I have an index number which is nether strictly cardinal nor strictly ordinal. Instead, the ratio between two differences is meaningful, although neither the differences nor the ratio between two index numbers are meaningful in themselves.
An index with the property that a person engaged in constrained maximization of the expected value of the index will choose what they do choose will have this property. You can see this by observing that neither multiplication of the index by a constant, nor addition of a constant to the index, will alter in any way what is chosen. The index is merely a tool to compactly summarize an individual's choices
I have two closely related questions. The first, which I hope is relatively easy, is whether this form of ordering (my first draft called this a quasi-order, until I learned that this has a precise technical meaning) has a name that can lead me to a literature. The second, and more difficult, is whether there is any straightforward mathematical object which has this property. I wish to employ the same approach as physicists did when they found it was impossible to simultaneously learn the momentum and location of a particle. declaring that a particle with a precisely known momentum did not have a position. This imaginative leap, which has always struck me as an effort to sweep the problem under the rug rather than solve it, has proven surprisingly fruitful, as when tunnelling – particles leaping through a gap apparently without crossing the space between – was shown to occur.
I wish similarly to model the belief that, although you may assign the index of different choices to a number as an intermediate step when modelling choice, nonetheless neither differences nor the ratios of these numbers exist, though ratios between differences do. The rhetorical problem is that so long as the index values are real numbers it is hard to persuade anyone that the ratios do not exist. It appears obviously to exist – just divide he one number by the other. I am hoping that the genuine nonexistence of ratios and differences, if properly expressed mathematically, may reveal unexpected structure in the index or the underlying choices, as the nonexistence of position in the mathematics of measurement revealed unexpected behavior in actual particles