Jason (jcreed) wrote,

Say an n-preference structure on the finite set X is a sequence of sets P_0 ... P_{n-1} such that P_0 = X subject to the following axioms:
1. for every i in [1,n-1], we have that P_i is a strict total order on P_{i-1}.
We write p < p' if (p,p') in P_i for some i.
Let * be the following partial operation of "merging" two ordered pairs
defined by (a,b) * (b,c) = (a,c), and (a,b) * (b',c) is undefined if b != b'.
Then for any i in [1,n-2] and any ordered pairs I,J,K,L in P_i X P_i we have
2. I * J > I if I * J is defined
3. I * J > J if I * J is defined
4. If I > J, then I * K > J * K if both merges are defined
5. If I > J, then K * I > K * J if both merges are defined
6. If I > J and K > L then I * K > J * L if both merges are defined
7. If I > J and K > L then K * I > J * L if both merges are defined

Then we can observe that "nearly all" mappings from X to the reals generates an n-preference structure for any n we might choose. In fact we could easily extend the above definition to an "omega-preference" structure. The order P_1 is given just by comparing the real numbers with ordinary < on the reals. The order P_2 asks us to rank all of the facts in P_1: so what we do is assign each pair (a,b) in P_0 the valuation b-a and sort them accordingly. Similarly we choose the order P_3 according to the rankings of elements ((a,b),(c,d)) by the valuation (d-c)-(b-a), and so on for greater n. We'll fail to have a strict order at some level, for instance, if the elements of X are equally spaced in the reals. In that case we are indecisive about ordering the elements of P_1. This is what I mean by "nearly all", that there are some degenerate cases.

Anyway, here's a question: is every omega-preference structure witnessed by some mapping from a finite set X to the reals? Certainly it won't be unique, because pointwise translations and scalings of the real-valued function on X wouldn't affect what order structure it generates.

The motivation for this is thinking about voting systems again. Analysis of voting systems begins with a good abstraction, namely asking everyone to rank their preferences for candidates. For comparison, a bad abstraction is asking everyone to give (open-ended) numeric scores to all candidates, which would degenerate into a contest of "who can name the biggest number" to express their approval or disapproval. But, and this is my point here, I think there might be other interesting good abstractions of preference, maybe not the one I just outlined, that capture strictly more information than ranking. The first step was observing that one might allow the voting guinea pigs in some hypothetical system to express the relative importance of their pairwise decisions. Like, "I like Kerry better than Nader, and Nader better than Bush, and the distance between Kerry and Nader is smaller than the distance between Nader and Bush". And then that lends itself to an obvious n-ary and omega-ary generalization, as above.
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