Last night’s Olympic ladies’ figure skating raised some interesting lessons for many people. How, they ask, could it happen that Michelle Kwan was ahead of Sarah Hughes before Irina Slutskaya skated, and then behind after? How did one skater’s performance affect the relative placement of two others? These two questions actually boil down to two deeper ones:

  1. How did Sarah Hughes win the long program?
  2. How does her having won the long program translate into her winning the overall event, given that she had placed fourth in the short program?

Brian Cazeneuve of Sports Illustrated provides a pretty good answer to the second question so I’m going to leave it alone. In my opinion the first question is the more interesting of the two anyway.

I found the detailed results, including each judge’s score, on the official 2002 Winter Olympics website. What you should be looking at are the “ordinals” for the long program for the first three athletes, and then adjust them to show rankings only between those three. I’ve condensed the result into a small table for easier reference:

Hughes 1 3 3 3 1 2 1 1 1
Slutskaya 3 1 1 1 3 1 2 3 2
Kwan 2 2 2 2 2 3 3 2 3

Figure-skating enthusiasts who check the full results will notice that in several cases a judge gave the same total score to two contestants (judges 1 and 5 for Slutskaya and Kwan, judge 2 for Hughes and Kwan, judge 7 for Hughes and Slutskaya). Those same fans would probably already know that ties are broken in favor of the skater with the highest presentation score; maybe one of them can tell me what happens if both scores are identical, which didn’t happen in any of those four cases.

So why is any of this interesting to anyone besides figure-skating fans? Why did I file this under “politics”? It’s because the question of how these numbers translate into a win for Hughes is a question of how to design a fair voting system. As it happens, Hughes had five out of nine first-place ordinals for a clear overall win. Imagine for a moment, though, that judge 8 had given Hughes a 5.7 instead of a 5.8 for presentation. This would have left her tied with Kwan for the top overall score according to that judge, with Kwan getting the nod based on the presentaion-score tiebreaker. Now nobody has a clean win, because the first-place votes would be split 4/4/1. What happens then?

This Voting Systems FAQ is a good source for background information about such issues. One common approach in these sorts of situations is runoff voting, but then we run into another problem. Each of the three contestants was judged, by a majority of the judges, to be in one of the top two spots. This can happen because there are eighteen total “first or second” votes, and each contestant only needs five to get the majority needed to qualify for a runoff. Runoff voting, therefore, does not help in this situation. It is (or, more precisely, would have been) a perfect illustration of runoff voting’s inadequacy for resolving certain types of conundrums.

An even more interesting situation would have occurred if we had combined the previous thought experiment with the arbitrary selection of only three judges – let’s say judges 2, 7, and 8. No, that was not a random choice. Let’s see what happens to the ordinals between these three contestants for these three judges, with the aforementioned change to judge 8′s presentation score for Ms. Hughes:

Hughes 3 1 2
Slutskaya 1 2 3
Kwan 2 3 1

Look at that for a minute, and try to figure out how you’d determine a winner. Ha ha, gotcha. Troubling, isn’t it? As it turns out, two out of three judges placed Hughes ahead of Slutskaya, two out of three placed Slutskaya ahead of Kwan, and – here’s the kicker – two out of three placed Kwan ahead of Hughes! It’s a “Condorcet rule” cycle, and there’s really no solution within any system based on ordinals; the only reason we don’t run up against this more often in real life is mere statistics, not any guarantee within the system that it can’t happen. I’m not going to use all of this as a reason to suggest changes to the way figure skating is judged, by the way. I don’t really care much. I just thought this current event provided a good starting point for a timeless exploration of general ideas about voting. I hope you enjoyed taking the trip with me.

P.S. Even though I just said I didn’t care, I extend my hearfelt thanks and congratulations to all three of these ladies – most especially to Sarah Hughes – for a well contested exhibition of grace and athletic prowess.