Hasse diagram of the 2008 Olympic medal table
tartarus.org
tartarus.org
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.
I think you can't do that on a partial order.
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.
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.
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.
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.
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.
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.