Normally, we think of Olympic judges as using range voting – the highest average score wins. (Albeit with some alterations such as dismissing the two highest and lowest "outlier" scores each performance, but the principle remains just averaging.)
But in 1995 at the World Figure Skating Championships, they used a different procedure. Here's Journalist Lila Guterman's recounting:
The U.S. champion, Nicole Bobek, had skated into second place behind Chen Lu of China. In third place after her final performance was Surya Bonaly from France. Then, a relatively unknown skater, 14-year-old Michelle Kwan, took the audience and judges by storm with a performance that catapulted her into fourth place.
Ms. Kwan's skate did not alter any of the judges' scores for Ms. Bobek or Ms. Bonaly. But after the votes were tallied, their positions flipped. Ms. Bonaly went home with the silver, and Ms. Bobek won bronze!
That's because the scoring rules called for each judge to rank the skaters and then for the group to determine winners by using a modified plurality vote based on those rankings. Ms. Bobek had had more second-place rankings than Ms. Bonaly until Ms. Kwan skated, but got bumped out of second in enough judges' rankings to give the silver medal to Ms. Bonaly.
Moral: switch from just using scores and averaging (range voting) to using ranking (first, second, third)? Big mistake, which made the figure skaters' association look stupid.
[The following postscript comes from W.Poundstone's book Gaming The Vote:] Then something very similar happened at the 1997 European men's championships. The top three were Urmanov, Zogorodniuk, and Candeloro, in that order. They all were done with all their skating. Then Vlascenko skated and came in sixth. Surprise! Vlascenko, just by skating badly, reversed Zogorodniuk versus Candeloro, causing Candeloro to end up with the silver medal. After that, the International Skating Union had had enough. Its chair Ottavio Cinquanta rolled out a new scoring system in 1998, promising "if you are in front of me, then you will stay in front of me!"
But that was a lie – counterexamples were constructed almost immediately. But note, there actually was no need to construct a counterexample – any voting theorist could tell immediately, with almost no thought whatever, that it had to be a lie, because of Arrow's theorem.
Condorcet, IRV, and Borda are examples of voting systems in which removing or adding a candidate C, can cause the relative order of two other candidates A, B to flip even though not a single voter changes her relative order for A and B. With Range Voting, that can't happen.
To learn more about the insane evolution of voting systems used for scoring skaters, we recommend
Maureen T. Carroll, Elyn K. Rykken, Jody M. Sorensen: The Canadians Should Have Won!?, MAA Math Horizons 10 (February 2003) 5-7 (pdf).
For skating, I personally would recommend the following "trimmed mean" system (which actually, I had thought, based on casually listening to talking heads on TV yakking about the Olympic skating, diving, and/or gymnastics, was the system they were using, but in fact at least sometimes, it wasn't...):
This system is exactly the same as range voting, except that "trimmed mean" is used instead of mean, i.e. "outlier" judges are discarded. It has these advantages:
Note that being "immune to coalitions of K strategic voters" is totally antithetical to "democracy" (!!) and hence I do not recommend trimmed-mean range voting for political applications. (Imagine saying to black, or Jewish, or gay, or third-party voters "sorry, the voting system has decided to ignore your votes because you are 'outliers'.") But it does seem a good idea in the skating application. The value of K can be adjusted anywhere from 0 to ⌊(N-1)/2⌋ where there are N judges.
We give a fairly precise description of the 1995 BOM and 1998 OBO scoring systems here – with "flip-flop" examples for each from the 1998 Olympics.
For two popular articles about skating scoring difficulties, see
Lila Guterman: When Votes Don't Add Up, Chronicle of Higher Education 47,10 (3 Nov. 2000) page A18 (2 pages plus 2 photos).The Bobek-Bonaly "great flip-flop" is also mentioned in
Richard Monastersky: Mathematicians Find Problems With New System for Scoring Figure Skating, Chronicle of Higher Education 49,2 (31 Jan. 2003) page A16.
Farhad Manjoo: Your presidential candidate: Hot or not?, for Salon.com.
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