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

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

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

Figure Skating Competitions The Four Disciplines Men Ladies Pairs Ice Dancing

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

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 0.0 6.0 in steps of 0.1 first mark: technical merit second mark: presentation

The Judge s Scores 1 2 3 4 5 6 7 8 9 WINKLER/ 5.4 5.3 5.2 5.0 5.3 5.3 5.3 5.4 5.3 LOHSE 5.5 5.3 5.4 5.1 5.4 5.6 5.5 5.6 5.4 GRUSHINA/ 5.3 5.2 5.3 5.1 5.3 5.1 5.2 5.4 5.3 GONCHAROV 5.4 5.4 5.5 5.0 5.2 5.4 5.4 5.4 5.3 KRYLOVA/ 5.8 5.8 5.8 5.8 5.6 5.7 5.8 5.8 5.8 OVSYANNIKOV 5.8 5.7 5.8 5.8 5.8 5.8 5.8 5.8 5.8 FUSAR-POLI/ 5.6 5.5 5.4 5.6 5.4 5.6 5.5 5.5 5.4 MARGAGLIO 5.6 5.6 5.6 5.7 5.6 5.6 5.6 5.7 5.6 ANISSINA/ 5.8 5.7 5.9 5.8 5.7 5.8 5.7 5.7 5.6 PEIZERAT 5.8 5.7 5.9 5.9 5.9 5.9 5.8 5.8 5.9 LOBACHEVA/ 5.6 5.5 5.7 5.7 5.7 5.8 5.8 5.7 5.5 AVERBUKH 5.7 5.7 5.7 5.8 5.8 5.8 5.9 5.7 5.6 DROBIAZKO/ 5.5 5.5 5.5 5.7 5.5 5.5 5.6 5.5 5.4 VANAGAS 5.6 5.5 5.6 5.6 5.6 5.6 5.7 5.6 5.5

The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse 5.0 5.1 10.1 Grushina/ Goncharov 5.1 5.0 10.1 Krylova/ Ovsyannikov 5.8 5.8 11.6 Fusar-Poli/ Margaglio 5.6 5.7 11.3 Anissina/ Peizerat 5.8 5.9 11.7 Lobacheva/ Averbukh 5.7 5.8 11.5 Drobiazko/ Vanagas 5.7 5.6 11.3

The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse 5.0 5.1 10.1 Grushina/ Goncharov 5.1 5.0 10.1 Krylova/ Ovsyannikov 5.8 5.8 11.6 Fusar-Poli/ Margaglio 5.6 5.7 11.3 Anissina/ Peizerat 5.8 5.9 11.7 1 Lobacheva/ Averbukh 5.7 5.8 11.5 Drobiazko/ Vanagas 5.7 5.6 11.3

The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse 5.0 5.1 10.1 Grushina/ Goncharov 5.1 5.0 10.1 Krylova/ Ovsyannikov 5.8 5.8 11.6 2 Fusar-Poli/ Margaglio 5.6 5.7 11.3 Anissina/ Peizerat 5.8 5.9 11.7 1 Lobacheva/ Averbukh 5.7 5.8 11.5 Drobiazko/ Vanagas 5.7 5.6 11.3

The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse 5.0 5.1 10.1 Grushina/ Goncharov 5.1 5.0 10.1 Krylova/ Ovsyannikov 5.8 5.8 11.6 2 Fusar-Poli/ Margaglio 5.6 5.7 11.3 4 Anissina/ Peizerat 5.8 5.9 11.7 1 Lobacheva/ Averbukh 5.7 5.8 11.5 3 Drobiazko/ Vanagas 5.7 5.6 11.3

The Hungarian Judge Team Tech Pres Tot Ord Winkler/ Lohse 5.0 5.1 10.1 6 Grushina/ Goncharov 5.1 5.0 10.1 7 Krylova/ Ovsyannikov 5.8 5.8 11.6 2 Fusar-Poli/ Margaglio 5.6 5.7 11.3 4 Anissina/ Peizerat 5.8 5.9 11.7 1 Lobacheva/ Averbukh 5.7 5.8 11.5 3 Drobiazko/ Vanagas 5.7 5.6 11.3 5

Judge s Ordinals to Places Couple Winkler/Lohse 6 7 7 6 6 6 6 6 6 Grushina/Goncharov 7 6 6 7 7 7 7 7 7 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 Fusar-Poli/Margaglio 4 4 5 4 5 4 5 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 Drobiazko/Vanagas 5 5 4 5 4 5 4 5 5

