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Monday, August 13, 2012

Why So Few Gold Medals?

Canada's low gold medal count in London - one - is raising eyebrows: it is lower than our "usual" total of three (achieved in every summer Games after 1984, except in London and in Barcelona, where seven were won). However, even three is low compared to how much silver and bronze our athletes bring home. Indeed, gold accounted no more than a third of Canada's medals in every postwar summer games except Barcelona! This Vancouver Sun blog post has more stats.

As it turns out, this can be explained by simple math and the following two facts:
- Canada does not have a very large population; and
- Canada is competitive in many sports.

To see this, consider the following situation: a sport has 5 elite athletes, two from China, two from the US, and one from Canada. For simplicity, we will assume there there's no luck involved at the Olympics - the final results perfectly reflect the quality of athletes.

What happens if every athlete receives the same quality of support and training, so that everyone has an equal opportunity? Our Canadian then has an equal chance of being in each position from 1st through 5th - 20% chance of gold, 20% chance of silver and 20% chance of bronze.

But now suppose that the sporting federation imposes a rule: each country can only enter one athlete. To win gold, our Canadian athlete would still need to be the best. But she can win silver either by being second, or by being third if the top two athletes are from the same country. And of course, she is now guaranteed at least a bronze. Simply by virtue of being a smaller country, Canada now has a 20% chance of gold, ~27% chance of silver and ~53% chance of bronze!

In other words, having limited entries hurts large countries chances of winning bronze much more than it hurts their chances of winning gold: they still get to send their best athlete(s), but might not get to send all of their medal contenders. Thus, small countries will tend to win disproportionately many bronze medals.

Of course, even though Canada isn't the US or China, we're still larger than most countries. But the same effect applies: suppose a sport has 100 elite athletes, with 50 from the US, two from Canada, and one each from 48 other countries, so that Canada is actually the second largest country out of 50. Again, assume that each country has one entry. Canada's chances of winning gold are 2%. We don't necessarily win silver if one of our athletes is second: we won't send her to the games if the other one is first. But it is very unlikely that the two Canadian athletes happen to be the top two in the world (~0.02% chance). It is much more likely that the top Canadian, even if she is third, still ends up with a silver because the top two are American (~0.50%). She could even rank worse and still get silver if all of the superior athletes are American. The probability of winning silver turns out to be about 2.97%, and the probability of winning bronze, roughly 3.92%. So if there were 100 such events, you would expect Canada to win roughly two gold, three silver and four bronze. Thus, limited entries help even relatively large countries like Canada, as long as some other country/countries are much larger. Canada is a significant player in diving, but think about what would happen to Canadian medal hopes in diving if all countries including China were allowed to send all medal contenders.

By the way, this also explains why, even excluding Vancouver, Canada has no gold medal problems in the winter: we are a relatively large winter sports nation. (We don't actually dominate any winter sport except for hockey and curling, which yield few medals - this is why you also wouldn't expect a disproportionate amount of gold. But we are a significant presence in all the ice sports, freestyle skiing and snowboarding, where we win most of our medals.)

What about small countries like New Zealand that rack up the gold medals? As it turns out, these countries focus on a small number of sports. New Zealand might have a smaller population than BC, but it has lots of rowers focusing on small boats (i.e. not Eights) and sailors. At these Olympics, we also saw North Korea and Kazakhstan focus on weightlifting, which brought results. On the other hand, Spain, whose talent is also spread out (outside male soccer and basketball, which don't give many medals), has the same problems as Canada: apart from Barcelona and two pre-WWII games where it only won one medal, Spain's gold medal count has never even reached one third of its total haul. In summary, what matters is a country's "effective size" within each sport. So small countries can earn a disproportionate amount of gold if they focus on certain sports. Canada, by contrast, is spread quite thin.

Does this mean that Canada should start focusing on few sports? No: spreading out makes our gold count low relative to our total medal count because it enhances our total medal count by allowing us to be a "small" country benefiting from the limited entries effect. (To be sure, there are other arguments, such as economies of scale, for focusing on a small number of disciplines - especially cheap ones like weightlifting.) Also, being competitive at a large number of sports may be more effective in encouraging athletic activity, which is the larger goal of participating in the Olympics.

By the same token, this analysis only explains why we have few summer gold medals relative to silver and bronze. It does not explain why our total count is high or low, though as I hypothesized in my previous post, our total count is probably right around where it should be.

That said, the single gold performance at these Olympics is very poor: just 0.3% of total gold medals, while we account for 0.5% of the global population and are comparatively very well off. Even accounting for the mathematical effect described here, you'd expect more than one gold out of 18 medals. Of course, throw in luck, and such a thing will happen once in a while - so let's hope that this is just a one-off.

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