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.
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.
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.
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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.
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.
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.