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

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Performance Evaluation and Ranking of Drivers in Multiple Motorsports Using Massey’s Method RYOGA YAMAGUCHI*, EIJI KONAKA (MEIJO UNIVERSITY, JAPAN) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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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

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Motorsport: Classification by vehicles ⚫Types of vehicle MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Formula Car Ref. D) Ref. E) Ref. F) Prototype Car Touring Car

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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

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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

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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

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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)

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Driver moves crossing the sea ⚫Driver moves between championships ⚫2021-2023(Left figure) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG US EU JP

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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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

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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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

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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

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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

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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

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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

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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

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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

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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

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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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)

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Driver moves crossing the sea ⚫Driver moves between championships ⚫2021-2023(Left figure) MATHSPORT INTERNATIONAL 2025@LUXEMBOURG US EU JP

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Outline 1. Research target: Motorsport 2. Objective 3. Proposed method 4. Data 5. Result and discussion MATHSPORT INTERNATIONAL 2025@LUXEMBOURG Ref. A)

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Main result ⚫Unified drivers’ ranking can be calculated MATHSPORT INTERNATIONAL 2025@LUXEMBOURG

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Main result ⚫Unified drivers’ ranking can be calculated ⚫Hierarchical structure between feeder series could be found MATHSPORT INTERNATIONAL 2025@LUXEMBOURG

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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

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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

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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

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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

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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

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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