Mathematical Olympic silliness

[simont]

At some point last week, while the Olympics were on TV, there was a five-minute segment mentioning that although China was at the top of the medals table by the IOC's official ranking, the USA's internal news services all put it at the top of the table – because the IOC likes to count gold medals and use the others to break ties, whereas the USA prefers to count total medals first and break ties by means of how many of them are which colour. (It was claimed that the USA has always counted this way, and that it was pure happenstance that on this occasion it happened to be a method of counting which put it at the top.)

So, just out of interest, I've prepared an alternative view of the final medals table for the 2008 Olympics, which simply does not take sides in debates of this sort: it shows which countries must be considered to have got a better medal haul than which other countries by any sensible ranking policy, and doesn't try to make arbitrary judgments between the rest.

http://tartarus.org/simon/2008-olympics-hasse/

I'm slightly surprised at how that turned out; I'd have guessed there'd be at least a few more unambiguous pinch points. As it is, the only countries in the entire table which can be sure of their position in the ranking are Russia, Great Britain, and the group at the bottom with one bronze medal each.

Comments

[the-alchemist.livejournal.com]

I think I possibly disagree with your definition of a 'reasonable' ranking system. I would be interested to see one which took into account the population of each country, on the grounds that medals (of each type) per person would be a better way of finding out which countries are more likely to produce winners.

[gerald_duck]

Unfortunately, once you go down that route the next thing people will point out is that it's population in the appropriate age range. In Swaziland, for example, something like 40% of the population is aged 16-30, where in Japan the proportion is nearer 20%. Then there will be attempts to normalise for development status, provision of sports facilities, etc.

[simont]

I agree, but I did specifically say on the page that for these purposes I was considering reasonable absolute rankings of just the medal count, and that rankings taking into account external factors of that nature were beyond the scope of the analysis.

(Apart from anything else, rankings scaled per head of population or per dollar GDP have a tendency to stop making sense once the medal counts get small enough to be statistically insignificant, which means that most of the interesting bits of your table are full of unhelpful anomalies.)

[the-alchemist.livejournal.com]

Sorry! Must have missed that bit.

[marnanel.livejournal.com]

I read your journal (among other things) for just such posts as this.

[douglas-reay.livejournal.com]

Medals by continent:
http://www.medaltracker.eu/

Medals by population and by GDP:
http://www.youcalc.com/pubapps/1219221778285/?cswid=48abc6bc903b61d0

[aldabra]

I-think-freakonomics did medals by population, and someone said in the comments that Jamaica would still have come top if it had won one gold and the US had got all other medals.

[emperor]

What a pleasing diagram :)

[gerald_duck]

I'm assuming there were also some countries that got no medals whatever and have been omitted from the diagram?

[simont]

I think so, yes, but they were also omitted from the raw data on which I based the diagram and I didn't feel like going and looking up all their three-letter codes.

[cartesiandaemon.livejournal.com]

Oh, that's lovely. About half way through I wondered if that's what you were going to do, and I'm glad it was.

I always love things like this, that show what ought to be unarguable, and show into sharp relief what you might disagree on. Looking at your diagram, it seems clear the question is not "Why is country X distorting the data?" but "Hey, China got more Golds and US more medals. I wonder why, and if there's any clear conclusion to be drawn about which did 'better'?"

(I did wonder if you could go any further and make any reasonable but but not unavoidable assumptions. Eg. would a country that won silver in every event be better than a country that won only one, gold, medal at all[1]? But obviously, you can't at all: the count-golds and count-medals algorithms are already at two of the extremes, and you can't make any such assumption without breaking one of them.)

[1] Even apart from that such an assumption may sound attractive, but couldn't be accepted as "the only fair way to do it" even if you wrote the gold-counting scoring of it off as an anomaly: remember we make an arbitrary cut off at third as a country that comes third once beats a country that always comes fourth by most metrics, so "all X+1 medals beats one X medal" may or may not be what you'd like to assume, but isn't unambiguously accepted as it is contrary to the current system, fair or not.

[simont]

The thing that I was vaguely thinking about taking it further is that not every topological sort of that graph (or, if you prefer poset terminology, not every total order consistent with that partial order) represents a fair ranking system. For instance, you could choose to rank a country with two silvers above one with a single gold, but further up the table you might choose to rank a country with one gold and a bunch of other stuff above a country with two silvers and the same other stuff. Locally, each decision would be consistent with the partial order, but globally one has to look at it and say there's something fishy about both decisions occurring that way in the same table.

So one might introduce a "fairness" or "consistency" criterion along the lines of insisting that this sort of decision has to go the same way throughout the table. Formally, it would go something along the lines of: suppose you have some function which maps a difference tuple of gold, silver and bronze medals to {positive,negative,zero}, in such a way that it's consistent with the previous rules (non-negative whenever g,g+s,g+s+b are all non-negative, non-positive when they're all non-positive) and also consistent with obvious ordering constraints (e.g. adding two positive things gives a positive thing). Then a total order can be derived from such a function by ranking each pair of countries according to the vector difference of their medal counts. Any such total order is "fair", in that there's some consistently applied comparison rule which gives rise to it; a total order which can be generated by no function in the above way is unfair and we want to discount it.

