Olympic medal table

From time to time, I plan to post an alternative version of the Olympic medal table.  Conventionally countries are ranked by the number of gold medals, followed by silver if the gold tally is equal, followed by bronze.  This means that a country with five silver medals and no gold medals would be ranked behind a country with one gold medal.  This seems a bit unsatisfactory.  There is in http://www.topendsports.com/events/summer/medal-tally/rankings-weighted.htm a list of some different sets of weightings that have be applied to the three medal counts to produce an overall score.  In the conventional system the weightings are effectively 1,000,000 or some suitably high number for gold, 1,000 (say) for silver, and 1 for bronze, at least for the purpose of ranking countries (not estimating their relative performance).  That's how you express the fact that no matter how many silvers you have (within reason) you will rank lower than a country with one gold.

The weighting system I am about to suggest doesn't appear in that list.  It comes from an article I read in Mathematics Today some years ago.  This is how it's derived:

Let's say a gold medal is worth G points, a silver medal S points and a bronze B points.  Now all that is going to matter is their relative scores, so let's completely randomly (!) say that G + S + B = 18 (say).  Now let's have a 3-dimensional coordinate system with axes labelled G, S and B.  A set of three weights will then be a point in this 3-dimensional space, and the job is to choose the "best" point in some sense.  The fact that G + S + B = 18, together with the obvious facts that G > 0, S > 0 and B > 0, mean that the point will actually be on an equilateral triangle between the points (18, 0, 0), (0, 18, 0) and (0, 0, 18), as shown here.

Now let's put this equilateral triangle flat on the paper and ask ourselves where our chosen point may be.  We can take some new facts into account.  One is that a silver is not going to be worth more than a gold.  This cuts the triangle in half, and our chosen point will be somewhere in the grey triangle BGP:

The next fact is that a bronze is not going to be worth more than a silver (or a gold, of course).  This cuts down our region of possible points still further to the triangle RGP:

Beyond that, it becomes a matter of subjective taste where we place our point.  The conventional weighting-for-ranking system is close to point G.  The system where you just count the total number of medals is point R.  But our reasoning is that, in the absence of any further guidance, we'll place our point in the "centre" of the grey triangle.  Now triangles have many different "centres" but the most reasonable one for us to use here is the centroid, the centre of gravity, which is the red spot with coordinates (11, 5, 2).  (Now you see why I chose them to add up to 18!)

So there is an answer.  We have a points system with 11 for gold (echoes of This is Spinal Tap), 5 for silver and 2 for bronze.  My impression is that this remains quite an "elitist" scoring system because each medal is worth more than double the one below, but it is going to be a bit fairer for ranking than the conventional system.

Other constraints could be applied: for example, we could impose that the ratios G/S and S/B are the same (though with this system they are already quite close, at 2.2 and 2.5 respectively).  Or we could say that a gold is worth no more than a silver and a bronze, so G<=S+B (this would lead to G=8, S=6.5 and B=3.5, which would be better expressed as 16, 13 and 7 - a system which seems very generous to silver medal holders).