Triple Flips. How Figure Skating Determines its Winners. Michael Stob. September 27, First-Year Seminar

Transcription

1 Triple Flips How Figure Skating Determines its Winners Michael Stob First-Year Seminar September 27, 2006

2 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

3 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

4 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

5 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

6 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

7 Outline Figure skating competitions The traditional method The problem The OBO method The mathematics Concluding Remarks

8 Figure Skating Competitions The Four Disciplines Men Ladies Pairs Ice Dancing

9 Components of a Competition The Men, Ladies and Pairs Qualifying Round (possibly) Short Program Long Program Ice Dancing 2 Compulsory Dances Original Dance Free Dance

10 Example 1999 European Championships Ice Dancing, Free Dance Program dance program of at least four minutes music has an audible, rhythmic beat includes lifts (5 to 7), spins (3 to 5) and other dance moves Judging nine judges two marks on scale of in steps of 0.1 first mark: technical merit second mark: presentation

11 Example 1999 European Championships Ice Dancing, Free Dance Program dance program of at least four minutes music has an audible, rhythmic beat includes lifts (5 to 7), spins (3 to 5) and other dance moves Judging nine judges two marks on scale of in steps of 0.1 first mark: technical merit second mark: presentation

12 The Judge s Scores WINKLER/ LOHSE GRUSHINA/ GONCHAROV KRYLOVA/ OVSYANNIKOV FUSAR-POLI/ MARGAGLIO ANISSINA/ PEIZERAT LOBACHEVA/ AVERBUKH DROBIAZKO/ VANAGAS

13 The Judge s Scores WINKLER/ LOHSE GRUSHINA/ GONCHAROV KRYLOVA/ OVSYANNIKOV FUSAR-POLI/ MARGAGLIO ANISSINA/ PEIZERAT LOBACHEVA/ AVERBUKH DROBIAZKO/ VANAGAS

14 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

15 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

16 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

17 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

18 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

19 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

20 The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse Grushina/ Goncharov Krylova/ Ovsyannikov Fusar-Poli/ Margaglio Anissina/ Peizerat Lobacheva/ Averbukh Drobiazko/ Vanagas

21 Judge s Ordinals to Places Couple Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

22 Judge s Ordinals to Places Couple OM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

23 Judge s Ordinals to Places Couple OM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

24 Judge s Ordinals to Places Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

25 Judge s Ordinals to Places Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

26 Judge s Ordinals to Places Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Order of Finish 1. Anissina/Peizerat 2. Krylova/Ovsyannikov 3. Lobacheva/Averbukh

27 The Competition Unfolds Winkler/Lohse skate first:

28 After Winkler/Lohse Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

29 Then Grushina and Goncharov

30 After Grushina and Goncharov Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Winkler/Lohse 2. Grushina/Goncharov 3.

31 After Grushina and Goncharov Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Winkler/Lohse 2. Grushina/Goncharov 3.

32 Eventually, Anissina and Peizerat skate

33 After Anissina and Peizerat Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Krylova/Ovsyannikov 2. Anissina/Peizerat 3. Fusar-Poli/Margaglio

34 After Anissina and Peizerat Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Krylova/Ovsyannikov 2. Anissina/Peizerat 3. Fusar-Poli/Margaglio

35 And then Lobacheva and Averbukh skate

36 After Lobacheva and Averbukh Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Anissina/Peizerat 2. Krylova/Ovsyannikov 3. Lobacheva/Averbukh

37 After Lobacheva and Averbukh Couple OM JM Winkler/Lohse Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Anissina/Peizerat 2. Krylova/Ovsyannikov 3. Lobacheva/Averbukh

38 The Flip-Flop Before Lobacheva and Averbukh Skate 1. Krylova/Ovsyannikov 2. Anissina/Peizerat 3. Fusar-Poli/Margaglio After Lobacheva and Averbukh Skate 1. Anissina/Peizerat 2. Krylova/Ovsyannikov 3. Lobacheva/Averbukh

39 Flip-Flops are Bad If one skater is in front of another, he should remain there. Ottavio Cinquanta President of the International Skating Union

40 The OBO (One-by-One) Method Key Idea Treat the competition as a series of head-to-head matches

41 Krylova/Ovsyanikov versus Lobacheva/Averbukh Krylova/Ovsyannikov Lobacheva/Averbukh Krylova and Ovsyannikov have a WIN over Lobacheva and Averbuch with 6 Judges in Favor (JIF).

42 The Wins and JIF Table WL GG KO FM AP LA DV W JIF Win/Loh Gru/Gon Kry/Ovs Fus/Mar Ani/Pei Lob/Ave Dro/Van Order of Finish 1. Krylova/Ovsyannikov (6) 2. Anissina/Peizerat (5) 3. Lobacheva/Averbukh (4)

43 The Wins and JIF Table WL GG KO FM AP LA DV W JIF Win/Loh Gru/Gon Kry/Ovs Fus/Mar Ani/Pei Lob/Ave Dro/Van Order of Finish 1. Krylova/Ovsyannikov (6) 2. Anissina/Peizerat (5) 3. Lobacheva/Averbukh (4)

44 Observations and Questions Observations on the example OBO does not result in a flip OBO and the Traditional Method choose different winners There are reasonable arguments for choosing each winner Questions about OBO Does OBO eliminate flips? Is OBO a better method?