However, it wouldn't change the Hasse diagram as shown. I've convinced myself that any pair of countries marked as incomparable on my diagram can be ranked in an order of your choice by choosing an appropriate additive scoring system (x points for a bronze, y > x for a silver, z > y for a gold), and ranked in the other order by a different one. Any such additive scoring system is fair by the above definition, so no additional pair of countries becomes comparable due to the introduction of a fairness requirement.

[cartesiandaemon.livejournal.com]

Ah! Yes, that would be interesting.

[simont]

remember we make an arbitrary cut off at third as a country that comes third once beats a country that always comes fourth by most metrics

Indeed, and I think this is one of the strongest arguments for the IOC's lexicographic ranking being a more natural one than the USA's total-medals-first approach; in the IOC scheme, the fact that medals stop at bronze has the effect of making the list less precise (sets of countries which would have been contiguous in the rankings now become equal), but in the USA scheme the arbitrary cutoff after third place is actually critical to what order the countries end up in.

[cartesiandaemon.livejournal.com]

Yes, that's interesting, and I think it makes sense. It may or may not be fair, but it's certainly more consistent :)

[cartesiandaemon.livejournal.com]

I'm surprised GBR did so well, given the variety of the Olympics, and our comparative tinyness.

I think your method of explaining the ranking is best, but for the record, I wasn't sure if it fit my intuition better to say any scoring system must assign x, y and z points for each gold, silver and bronze medal respectively, and have:

0<x 0<=y<=x 0<=z<=y Which I think is completely equivalent, but made it immediately obvious that you really, really must do it that way.

[cartesiandaemon.livejournal.com]

No, wait, cancel that. That's an obvious way of doing it, but yours is more general, because it would (I think) cover any formula, whereas mine covers only linear ones.

[simont]

Also, mine covers non-Archimedean orders. The IOC's ranking system, for instance, is a lexicographic order in which one gold beats any number of silvers; that can't be expressed at all as an additive scheme with a positive real score for each of gold, silver and bronze, because for any additive scheme there must be some number of silvers large enough to beat one gold.

[cartesiandaemon.livejournal.com]

Good point. I think I assumed x,y and z would be rational numbers (equivalent to integers), and ignored tie-breaking as some extra step that might or might not involve counting red-haired witch-doctors in each team :)

(Although, I could retrospectively pretend that I intended the domain to include infinite ordinals.)

[simont]

And yes, everyone's a bit surprised GBR did so well. That's why all the news has been saying "wow, we've had a great Olympics"!

[cartesiandaemon.livejournal.com]

Oh, right. I haven't been watching any Olympic news whatsoever. (Which I haven't put a lot of effort into avoiding, but I guess I must concede must have been difficult :)) I assumed that was normal and was surprised we usually did that well, rather than surprised we did that well this time :)

I know where some of the Olympics have been, but I slightly embarrassedly admit I don't think I know any of the results, ever.

[cartesiandaemon.livejournal.com]

if you allocated π2 for a gold, π for a silver and 1 for a bronze, there could never be any tie in the total scores except when two countries had exactly the same medal counts in all three categories!

LOL. That's nice.

I expect that's true. Can any olympic sports result in a draw? If a timed event is a dead heat according to the best measurements, or a scored event has two players receive (a permutation of) the same scores from each judge, I don't know if: they can share a medal; or if they have to do it again; or if there's a coin-toss; or if it varies by event.

I assume no sport would divide a medal in any proportion but equally (since, obviously the player with a larger portion did better and ought to win outright), but I wouldn't be absolutely certain -- there are some very freaky sports out there somewhere :)

I guess the only other way pi-scoring could fail is if someone had the bright idea of partly introducing a game to the Olympics. Like Cambridge University awards (iirc) half- and quarter- blues to some varsity teams, instead of a blue[1]. If a gold at caber-toss counted as (pi-2) of a gold... But that would be silly :)

[1] That sounds unfair, but some varsity matches (eg. Tolkien Quiz, Tolkien Croquet, don't get blues at all! :)

[simont]

Can any olympic sports result in a draw?

I believe there are a few exact draws, yes, and what happens is that the medal does get shared. A draw in third place leads to that event awarding a gold, a silver and two bronzes; a draw in second place leads to two silvers and (I think) no bronze. In fact, apparently each of those happened once this year. I don't think there was a draw in first place at any point, but it seems clear that there'd be two golds and a bronze awarded if there were. (Or there might be emergency sudden-death playoffs of some sort, but if those failed then that'd be the obvious fallback position.)