After unambiguously specifying the partial ordering for the triple, its very gratifying to note how each node gets an ordinal rank in the Hasse. So you can talk about
USA > Russia ( rank 0 vs rank 1) but can't say anything about USA vs China ( both rank 0) - you can do this simply in terms of the ordinal rank and forget all about the medal tally.
A few years back, there was a data science problem to predict which startup had the best chances of exit, based on similar set of presumably unquantifiable predictors. There were some 10000 startups, and for each startup you knew who the VC's were, number of rounds of funding, valuation etc. Though it seemed quite an amorphous problem, we hit upon a very similar partial order, that made all the startup nodes fall into a Hasse and you could practically read off from top to bottom which startup would succeed and which ones would fail.
At that time, I remember thinking how you could do the same exact thing for humans too. Quantitative reductionism at its finest, sure, but when it works it's pretty amazing.
> After unambiguously specifying the partial ordering for the triple, its very gratifying to note how each node gets an ordinal rank in the Hasse. So you can talk about USA > Russia ( rank 0 vs rank 1) but can't say anything about USA vs China ( both rank 0) - you can do this simply in terms of the ordinal rank and forget all about the medal tally.
You're correct, and on this particular example you can see the problem immediately: is France in the same "rank" as Korea or Italy? Neither, the question is ill-posed.
since france isn't connected by arrow to korea, or vice-versa, the article rightly says you can't determine who'se ranked higher. you have (13>7, 23=23,31<40) in this case. you clearly want the first inequality to prevail.
I think you've missed the point that willun is making. It's in general impossible to assign a numeric "rank" to a partially ordered set such that the rank(A) > rank(B) iff A > B and rank(A) = rank(B) iff A is incomparable to B. I'm pointing out a concrete example where you cannot assign a rank to France that is consistent with the rank of Italy and the rank of Korea. Since France is incomparable to both Italy and Korea, it must have the same rank as both, but Korea is better ranked than Italy. Contradiction.
The point is that your idea of "ordinal rank with ties" isn't specified.
Your ordinal rank is essentially a question of "how many teams are ranked above this one". ie US and China are both 0, Russia is 1 because there is no 1 otherwise, GBR is 2, AUS/GER are tied for 3, Korea and France are 4, and Italy and Japan are 5. But there's no way to claim that Italy and Japan should be below France.
That sounds familiar to the various voting systems that satisfy the Condorcet criterion (basically, ignoring all votes for other parties, do people prefer A to B?). There are a lot of different ways of picking a winner when two or more candidates are tied under Condorcet -- ranked pairs, beat paths, etc, including rules like the one you suggest.
But in voting systems, Arrow's Theorem says none of those methods can be truly satisfactory, at least if you accept Arrow's list of requirements.
Similarly for Olympics ranking, I don't think any method for breaking ties will be able to satisfy all the requirements of a good ranking (e.g., removing a country from competition shouldn't change any other rank orderings).
So personally I don't think any of those tie-breaking methods really have any meaning. You might as well just toss a coin.
> Different people have different conventions for how to rank the Olympic medal results.
The newspaper in my country sometimes used another method, where they divide the medal count by the population of the country. In 2008 that put Jamaica at the top, followed by New Zealand. Both China and the U.S. were well down in the chart.
That partly measures a policy choice "how good must athletes be for us to send them to the Olympics?"
Different countries make different choices there. Small poorer countries, for example, aren't likely to have any athlete who can get to the final in the 100 meters, but yet may want to send an athlete to that event, simply because it is one of the cheapest events to send someone for, and they don't want to send a zero sized team (for example because they want to earn 'legitimacy points' as a country or because their Olympic committee wants to make a trip to the Olympics)
The USA, on the other hand, isn't likely to send anybody who can't make the final in that event. Part of the reason for that is that they try hard good candidates, but part also simply is that they can make way more throws of the genetic dice, and thus are more likely to hit on an outlier who performs exceptionally well.
If you have a notion of "country" that you're measuring things against, then the genetic darts they have is just an intrinsic property. Normalizing by total population is almost certainly not the right metric, but to your point, there could exist more sophisticated distribution-based metrics that better capture the distribution mean which is what you might be after.
I did this exact thing (even the part where I displayed the results using Graphviz) to come up with a partial ordering of General Managers of Major League Baseball teams. The partial order was simply:
GM_of_Team A > GM_of_Team B iff (endofseason_ranking (A) > endofseason_ranking(B)) and (budget(A) < budget(B))
So this would capture how much "bang for the buck" a GM is getting. I did it for a couple of years, and not surprisingly, there's quite a bit of year-to-year variability. Teams go through cycles where they increase the budget to capitalize on a favorable situation, but then they get saddled with bad contracts and are disadvantaged in the draft, etc. Still, if you do this over the long term, there could some teams/GMs that are consistently good.
I remember that my first attempt generated really busy diagrams because the ordering is transitive, so if you simply draw an arrow whenever the ordering holds you end up with the transitive closure. A couple of hacks later (basically, add a test to check if this ordering can be obtained by triangulation, and if so, don't display it) I managed to end with the same kind of diagram as the OP.
The theoretical property those hacks were aiming at showing is called the transitive reduction. And you don't need any hacks to calculate it, graphviz comes with the `tred` tool to do so.
Where I'm from, it's "inarguable" that [1,0,1] > [0,2,0]; that is, that a gold and a bronze is better than two silvers. This would break some of the ties shown in the diagram.
Your statement is unclear; are you asserting the truth or false of the inequality? (The unclear word is "inarguable" - do you mean there is no argument in favour or no argument against)
This reminds me of the beatpaths graphs I used to do for the NFL (http://beatpaths.com) - I'd then come up with power rankings by applying a topological sort. It was all sort of silly and tongue in cheek, but fun.
why does allocating "pi^2 for a gold, pi for a silver and 1 for a bronze" mean "there could never be any tie in the total scores except when two countries had exactly the same medal counts in all three categories"?
I'm sure someone will pipe up with a more exact answer (likely using permutation and/or combination in their specific mathematical senses), but I believe it's because summing medals is algebraic, and with the given weights there's no way for the sum of any combination of non-identical medals that equals any other. The values of 𝛑 and 𝛑² (and the patient, unacknowledged workhorse 1) provide this. Looking at a simple case where s is the number of silvers and b the number of bronzes, there's no value of b that can equal any value of s·𝛑. You could use another triple of weights, such as (𝛑, e, 1) as well.
Medals are awarded in integer quantities, and you don't want j·g+k·s+l·b=0 to be possible for integer j,k,l (except when all three medal counts are 0).
This is actually a slightly stronger condition than is needed, because no country can earn negative medals and there are a finite number of medals available, so you could actually have the weights for eg. gold and silver medals differ by a rational factor as long as the denominator was large enough.
A few years back, there was a data science problem to predict which startup had the best chances of exit, based on similar set of presumably unquantifiable predictors. There were some 10000 startups, and for each startup you knew who the VC's were, number of rounds of funding, valuation etc. Though it seemed quite an amorphous problem, we hit upon a very similar partial order, that made all the startup nodes fall into a Hasse and you could practically read off from top to bottom which startup would succeed and which ones would fail.
At that time, I remember thinking how you could do the same exact thing for humans too. Quantitative reductionism at its finest, sure, but when it works it's pretty amazing.