Olympic skaters complete two programs in their competition, one short and one lo

Question

Olympic skaters complete two programs in their competition, one short and
one long. In each program, the skaters are scored and then ranked by a panel
of nine judges. The skaters' positions in the rankings are used to determine
their final scores. A skater's ranking depends on the number of judges placing
her first (or second or third); the skater judged to be best by the most judges is
ranked number one, and so on. In the calculation of a skater's final score, the
short program gets half the weight of the long program. That is, Final score = 0.5
(Rank in short program) + Rank in long program. The skater with the lowest
final score wins the gold medal. In the event of a tie, the skater judged best in the
long program by the most judges takes the gold. In the 2002 women's individual
figure-skating competition in Salt lake City, Michelle Kwan was in first place
after the short program. She was followed by Irina Slutskaya, Sasha Cohen, and
Sarah Hughes, who were in second, third, and fourth places, respectively. In the
long program, the judges' cards for these four skaters were as follows:

(a) At the Olympics, Slutskaya skated last of the top skaters. Use the information
from the judges' cards to determine the judges' long-program
ranks for Kwan, Cohen, and Hughes be/oreSlutskaya skated. Then, using
the standings already given for the short program in conjunction with
your calculated ranks for the long program, determine the final scores
and standings among these three skaters before Slutskaya skated. (Note
that Kwan's rank in the short program was 1, and so her partial score
after the short program is 0.5.)
(b) Given your answer to part (a), what would have been the final outcome
of the competition if the judges had ranked Slutskaya's long program
above all three of the others?
(c) Use the judges' cards to determine the actual final scores for all four
skaters after Slutskaya skated. Who won each medal?

(d) What important principle, of those identified by Arrow, does the Olympic figure- skating scoring system violate? Explain.

judge judge judge judge judge judge judge judge judge 1 2 3 4 5 6 7 8 9 kwan points 11.3 11.5 11.7 11.5 11.4 11.5 11.4 11.5 11.4 kwan rank 2 3 2 2 2 3 3 2 3 slutskaya points 11.3 11.7 11.8 11.6 11.4 11.7 11.5 11.4 11.5 slutskaya rank 3 1 1 1 4 1 2 3 2 cohen points 11.0 11.6 11.5 11.4 11.4 11.4 11.3 11.3 11.3 cohen rank 4 2 4 3 3 4 4 4 4 hughes points 11.4 11.5 11.6 11.4 11.6 11.6 11.3 11.6 11.6 hughes rank 1 4 3 4 1 2 1 1 1

Explanation / Answer

Answer to part a:

First, we have to count how many judges put every competitor in a ranking:

Now, we need to settle every competitor in a position for the long tournament. We put first the one who got more judges placing her in position 1st, and this is Hughes with 5 judges, the second is Slutskaya with  4 judges. The third and fourth positions, Kwan and Cohen, respectively, did not have any judges placing them as first, so they got tied for now, but Kwan was placed second by 5 judges, while Cohen was placed second by one judge, that is how these two girls got untied in the final score of the long tournament, it implies, Kwan is third and Cohen is fourth.

The final score from the short and long tournaments, is calculated as:

Final score = 0.5*(Rank in the short Tournament) + Rank in the short Tournament

The lower scores (of 3) were for Hughes and Slutskaya, but, again, 5 judges placed the former first in the long tournament, while 1 did for the latter, therefore, the sole winner is Hughes, it means, she gets the golden medal. The higher scores (of 4.5) were for Kwan and Cohen, but the former had 0 judges placing her in the first place and 5 judges placing her as second in the long tournament, while the latter had no judges placing her as first, and just one placing her as second, so, the winner of the third place is Kwan.

The following table shows the results of those calculations explained before:

judge judge judge judge judge judge judge judge judge 1 2 3 4 5 6 7 8 9 kwan points 11.3 11.5 11.7 11.5 11.4 11.5 11.4 11.5 11.4 kwan rank 2 3 2 2 2 3 3 2 3 Count 5 4 5 5 5 4 4 5 4 slutskaya points 11.3 11.7 11.8 11.6 11.4 11.7 11.5 11.4 11.5 slutskaya rank 3 1 1 1 4 1 2 3 2 Count 2 4 4 4 1 4 2 2 2 cohen points 11.0 11.6 11.5 11.4 11.4 11.4 11.3 11.3 11.3 cohen rank 4 2 4 3 3 4 4 4 4 Count 6 1 6 2 2 6 6 6 6 hughes points 11.4 11.5 11.6 11.4 11.6 11.6 11.3 11.6 11.6 hughes rank 1 4 3 4 1 2 1 1 1 Count 5 2 1 2 5 1 5 5 5

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