29 August 2008

The Olympic poset

During the Olympics (yeah, I know, you've all forgotten about the Olympics), there was much argument about whether the right thing to do, if we want to determine which country "won the Olympics", is to count the country that got the most gold medals (China), the most medals (the US), or something else -- say a points system that allocates three points for a gold, two for a silver, and one for a bronze.

Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:
So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades.
This gives a partial order on the countries.

Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.

One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?

8 comments:

Anonymous said...

What is the point of meta-accolades anyway?

John said...

I will leave aside the question of whether to adjust medal counts for population or popularity, but here's a post that suggests a start for how you might do so. The post was written to examine what you would expect if men and women had equal ability but differing levels of interest in some sport, but the same analysis would apply to any two populations with equal ability distributions but differing sizes.

Anonymous Rex said...

Yay, majorization!

David said...

I second John's idea. I'd like to see every nation given a "fitness" distribution, and then use order statistics and medal results estimate results.

Of course, I wouldn't want to do such things myself...

Jon Ingram said...

Thank you for your link to that visualisation of the Olympic medal table. It roused me to write about a visualisation of the first book of Euclid's Elements which I created using the same piece of software (Graphviz) a while back.

Knut M. Wittkowski said...

Hi Simon,

I like your graph a lot! A bit more on computational aspects of this approach is available in a recent publication "U-Scores for Multivariate Data in Sports" http://www.bepress.com/jqas/topdownloads.html, see also http://newswire.rockefeller.edu/

Knut

Simon Tatham said...

"This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it."

I have :-) It is.

Under any scoring system of the type you describe, a country's total score is xG+yS+zB, which can also be written as (x-y)G + (y-z)(G+S) + z(G+S+B), which is now helpfully written in terms of exactly the three quantities which are product-ordered by my order relation - and each of those quantities is being multiplied by one of the non-negative quantities (x-y),(y-z),z. Hence, if country A has all of G, G+S and G+S+B at least as good as country B, then its score as calculated above must be greater than that of country B.

Conversely, suppose country B has at least one of G, G+S and G+S+B better than A. If it has higher G, then it beats A under the scoring system x=1,y=z=0; if it has higher G+S then it beats A under the system x=y=1,z=0; if it has higher G+S+B then it wins under the system x=y=z=1.

Hence, exactly as you say, my partial order ranks country A as strictly better-or-equal to B if and only if A's score under any scoring system of the above type would be at least B's score.

Knut said...

... and it is equivalent to the orders proposed for the Olympics and other 'graded' event categories, like medical side effects ( CSS 2003 ), Tour-de-France jerseys ( JSE 2006 , 2007 ) and many more (see JQAS 2008 ).