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Olympic thoughts, part 2

Like I mentioned in a previous post, I’m a big fan of the Olympics. You may put some question marks around the whole event, the usefulness and effect on the hosting country is perhaps exaggerated, the city selection seems to be somewhat peculiar, the Olympic Committee is a little too powerful, etc. However, the performances of the athletes are amazing. World records are broken, world champions are dethroned, outsiders and underdogs win gold medals, the whole shebang.
Another aspect of the Olympics is that it is a clash of nations of epic proportions. All over the news and internet one can find examples of whole nations claiming some credit from the medals won by their athletes. Even in the introduction of each athlete, we are told they  are representing their country. And in some sense this is justified, as usually the governments fund their Olympic delegation, so that it is ultimately the tax payer who contributed to the performance. I, for one, am very happy to do so.
Yet, if the Olympics is to be considered a tournament between nations where each athlete contributes a small bit to the whole, how are we to measure and judge? The answer seems obvious, by the medals of course! Again, there is some subtle math going on behind the scenes.

Dictionary ordering

The standard approach to the nation medal rankings is to apply a dictionary ordering. Like a dictionary, we first count the gold medals and order the countries accordingly. Next, if there’s two countries with the same number of gold medals, we order those two with respect to the silver. Finally, if there’s still some countries on equal footing, we order those according to the bronze.
This approach is sometimes used in mathematics to order multidimensional spaces. Indeed, if we assign a dimension to each color of metal, we have x = gold, y = silver and z = bronze. The standing of a country is then represented by point in three-dimensional space (x,y,z). Ordinarily, in multidimensional spaces we are faced with the same problem if we fancy to order the points. The dictionary ordering offers a way to, arbitrarily, impose an ordering to the space.
This type of ordering works well if the dimensions are more or less independent and if they are, it doesn’t really matter which dimension is ordered first, second, etc. However, medals aren’t really independent, as one country may simply rack up the gold and silver on a single discipline. That’s an amazing achievement, but it shows that our dimensions aren’t necessarily independent.
Another objection is that one gold medal is worth more than any number of silver or bronze medals. Even if one country stacks up a million silver medals, they’ll be behind the country having only a single gold medal in the ranking. This doesn’t really appeal to me, and I suppose it wouldn’t really appeal to anyone. A gold medal is worth more than a silver one, but not by a factor of a million.

Weighted ordering

This suggests a different way of measuring the standings, and that is to assign weights or points to each type of medal. In Beijing 2008, the final tally counted 51 gold, 21 silver and 28 bronze for China. For the USA, the final score was 36 gold, 38 silver and 36 bronze. By the dictionary ordering, China was first in the rankings by a fairly comfortable margin. However, careful examination of the numbers shows that China scored 100 medals exactly, whereas the USA scored 110. Simply counting medals, it was actually the States that came in first.
This type of counting system is an example of a weighted ordering system, one where each medal has the same value, say one point. However, this again doesn’t satisfy, as then ending up second or third is just as good as being first. Our gut feeling tells us that we should assign different weights to each medal. A number of such systems has been suggested, for instance the New York Times of 4 (gold), 2 (silver) and 1 (bronze).
However, we face another problem here: the weight we assign to each medal is fairly arbitrary. Is a gold medal worth two or three or perhaps even four silvers, and how many bronzes should we win to match the gold? Using another weight system might result in another nation capping the ranking. This system seems just as flawed!

Per capita

Another objection we might raise about the medal rankings is the fact that some countries are simply bigger than others in terms of the number of inhabitants. The USA has somewhat over 324 million people, whereas a small country like New Zealand has 4.7 million people living in it, which is about 1.5% of the population of the States. One might therefore expect the same ratio in the athlete population, and as such, New Zealand has a selection pool which is that much smaller. The chances of having a gold-winning medalist are then likewise a lot smaller.
This suggests we scale the number of medals with the population, reaching a medals-per-capita ranking. And indeed, we see that New Zealand is second in the current ranking, much better than the USA, which is at 40th place. However, this is again somewhat unsatisfactory, as this ranking favours smaller countries. Several island states have managed to win some medals and they are leading the per-capita rankings.
A similar thing happens if we scale the Nobel Prizes per capita, since then Iceland becomes the number one supplier of Nobel laureates. Halldór Laxness is the first-and-only Icelandic to win a Nobel prize, but Iceland has only 332 thousand people. Again the per capita ranking favours the countries with small populations, rather than giving a fair overview of the rankings.
An even more extreme example is the Independent Olympic Athletes. There are, at the moment, nine independent Olympians who are all from Kuwait since that country was banned. There are a further ten refugee athletes who compete under the same flag. This group of people has won two medals, one gold and one bronze. Both medals were in shooting disciplines incidentally. Due to the incredible medal-to-population ratio, 2 in 19, they will never be beaten by any nation as the next best is Grenada with 1 in 110 thousand.
Once again, we are foiled in our quest for finding a better ranking system. Counting per capita, we are favouring the statistical quirks of the extremely small nations. Since there are quite a few small nations, there is a rather large probability of one of them winning a medal.

So now what?

Finally, after examining all the different ways of ranking medals, we can conclude two things. First, the current system is not perfect as it really only counts the gold. Second, the other systems we might come up with aren’t much better. Of the three suggestions we looked at, perhaps the weighted ordering is best, but it quickly becomes very arbitrary. Already there are multiple weighted ordering systems in play, all of them giving a slightly different tally. So, we’ll just have to make due with what we got.

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