An ordinal ranking that satisfies some basic and intuitive criteria has a continuous cardinal representation. As for differentiability, this is often not neccessary for many economic models. Therefore, even if you think derivatives are suspicious, many mainstream results would still stand.
An ordinal ranking that satisfies some basic and intuitive criteria has a continuous cardinal representation.
What about the simple ranking of 4 objects in table 4 [page 257] of this article http://www.econ.ohio-state.edu/jhm/papers/AustrianOrdinalMU.pdf, and the simple proof on page 274 that there is no cardinal ordering possible?
As for differentiability, this is often not neccessary for many economic models. Therefore, even if you think derivatives are suspicious, many mainstream results would still stand.
But what is the point, then, of using a cardinal ordering, something mainstream economists concede is disconnected from reality [=nobody has a cardinal ordering in his head of anything]?
I’ve read McCulloch’s work before; thanks for reminding me of it. It deserves more attention, since it’s method of utility analysis is very unique.
McCulloch’s counterexample is not really a counter example. He is imposing very strict and unusual assumptions about the utility function, which most theorists do not impose.
Cardinality makes things much easier to work with, regardless of differentiability. I won’t get into the details, but many theorems have been discovered that allow one to get the same results as differentiable functions without this feature.
McCulloch’s counterexample is not really a counter example. He is imposing very strict and unusual assumptions about the utility function, which most theorists do not impose.
Couls you be more specific, please? What strict and unusual aasumptions does he impose?
Well, he directly states his assumption at the begining of the section. He wants a function that ranks subsets using the relative rankings of their elements. That’s not exactly standard. Most utility functions are not measures. It may be plausible in some cases, but it does not serve as a universal counterexample.
He wants a function that ranks subsets using the relative rankings of their elements. That’s not exactly standard.
But it seems quite realistic, something that might happen constantly in a real life situation. So are you saying that the standard just doesn’t bother with those ordinals because they are inconvenient to deal with? Sort of like a doctor who doesn’t treat broken bones because it’s too messy and not amenable to pills?
…it does not serve as a universal counterexample.
Not sure what you mean by a universal counterexample. Something is either a counterexample or it isn’t.
My point was just that it’s a very strict assumption, which is clearly not true for many situations in life. If you read the paper he cites by Pratt and Kraft, you can see more clearly that this is a strange assumption, since, in essence, he is assuming that preference representations look like measures. Why is this realistic?
What I originally said still stands. His example is not a counterexample. It is only a counterexample to his own question, as posed at the beginning of that section.
My point was just that it’s a very strict assumption, which is clearly not true for many situations in life.
But it is true in many other situations in life.
In any case, we aren’t looking for something that true in all situations in life, or even many situations in life. We are looking for a counterexample to the statement that all the interesting and realistic ordinal rankings can be turned into cardinal rankings, and he has found one [in fact, two. Table 4 and table 16 of his paper].
Why is this realistic?
Because it quite possible that four real life objects, in the minds of real life people, have the ordinal ranking he assigns them. AKA realistic. What is unrealistic about it?
Perhaps you could provide a link to the paper by Pratt and Kraft.
What I originally said still stands. His example is not a counterexample.
That’s not what you said earlier. You said it is indeed a couterexample, but not a “universal” one. But you did not explain what a universal counterex is.
It is only a counterexample to his own question, as posed at the beginning of that section.
But his question, as posed in the beginning of the section, is whether one can find an ordinal ranking that cannot be made cardinal. Which is exactly what we are looking for, and that he displayed.
If you don’t think I’m correct, I encourage you to email McCulloch. I’ve corresponded with him before, and he answers emails.
That said, I’ll try to explain this again. McCulloch is asking whether a particular kind of cardinal representation exists for a given ordinal ranking. This is similar to asking whether a differentiable representation exists. In either case, the answer may be in the negative, as McCulloch shows.
That does not mean that there does not exist a cardinal representation of some kind. Therefore, my original comment is correct.
OK, I finally see what you are getting at. You are saying McCullough is placing a restriction on the cardinal function, not on the ordinal function [as I thought you meant until now].
Will look into this. TY