On Counting (or Weighting) Olympic Medals
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.
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Comments:
Sep '10
Re: On Counting (or Weighting) Olympic Medals
My only exposure to Kenneth Arrow was a Marxist economics professor describing "Arrow's Impossibility Theorem" which "proved" that "democracy doesn't work."
My retort?
"Apparently your ordinal ranking left you stuck at a third tier university dump like this one, so I can see why you'd be bitter."
Edited on August 14, 2012 at 9:23pmMay '12
Re: On Counting (or Weighting) Olympic Medals
I've been hearing a lot of the 'Gold medals matter, silver & bronze don't matter' argument from the Brits lately. I'm not sure what's happened to Britain, but they've become even more sour and unpleasant than the French. Teddy Roosevelt's 'Man in the Arena' speech should recited daily to every Brit.
I really dislike parsing medal worth, but if the Brits seem to want to go that route then let's parse their medals in detail. If one compares the Russian medals & the British medals one notices that Britain's medal tally is due to medals from sports where the demands of competition, physical talent, and skill are not as high. Their tally is dependent upon what I call 'geek' sports where the cost to even enter them is really high which limits the number of competitors & quality of competition. Which is more impressive? Medals in track & field, wrestling, and swimming, vs. canoe, rowing, and sailing? The former obviously, because there are fewer barriers to entry eg costs and thus far more competitors.
If one judges the quality of medals on those criteria, Britain's medal tally looks far less impressive.
Edited on August 14, 2012 at 9:55pmSep '10
Re: On Counting (or Weighting) Olympic Medals
Second place is the first loser.
That's all.
Mar '11
Re: On Counting (or Weighting) Olympic Medals
Meh.
To quote Abba:
"The winner takes it all
The loser's standing small
Beside the victory
That's a destiny"
Apr '11
Re: On Counting (or Weighting) Olympic Medals
While I will always take the view that anything and everything should be done to rub salt in Russian wounds, I feel that to a certain degree the national posturing that the Olympics bring about is actually a disservice to the spirit of Athleticism and Sport. It is hard to disentangle national pride from international events of course, but I have felt that athletics at its heart should be about individual achievement unmoored from collectivist impulses of any sort. Of course I say this as a fan of team sports and a dedicated backer of US teams everywhere.
But, really do such medal contests serve to better ennoble us? The beauty of sport is in the constant competition and struggle. Some times you are up, some times you are down, but you are always playing your best and hardest. The tally only matter if you don't plan to keep playing. But, hey I'm a Chicagoan, so maybe I'm biased.
Mar '11
Re: On Counting (or Weighting) Olympic Medals
More Abba:
"The gods may throw a dice
Their minds as cold as ice
And someone way down here
Loses someone dear
The winner takes it all
The loser has to fall
It's simple and it's plain
Why should I complain."
Besides, the BBC decides the weighting, and they are always right.
Mar '11
Re: On Counting (or Weighting) Olympic Medals
And this, boys and girls, is why we don't have, never will have, indeed can't have, a Bowl Championship Series that is worthy of the name. The apple of an SEC Championship versus the orange of a Pac Ten Championship versus the lemon of being Notre Dame...the circle will not square.
Jan '11
Re: On Counting (or Weighting) Olympic Medals
Which states supply America's medals?
I live in Maryland, home of Michael Phelps, so I'm feeling pretty smug right now.
Feb '12
Re: On Counting (or Weighting) Olympic Medals
Richard VanderHoek: Second place is the first loser.
That's all. · 6 hours ago
Best line in "The Rock"
"Losers whine about their best; winners go home and [redacted] the prom queen."
Jan '11
Re: On Counting (or Weighting) Olympic Medals
KC, Utah supplied most of all of the medals, via KUCC!
Yes, it's a trivial "first world problem", but for mine it's 3,2,1, then divided by population.
May '11
Re: On Counting (or Weighting) Olympic Medals
I have a more simplistic view of it. The country that wins the basketball gold wins. Period. Well, maybe soccer. Fencing, table tennis, rhythmic gymnastics, blah, blah, blah... Who cares?
Also a winner, the coach of the sprinters on the Jamaican team, whose name I have never heard mentioned in the media coverage. Either him, or some chemist down there who is adept at creating undetectable drugs, but I doubt that. Have you ever BEEN to Jamaica? I'm not sure there is a chemist on the entire island.
A modest proposal for increasing the U.S. medal count: Include new events in swimming: The 101 meter frestyle, 102 meter frestyle, 103 meter frestyle, 104 meter frestyle... You get the gist. Phelps would have come home with 73 gold medals.
Jun '12
Re: On Counting (or Weighting) Olympic Medals
Personally, if rhythmic gymnastics and synchronized swimming are Olympic sports, why not bowling?
Jan '11
Re: On Counting (or Weighting) Olympic Medals
Tony Martyr: KC, Utah supplied most of all of the medals, via KUCC!
Yes, it's a trivial "first world problem", but for mine it's 3,2,1, then divided by population. · 34 minutes ago
You got me - can't argue there!
Dec '11
Re: On Counting (or Weighting) Olympic Medals
Richard VanderHoek: Second place is the first loser.
That's all. · 16 hours ago
And really, this whole discussion is to declare who came third (or the runner-up to the first loser position), as China was clearly the first loser.
Jun '12
Re: On Counting (or Weighting) Olympic Medals
Professor Epstein, something just occurred to me.
How to the UK and Russia rate if you calculate it based on the value of the precious metals in the medals (at, say, the time of the official closing of the London Games) and the physical weight of the medals?
Nov '10
Re: On Counting (or Weighting) Olympic Medals
Personally, I've always just thought that gold was what matters, and silver and bronze counted primarily as tie-breakers.
May '12
Re: On Counting (or Weighting) Olympic Medals
nm
Edited on August 17, 2012 at 9:09amMay '12
Re: On Counting (or Weighting) Olympic Medals
Amy Schley
Richard VanderHoek: Second place is the first loser.
That's all. · 6 hours ago
Best line in "The Rock"
"Losers whine about their best; winners go home and [redacted] the prom queen." · Aug 14 at 7:49pm
Spoken like people who have never played the game...
From my experience, the fans who usually smack talk the most usually fall silent when then run into the team or players that lost. I really doubt anyone on this thread would call a silver or bronze medalist a 'loser' to their face.
Edited on August 17, 2012 at 9:11amMay '12
Re: On Counting (or Weighting) Olympic Medals
David Williamson
Meh.
To quote Abba:
"The winner takes it all
The loser's standing small
Beside the victory
That's a destiny" · Aug 14 at 1:42pm
Bill Shakey>Abba