The canonical example is the 2000 U.S. Presidential election between Al Gore and George Bush. It all came down to Florida, where Gore was clinging to a small lead, let us say 1,000,000 to 999,000. Along comes liberal candidate Ralph Nader, to the left of Gore. He siphons off 2000 votes from Gore. The final tally is
Bush 999,000; Gore 998,000; Nader 2000.
Bush (after a few hanging chads and a five to four vote in the U.S. Supreme court) wins all of Florida’s electoral votes and the presidency.
This sort of thing is regarded as anti-democratic because the will of the people (they preferred Gore to Bush) has been frustrated by an “irrelevant alternative” (Nader).
In figure skating this problem came to a head at the 1997 European men’s competition, where nobody skated well and the ordinals were all over the place. The ISU rushed a new system in place (OBO), but it did not address this particular problem. (All it did was make it harder for Michelle Kwan to win the 2002 Olympics. ) Here is an excellent article, by Sandra Loosemore of Frogs on Ice about all this.
Anyway, in figure skating the problem arises in a situation like this. Here are the ordinals given by nine judges after two skaters, A and B have gone.
Skater A: 1 1 1 1 1 2 2 2 2
Skater B: 2 2 2 2 2 1 1 1 1
Skater A is winning. She is preferred over skater B by a majority of the panel.
Now skater C goes. The new rankings are
Skater A: 1 1 1 2 2 2 2 3 3
Skater B: 2 2 2 3 3 1 1 1 1
Skater C: 3 3 3 1 1 3 3 2 2
Skater B wins with four first place ordinals, three seconds, and two thirds. Skater A must be satisfied with silver even though head-to-head she beat skater B by a score of five judges to four and she beat Skater C by a score of five judges to two. Skater A beat everybody (she is the “Condorset winner”), but Skater B won the gold medal.
So the question is, is this the wrong outcome? If we do not announce any intermediate results, but wait until the end to tally all the votes, Skater B has a good claim. Her 4 firsts and 3 seconds is better that Skater A’s 3 firsts and 4 seconds, both receiving 2 thirds. The only reason that Skater A is mad is because she thought she was winning before Skater C snuck in there and stole some first place ordinals from her. I don’t know that this is so terrible in figure skating, despite the fact that Prof. Arrow (a Nobel Prize winning economist) didn’t like it.