45 Observations and Questions Observations on the example OBO does not result in a flip OBO and the Traditional Method choose different winners There are reasonable arguments for choosing each winner Questions about OBO Does OBO eliminate flips? Is OBO a better method?

46 Social Choice Functions Social choice function A social choice function F has n inputs O 1, O 2,..., O n each of which is an ordering of m objects. The output F(O 1,..., O n ) of F is also an ordering of those same m objects. The ice dance example n = 9 (judges) m = 7 (pairs) O i is the ordering of the skaters of the i th judge F(O 1,..., O n ) is the final ranking of the skaters

47 Social Choice Functions Social choice function A social choice function F has n inputs O 1, O 2,..., O n each of which is an ordering of m objects. The output F(O 1,..., O n ) of F is also an ordering of those same m objects. The ice dance example n = 9 (judges) m = 7 (pairs) O i is the ordering of the skaters of the i th judge F(O 1,..., O n ) is the final ranking of the skaters

48 Marquis de Condorcet Wrote extensively on the integral calculus Essay on the Applications of Analysis to the Probability of Majority Decisions, 1785 Executed during the French Revolution

49 Desirable Properties of a Skating Ranking Method Property (Winning Majority) If skater A is ranked first by a majority of the judges, the method ranks skater A first. Property (No Flips) If the method is applied sequentially as the skaters skate, then if one skater is in front of another, he should remain there. Property (Anonymity of Skaters) If the marks of two skaters are exchanged, then the method exchanges their ranks.

50 Desirable Properties of a Skating Ranking Method Property (Winning Majority) If skater A is ranked first by a majority of the judges, the method ranks skater A first. Property (No Flips) If the method is applied sequentially as the skaters skate, then if one skater is in front of another, he should remain there. Property (Anonymity of Skaters) If the marks of two skaters are exchanged, then the method exchanges their ranks.

51 Desirable Properties of a Skating Ranking Method Property (Winning Majority) If skater A is ranked first by a majority of the judges, the method ranks skater A first. Property (No Flips) If the method is applied sequentially as the skaters skate, then if one skater is in front of another, he should remain there. Property (Anonymity of Skaters) If the marks of two skaters are exchanged, then the method exchanges their ranks.

52 Stephanie Baar Potoka

53 Baar s Impossibility Theorem Theorem There is no method that satisfies Winning Majority, No Flips, and Anonymity of Skaters (if there are more than two skaters). Corollary Flips are possible in OBO. Corollary No acceptable method of ranking skaters will avoid flips.

54 Baar s Impossibility Theorem Theorem There is no method that satisfies Winning Majority, No Flips, and Anonymity of Skaters (if there are more than two skaters). Corollary Flips are possible in OBO. Corollary No acceptable method of ranking skaters will avoid flips.

55 Baar s Impossibility Theorem Theorem There is no method that satisfies Winning Majority, No Flips, and Anonymity of Skaters (if there are more than two skaters). Corollary Flips are possible in OBO. Corollary No acceptable method of ranking skaters will avoid flips.

56 Kenneth Arrow Ken Arrow of Princeton University won the Nobel Prize in Economics in 1972 partly for his work on the problem of social choice

57 Arrow s Properties Property (Unanimity) If skater A is placed ahead of skater B by every judge, then the method should place skater A ahead of skater B. Property (Independence of Irrelevant Alternatives) The ranking of skaters A and B depends only on the relative ranking of skaters A and B by the judges. Property (No Dictator) There is no one judge J such that the method always produces the same ordering as that of judge J.

58 Arrow s Theorem Theorem (Arrow s Theorem) No method satisfies each of No Dictator, Independence of Irrelevant Alternatives, and Unanimity.

59 Is OBO Better? Question Should the ISU have changed from the traditional method to the OBO method? OBO does not eliminate flips. Does OBO reduce the number of flips? Does OBO choose the right winner?

60 OBO results in fewer flips Historical results In a study of a large set of past competitions, Baar found that OBO averaged 2.6 flips per competition the traditional method averaged 5.6 flips per competition

61 Baar s Program Question Why do judges disagree? measurement error systematic differences (bias) strategic considerations

62 Baar s Model Model Assumptions Suppose that each skater has a true performance Judges differ only because of random measurement error Question Which method chooses the best skater most often?

63 Baar s simulation studies OBO slightly outperforms the traditional method in choosing the correct winner and in choosing the top three skaters for a wide variety of assumptions of distributions of skater performance and judge error

64 Biased judges The method can be extended to biased and block judging For a wide variety of assumptions on bias and blocks, OBO again outperforms the traditional method (with more significant differences)

65 Baar s Conclusion OBO is better OBO is more likely to pick the right winner under a wide variety of assumptions on distribution of skaters and the nature of the judging. OBO can be improved substantially by changing its tie-breaking method.

66 Yet Another Change In 2004, the ISU changed the method again to Code of Points. The system is not based on ordinals. The skaters receive points for each skating element that they perform as well as some overall scores.

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68 There is a great undergraduate research project here.

69 The Moral(s) of the Story 1. Mathematics can be applied to almost anything. 2. Undergraduates can make new mathematics.

70 The Moral(s) of the Story 1. Mathematics can be applied to almost anything. 2. Undergraduates can make new mathematics.