Canada owns 40% of the Olympic podium

In this year’s Olympics, much has been made of the Canadian efforts to “own the podium“.  Canada has spent $118 million in training its athletes, far more than the US has spent ($55 million over four years).  Since it seems that, despite a late rush, the Canadian goal of winning more medals than any other country will not be met, the Own the Podium effort appears to be a failure.  But perhaps operations research can come to the rescue here.

The problem is, perhaps, in defining the goal.  By defining the goal in terms of overall medals, the Canadians were perhaps too modest.  If they had simply strived for excellence and defined their goal in terms of “Most gold medals”, then they would have succeeded:  they have 13 gold compared to the Germany’s 10 gold with two events to go.

It does seem kind of strange to define winning as “Most Medals”:  a bronze is not the same as a gold!  But it also seems pretty strange to only count gold:  the others seem to have some value.

Rather than look at any particular weighting of the medals, perhaps we should look at any reasonable weighting and see who wins.  If we give weights wg, ws, and wb to each of gold, silver, and bronze, and let ng, ns, and nb be the number of such medals won, then the score of a country is wg*ng+ws*ns+wb*nb.  The stated Canadian goal had (wg,ws,wb)= (1,1,1).  Counting gold only has (wg,ws,wb) = (1,0,0).  What other weights would be reasonable?

Clearly, gold is at least as valuable as silver which is at least as valuable as bronze, so we want wg>=ws>=wb.  Also, we can normalize so that wb+ws+wb=1 (since, for instance, (2,2,2) is the same as (1,1,1) which is the same as (1/3, 1/3, 1/3)).   With these requirements, there are only three teams that might be considered ahead at this point.  Consider the leading countries (from nbcolympics.com:

Country		Medalists				Total
	United States	See Names	9	14	13	36
	Germany	See Names	10	12	7	29
	Canada	See Names	13	7	5	25
	Norway	See Names	8	8	6	22
	Austria	See Names	4	6	6	16

Norway, Austria, and every other country (other than Germany and Canada) is dominated by the USA, so cannot be the winner, no matter the weight.

There are many weights other than (1,0,0) for which Canada is the winner.  For instance (.68, .16, .16) is also a win for Canada.  Even (.64, .32,.04) results in Canada in first.

Going through the grid of possible values, it seems that Canada is currently in first in about 40% of the cases;  the USA is in first in the remaining 60% of the weights.  Germany is never in first, being dominated by the combination of 74% USA and 26% Canada.  So perhaps it is fair to say that Canada owns 40% of the podium, trailing the USA with 60%.

If Canada were to beat the USA in hockey on Sunday, they would go up to 45%.  This assumes no further medals in the men’s 50km cross country.  But if a Canadian could also win the cross country, then the fraction of weights for which Canada wins goes up to 54.7%.  There is still a chance for Canada to “Own the Podium!”.