Impossible Ranking Systems
Even when approached with seemingly the most scientific methods, college football ranking systems have always seemed vague and completely subjective to me (and the end of the year awards presented to individuals too for that matter). Science News recently had an article outlining how such systems' ability to produce "reasonable results" are inherently impossible.
In a paper published in a recent issue of SIAM Review, Paul K. Newton and Kamran Aslam of the University of Southern California argue against the widespread belief that it is possible, with just the right tweaking, to come up with a ranking system that yields reasonable results and eliminates logical inconsistencies—and, hence, settles all arguments, leaving everyone satisfied.
At the heart of the argument is the challenge of assumptions made when coming up with the various ranking systems. Highlighted is the assumption that "when team A is ranked higher than team B, and team B is ranked higher than team C, then team A is ranked higher than team C...seems like a reasonable requirement". This assumption is shown to be faulty, particularly when votes are part of the process.
So how do the bowl games get determined, if not by some ranking process? That's the million dollar question (not that the collegiate atheletes get any of it, at least not legally...). Well, unless another option is presented, science be damned (uh?), as the current system is what we have that works best so far.
Tangentially, this reminded me of a (not-so-recent) post on InsomniousPolitico where there was an attempt to classify various popular dichotomies (the term is used loosely) into two distinct groups; an attempt met with many vociferous comments as the ultimate goal seemed to be grouping logic, men, and conservatism against emotion, women, and liberalism (go see and decide for yourself). In this Science News article, the aforementioned faulty assumption and the example they chose to illustrate it (the selection of the top men's tennis player in 2002) is also exactly why Jaz's attempt to make two mutually exclusive groups won't work.
Let's say you have 3 groups of 2 instead of 3 individuals, groups A, B, and C. Group A may match up with group B in a particular way, and group B may match up with group C in a particular way, but that does not say anything about the relationship between group A and group C, which must be handled seperatly (particularly when the matching up of groups is as subjective as was outlined in the post). As in the tennis example, it is possible to have, even in a sample space as small as 3, a circular state of relation between the groups. Consider the following pairings:
| Pairing 1 | Pairing 2 | Pairing 3 | |||
| A1 | A2 | B1 | B2 | A1 | A2 |
| B1 | B2 | C1 | C2 | C2 | C1 |
There are only eight possible ways the three groups can be grouped together, and all of them will go against how we defined the group pairings above in exactly one way.
| Potential Group | Bad Because of |
| A1, B1, C1 | Pairing 3 |
| A1, B1, C2 | Pairing 2 |
| A1, B2, C1 | Pairing 1 |
| A1, B2, C2 | Pairing 1 |
| A2, B1, C1 | Pairing 1 |
| A2, B1, C2 | Pairing 1 |
| A2, B2, C1 | Pairing 2 |
| A2, B2, C2 | Pairing 3 |
Well, you can't blame a guy for trying (to equate conservatism with logic). Anyway, sorry for what was I'm sure way too much information...I have occasional relapses into math education background. And I miss making tables.
3 comments:
“…is also exactly why Jaz's attempt to make two mutually exclusive groups won't work.”
For the billionth time, never did I claim that each grouping was “mutually exclusive”. This is most likely why you so ardently disagree with my premise. If my arrangement presupposed mutual exclusivity then it would follow that men are devoid of emotion and women, of logic. This is obviously absurd and not at all the point of my observation, which, for the record, would be more fairly and accurately characterized as women, in general, are predisposed to favor emotion when making a decision whereas men favor logic.
While I understand your general point about college football and tables, My “Everything and Everything” post is hardly a mathematical table, rather it is a philosophical general observation which, rather than mutual exclusivity, presupposes the “if you had to pick” general classification and condition. Therefore, if you disagree with my classification you should make the opposite argument that is, in fact left leaning individuals and women are more predisposed to rely on logic and men are primarily governed by emotion.
It is my position that in fact, if anything, the opposite is true.
I guess this round goes to me. "...you can't blame a guy for trying."
Round of what?
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"...it is a philosophical general observation which...presupposes the 'if you had to pick' general classification and condition. Therefore, if you disagree with my classification you should make the opposite argument that is, in fact left leaning individuals and women are more predisposed to rely on logic and men are primarily governed by emotion."
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You obviously missed the point of the post (again), so I'll try to be even more elementary:
Nowhere did I "disagree with your classification". In fact the post has nothing to do with whether men or women are more logical or more emotional. It is a visual demonstration of how a paradoxical case can be made for ANY given set of pairings greater that 2. You could be talking about apples-bananas, reds-yellows, and tastys-disgustings; it's all the same.
You are right about my incorrect use of the term "mutually exclusive" sets. I used this term because I assumed it was a property of the "pairings" listed off.
By the way, when you say "If my arrangement presupposed mutual exclusivity then it would follow that men are devoid of emotion and women, of logic.", you aren't entirely correct. Even thinking of the pairings as "mostly one thing" vs "mostly another thing" is a mutually exclusive definition.
So, since we aren't using mutually exclusive sets here, couldn't men and women, to use your most famous example, both be mostly logical? If "logical" means "what makes sense" then a case can be made that people, regardless of gender, always do what makes sense to them, and what you are debating is a difference in definition. You have your own (sometimes very unique) brand of logic and others have theirs.
If the case is, however, the "if you had to pick" classification, then it is a case for mutually exclusive sets, and the post's logic holds.
All that being said, you may in fact be right in your alignments; it is subjective by its very definition and is a matter of an individual's opinion.
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