## 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!”.

### { 6 } Comments

1. Suresh | February 28, 2010 at 5:37 am | Permalink

Excellent idea. it seems like this might even be well known as a way of imposing a total order on R^n !

2. Stanley E. Zin | February 28, 2010 at 10:23 am | Permalink

The IOC’s choice of medal metals (gold, silver and bronze) is not arbitrarily. Rather the ranking of medals corresponds to the economic-value ranking of their composite metals. We are comfortable using market prices to approximate economic value is a wide variety of settings, and aggregating across economic values is simple. Why not do the same here? Since the current price of gold is on the order of 100 times bigger than the price of silver (actually about 1100 to 16), and since the price of silver is about 200 times bigger than the price of bronze, based on economic-value weighting, Canada wins the Vancouver Olympics, and by a lot! (Germany finishes second, the US third.) But wait… it turns out that a “gold medal” is not actually gold metal. Gold medals made of silver with a little gold plating. The IOC stipulates, however, that this gold plating must contain at least 6 grams of gold (evidently they too like the economic interpretation of the relative value of a medal else why make such a stipulation). No worries, that’s plenty of gold, at current prices, for Canada’s purposes, and they still beat Germany by about \$600 and the US by \$800 (which, for the irony impaired, is even more in Canadian dollars!). And the victory over the US will crack a cool \$1000 after this afternoon’s hockey game.

3. Diana | February 28, 2010 at 10:41 am | Permalink

Really enjoyed this fun post.

4. Larry (IEOR Tools) | February 28, 2010 at 4:08 pm | Permalink

For those that have competed in Track&Field there is a pretty common scoring system for teams.

1st place: 10 pts
2nd place: 8pts
3rd place: 6pts
4th place: 5pts
5th place: 4pts
6th place: 3pts
7th place: 2pts
8th place: 1pt

This is used pretty universally for Track meets. I’ve always wondered why they never did this for the Olympics. Anyway this sounds like a pretty good measure.

5. Larry (IEOR Tools) | February 28, 2010 at 4:09 pm | Permalink

How about a measure of medals per capita (athletes competing or population). I think accounts for something also.

6. Joe@Small Business | March 2, 2010 at 3:03 am | Permalink

I won’t even pretend to comprehend what it is is you’ve done, but am glad their are folks who are here to figure this stuff out. Either way we dice it, the Canadians did well. One of my best friends is from Kethener WaterLoo and went home to watch the final game, they were lovin it!

Thanks

Joe@ Small Business Ideas

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