Further to yesterday’s entry, we can now determine exactly how much of the podium Canada owns. To determine the “winner” of the Olympics, you need to determine the relative values of gold, silver, and bronze medals (with the assumption that non-medalers do not count, which is arguably false, but necessary in order to stop me from spending the night compiling broader lists). The final medal standings are (from nbcolympics.com):
|United States||See Names||9||15||13||37|
So, if you count every medal equally, then the USA won; if you only count gold, Canada won. But what if you count things 5 for a gold, 3 for a silver, and 1 for a bronze? Then the USA wins. How about 10, 5, 1? That would be Canada. Is there a set of points for Germany to win? It turns out there is not: anyone with operations research training would fiddle around for a while and figure out that 3/4 of the US medals plus 1/4 of the Canadian medals dominates the German medal counts. Everyone else is dominated by the USA: only Canada and the USA might win for a given set of medal weights.
Now not every point system makes sense. Giving 10 points for a bronze and 1 point for a gold might match up with certain egalitarian views, but would not really be in keeping with a competition. So we can limit ourselves to point systems with gold >= silver >= bronze. Further, we can normalize things by making the weights add up to 1 (since multiplying a weighting by a constant number across the scores doesn’t change the ordering) and having the weights be non-negative (since getting a medal shouldn’t hurt your score).
This gives a base set of linear equalities/inequalities. If we let wg, ws, and wb be the weights for gold, silver and bronze, we are interested in weights which satisfy
wg >= ws >= wb
wg+ws+wb = 1
wg, ws, wb >= 0
Now, which weights favor Canada? It turns out that, with some basic algebra, you can deduce (using the medal counts above) that Canada wins whenever wg > 8/13 (and ties with wg=8/13). So as long as you put more than 61.5385% of the weight on gold, Canada wins. This amounts to about 45.25% of the feasible region. USA wins on the remaniing 54.75% of the region. If Canada had won one more silver medal, they would have prevailed on more than half the reasonable region.
The diagram illustrates the weights for the USA and Canada, giving only the weights for gold and silver (the weight for bronze is 1-gold-silver). The red region are the weights where Canada wins; the blue is for the USA. Point A is “all medals are equal”; Point B is “count only gold and silver”; Point C is “Count only gold”. The yellow line corresponds to the weight on gold equaling 8/13.
Bottom line: on this measure, the USA won the Olympics in an extraordinarily close race. Canada may not have “Owned the Podium” but they came darn close.