A recent story in the Wall Street Journal raised the question of how to rank the nations that participated in the London Olympics. The top of the draw was no problem, because the strong finish by the United States meant that it dominated China. By pulling down more gold, silver and bronze, the U.S. inevitably finished first. But the situation between Great Britain and Russia gets stickier, because the British had more gold medals than Russia —29 to 24 — while Russia had more overall medals than Great Britain -- 82 to 65. Naturally, the British have developed a deep attachment to gold as a result; the Russians, in turn, have developed a newfound affection for medals of all kinds.
Who is right? The only definitive answer is that both parties have taken the wrong approach. The root of the difficulty lies in the so-called Arrow Paradox, named after the economist Kenneth Arrow, who electrified the social science community by publishing a Nobel-Prize winning book with the dreary title “Social Choice and Individual Values” in 1951. Arrow showed the conceptual problems of aggregating individual preferences in a wide range of common social settings.
The root of the Olympian difficulty is that the only clear information about these medals lies in their ordinal ranking—gold is greater than silver and bronze, and silver greater than bronze. Applied here, Arrow's lesson would stress that there is no way in which to break the tie between Russia and Great Britain without attaching numerical weights to the three color medals -- a task for which there is no natural convention.
To show how difficult this was, Arrow gave a famous example of a swimming match with many different teams participating. In order to decide who would win the meet, its organizers had to assign numbers to the order of finishers. The International Olympic Committee studiously refuses to engage in a similar exercise, precisely because it does not want to put its imprimatur on national winners. But once this ranking is attempted, strange results happen.
Let's imagine a hypothetical 20-team swimming meet, in which points are awarded to the first five winning swimmers. It is a large competition and the intuitive judgment is thus made that a team whose swimmers achieve both a second and third-place finish have accomplished something more difficult than a team that simply had one swimmer finish in first. So the agreed point scale is (in order from first place finish to fifth), 5, 4, 3, 2, and 1. Note that under this system the team that gets silver and bronze gets seven points and the first place finisher only gets five.
But as Arrow shrewdly observed, if this were a dual meet (which uses conventional scoring, where only the top three are awarded points -- 5, 3, and 1, respectively), the order in the hypothetical reverses, with the team that wins the race coming out ahead by a score of five to four.
In these pairwise comparisons, it would be ideal to ignore all of what Arrow termed the “irrelevant alternatives.” But it turns out that outcome is just not possible. The legitimacy of the two separate rankings, each for its own context, comes not from any inexorable law, but from the antecedent consent of all the parties, who agree in advance to compete under these rules.
What is wrong therefore with both the British and Russian contentions is that neither purports to assign intelligible weights to all different scoring positions. The British ignore both silver and gold; the Russians ignore the differences in the relative weights of all three medals. So it is instructive therefore to retrofit the inquiry in accordance with the two different weighting systems set out above. Here is the tally:
Gold Silver Br. 5,3,1 5,4,3,2,1
RUSSIA 24 26 32 230 320
G. B. 29 17 19 215 260
The Russians squeak by in the first system, but win the second in a walk.
Of course, these numbers are not set in stone. Many would say that, because of the attendant influence and honor, gold deserves a larger premium in both systems. This again changes the outcome.
If we set the value of the gold medal at 6 points, we create a dead tie in a 6-3-1 scoring system. If we set it at seven, Britannia again rules the waves.
However, if we create a 6-4-3-2-1 system, the Russians still prevail by 55 points. At seven points, their lead drops to 50.
The upshot: it really matters which system we use. The British will treat this as a head-to-head competition. The Russians will place it in the context of a larger meet. This writer will beat a tactical retreat and let each person decide the matter as he or she sees fit -- exactly as the IOC wishes.