Apart from the calculating the medal rankings per population or GDP, I propose a new way of assessing how well a national team did at some given Olympic Games.
Let \(r\) be the rarity index of each athlete at a given event, defined as the reciprocal of the number of athletes who did equally well or better, including self.
Then the athlete can have \(r=1\) by winning a gold medal, \(r=1/2\) for a silver medal, etc.
An unusual example in the men's 200m freestyle swimming is the tie between Park and Sun which resulted in 2 silver medals and no bronze medals, in which case both swimmers get \(r=1/3\) while Agnel gets \(r=1\) for the gold medal. (If the top 3 players tie in a single event, each will get \(r=1/3\), so it's usually a bad idea to aim for a tie.)
Here we choose to include all competitors, so \(r\) doesn't stop at \(1/3\) (bronze medal) but continues as \(1/4, 1/5, \cdots , 1/N\), where \(N\) is the number of players in the event who are not disqualified. Those who leave the game due to injuries or any other circumstances beyond their control would have their score thus far compared against the other competitors in the event, or be given \(r=1/N\) in case an intermediate score cannot be given. An automatic \(r=0\) is given to a disqualified athlete. \(r\) for a team event would be calculated by counting each team as a unit, then multiplying it by the number of players in the team. For example, an archery team (of 3) which placed second (silver medals) would count as \(r = 1/2 \times 3 = 3/2\).
Next, define \(Z\) as the total of \(r\) values which an entire national team can have in the ideal scenario, ie. winning all gold medals for the events in which its players are competing. \(Z\) need not be a whole number if more than one player from a single country compete in the same event. For example, if there are two players from the same country in the same event, the ideal total of \(r\) values that they can get from the event is \(1+1/2=3/2=1.5\) for a gold and a silver medal.
Then I argue \[ \frac{\sum_\text{national team} r}{Z} \equiv A \] (call this the national achievement index, which ranges between 0 and 1) to be a fairer way to rank the national teams. Somewhat complicated, but it is more directly weighted by how much the country, within its powers, has put into having their athletes compete in the Games, without focusing too much on population (people in some well-populated countries are not very interested in the olympics) or GDP (the cost to train an olympian differs greatly across countries).
One obvious drawback would be that this can discourage certain national-level athletes from competing at all for their country, as in, "Don't even try if you know you're not going to place high." But I think this actually will become more of a motivator to the less developed countries and will encourage them examine their strengths and weaknesses more carefully in bringing up their next generation of athletes.
I will post the values for USA, KOR, CAN, PRK, plus some other countries of interest to me after the closing ceremony.
