.29 ⚫ The same procedure follows as Massey’s method
22. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
23. Data MATHSPORT INTERNATIONAL 2025@LUXEMBOURG FE FE F1 F1 F2 F2 F3 F3 EU JP SF SF SFL SFL US Indy Indy IndyL IndyL Championship Abb. Rounds Drivers Formula 1 F1 66 29 Formula 2 F2 71 48 Formula 3 F3 58 79 Formula E FE 47 33 Super Formula SF 26 32 Super Formula Lights SFL 53 32 Indy Car Indy 50 55 Indy Lights IndyL 34 25 TOTAL 405 275 ⚫ 3 Seasons (2021 to 2023)
24. Driver moves crossing the sea ⚫Driver moves between championships ⚫2021-2023(Left figure) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG US EU JP
25. Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)
26. Main result ⚫Unified drivers’ ranking can be calculated MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
27. Main result ⚫Unified drivers’ ranking can be calculated ⚫Hierarchical structure between feeder series could be found MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
28. Main result ⚫Unified drivers’ ranking can be calculated ⚫Hierarchical structure between feeder series could be found ⚫We can compare championships in different continents ⚫F2≃Indy>SF MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
29. Discussion ⚫We now can compare championships in different continents. ⚫Inconsistency with the Super License MATHSPORT INTERNATIONAL 2025@LUXEMBOURG 1st 2nd 3rd … F2 40 40 40 F3 30 25 20 FE 30 25 20 SF 25 20 15 Indy 40 30 20
30. Discussion ⚫We now can compare championships in different continents ⚫Inconsistency with the Super License MATHSPORT INTERNATIONAL 2025@LUXEMBOURG 1st 2nd 3rd … F2 40 40 40 F3 30 25 20 FE 30 25 20 SF 25 20 15 Indy 40 30 20
31. Discussion ⚫Inconsistency with the Super License ⚫Career direction, age effect ⚫F1->FE ⚫Main aim of Super License Points ⚫≠Ranking all drivers ⚫Open problem: Which is more correct ranking? MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
32. Summary and future work SUMMARY ⚫Objective: Unified drivers’ ranking in multiple motorsport ⚫Method: Modified Massey’s method ⚫Result: Rank 275 drivers participated in 8 different championships all over the world ⚫Contribution: We now can compare championships in different continents. FUTURE WORK ⚫Prediction of future race ⚫Accuracy of ranking ⚫Considering age effect ⚫Individual survey on outliers MATHSPORT INTERNATIONAL 2025@LUXEMBOURG
33. References A) Red Bull Racing's Sergio Perez, Ferrari's Charles Leclerc, Mercedes' Lewis Hamilton, Alpine's Fernando Alonso and McLaren's Lando Norris battling over the final podium place at the 2022 British Grand Prix at Silverstone., CC BY-SA 2.0, https://commons.wikimedia.org/w/index.p hp?curid=120717939 B) By Harpagornis - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.p hp?curid=48987265 C) https://formulad.jp/news/20241015-01/ D) https://global.honda/jp/F1/2025/shared/i mg/eb21_1.jpg E) https://www.astonmartin.com/ja/experien ces/le-mans F) https://press.jp.porsche.com/prod/presse_ pag/PressResources.nsf/Content?ReadFor m&languageversionid=685746 MATHSPORT INTERNATIONAL 2025@LUXEMBOURG