Month: February 2010

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):

Country		Medalists				Total
	United States	See Names	9	15	13	37
	Germany	See Names	10	13	7	30
	Canada	See Names	14	7	5	26
	Norway	See Names	9	8	6	23
	Austria	See Names	4	6	6	16

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.

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!”.