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> 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 ta
by wilun 9y ago
> 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.
I think you can't do that on a partial order.
- tgb 9y agoYou'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.
- dxbydt 9y agosince 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.
- joshuamorton 9y agoThe 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.
- tgb 9y agoI 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.
- dxbydt 9y agoI understand now. Thank you!
- kzrdude 9y agoI see that France dominates 2 fewer nations than South Korea. Does this number have any meaning? In a way, South Korea did better in this way.
- iainmerrick 9y agoThat 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.
- deleted 9y ago[deleted]