Judge s Ordinals to Places Couple OM Winkler/Lohse 6 7 7 6 6 6 6 6 6 6 Grushina/Goncharov 7 6 6 7 7 7 7 7 7 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 Fusar-Poli/Margaglio 4 4 5 4 5 4 5 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 Drobiazko/Vanagas 5 5 4 5 4 5 4 5 5

Judge s Ordinals to Places Couple OM Winkler/Lohse 6 7 7 6 6 6 6 6 6 6 Grushina/Goncharov 7 6 6 7 7 7 7 7 7 7 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 2 Fusar-Poli/Margaglio 4 4 5 4 5 4 5 4 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 2 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 3 Drobiazko/Vanagas 5 5 4 5 4 5 4 5 5 5

Judge s Ordinals to Places Couple OM JM Winkler/Lohse 6 7 7 6 6 6 6 6 6 6 Grushina/Goncharov 7 6 6 7 7 7 7 7 7 7 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 2 Fusar-Poli/Margaglio 4 4 5 4 5 4 5 4 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 2 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 3 Drobiazko/Vanagas 5 5 4 5 4 5 4 5 5 5

Judge s Ordinals to Places Couple OM JM Winkler/Lohse 6 7 7 6 6 6 6 6 6 6 Grushina/Goncharov 7 6 6 7 7 7 7 7 7 7 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 2 7 Fusar-Poli/Margaglio 4 4 5 4 5 4 5 4 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 2 8 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 3 Drobiazko/Vanagas 5 5 4 5 4 5 4 5 5 5

The Competition Unfolds Winkler/Lohse skate first:

After Winkler/Lohse Couple OM JM Winkler/Lohse 1 1 1 1 1 1 1 1 1 1 Grushina/Goncharov Krylova/Ovsyannikov Fusar-Poli/Margaglio Anissina/Peizerat Lobacheva/Averbukh Drobiazko/Vanagas

Then Grushina and Goncharov

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

Eventually, Anissina and Peizerat skate

After Anissina and Peizerat Couple OM JM Winkler/Lohse 4 5 5 4 4 4 4 4 4 4 Grushina/Goncharov 5 4 4 5 5 5 5 5 5 5 Krylova/Ovsyannikov 1 1 2 2 2 2 1 1 1 1 Fusar-Poli/Margaglio 3 3 3 3 3 3 3 3 3 3 Anissina/Peizerat 2 2 1 1 1 1 2 2 2 2 Lobacheva/Averbukh Drobiazko/Vanagas Leaderboard 1. Krylova/Ovsyannikov 2. Anissina/Peizerat 3. Fusar-Poli/Margaglio

And then Lobacheva and Averbukh skate

After Lobacheva and Averbukh Couple OM JM Winkler/Lohse 5 6 6 5 5 5 5 5 5 5 Grushina/Goncharov 6 5 5 6 6 6 6 6 6 6 Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 2 7 Fusar-Poli/Margaglio 4 4 4 4 4 4 4 4 4 4 Anissina/Peizerat 2 2 1 1 1 1 3 2 2 2 8 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 3 Drobiazko/Vanagas Leaderboard 1. Anissina/Peizerat 2. Krylova/Ovsyannikov 3. Lobacheva/Averbukh

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

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

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

Krylova/Ovsyanikov versus Lobacheva/Averbukh Krylova/Ovsyannikov 1 1 2 2 3 3 2 1 1 Lobacheva/Averbukh 3 3 3 3 2 2 1 3 3 Krylova and Ovsyannikov have a WIN over Lobacheva and Averbuch with 6 Judges in Favor (JIF).

The Wins and JIF Table WL GG KO FM AP LA DV W JIF Win/Loh 7 0 0 0 0 0 1 Gru/Gon 2 0 0 0 0 0 0 Kry/Ovs 9 9 9 5 6 9 6 Fus/Mar 9 9 0 0 0 6 3 Ani/Pei 9 9 4 9 8 9 5 Lob/Ave 9 9 3 9 1 9 4 Dro/Van 9 9 0 3 0 0 2 Order of Finish 1. Krylova/Ovsyannikov (6) 2. Anissina/Peizerat (5) 3. Lobacheva/Averbukh (4)

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?

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

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

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.

Stephanie Baar Potoka

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.

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

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.

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

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?

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

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

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?

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

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)

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.

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.

There is a great undergraduate research project here.

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