During the 2002 (Salt Lake City) Winter Olympic Games, the International Skating Union rules at the time did not satisfy the "Independence of Irrelevant Alternatives" criterion. Scores were calculated as the sum of half the rank in the short program plus the rank in the free skate. Prior to the last skater in the free skate, Irina Slutskaya (Russia), the leaders were Sarah Hughes, Michelle Kwan, and Sasha Cohen (all USA). Slutskaya finished 2nd in the free skate and overall. So Sarah Hughes won the Gold medal. However, had Slutskaya finished third (or worse), the Gold would have gone instead to Michelle Kwan. Read more about this here.
Now, at the 2012 (London) Summer Olympic Games, we get another teaching example. This time, it wasn't ice skating, but gymnastics (another "scoring" not timed event). There was no doubt as to who finished first in the Women's All-Around competition-- Gaby Douglas (USA) earned the Gold medal with 62.232 points, while Viktoria Komova (Russia) earned the Silver medal with 61.973 points). However, at the end of the four events (vault, uneven bars, balance beam, and floor exercise), two gymnasts (Aliya Mustafina - Russia and Aly Raisman - USA) were tied for third place with exactly 59.566 points. In swimming and other races ties like this happen from time to time and multiple medals are awarded. However, under the current rules for gymnastics, the lowest event score for each tied participant is removed. So, after removing the beam scores for each (Mustafina's 13.633 and Raisman's 14.200), Mustafina was declared the Bronze medal winner since she had the highest score on the other three events. At first glance, I thought this was an example of the "maximize the minimum" decision rule, but after further consideration do not believe it is. Not sure what the logic behind it is. Any thoughts from other decision analysts out there?