Weighted Medal Count, Explanation of Weighting

Gold medals are weighted by a factor of 6, silver medals by a factor of 3, bronze medals by a factor of 2.

The reasoning is simple. One person has won a gold medal. Two people have won at least silver, and three people have won at least bronze. The weightings simply reflect these factors.

Note that this weighting system is consistent with many golf tournament payouts, in which the 2nd place finisher receives half the prize money of the champion and the 3rd place finisher receives a third as much as the champion.
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