4 ms·

> The problem is that it makes no sense to even formulate a nonlinear utility function for an ordinal scale, because there will be a linear function that repres

> The problem is that it makes no sense to even formulate a nonlinear utility function for an ordinal scale, because there will be a linear function that represents the same preferences. I'm afraid I don't understand your point. There won't always be a linear function representing the same preferences and anyway 'ordinal' utility functions that represent the same preference order mean the same thing, so they make as much sense. > EU makes no sense with purely ordinal utilities either, but economists use it all the time. When they do so, they assume cardinal utilities They use cardinal utility but they don't just assume it. The assumptions (implicit in a lot of cases, admittedly) behind Expected Utility are generally the Von Neumann-Morgenstern axioms (relating to ordinal preferences over probability distributions over outcomes) [0]. > the Principle of Diminishing Marginal Utility makes no sense with purely ordinal utilities True, but I think very few if any economists would be confused about this. Also economists and econ text books (particularly after 101) are more likely to talk about marginal rates of substitution [1] or convex preferences [2] than diminishing utility. [0]: https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenster... [1]: https://en.wikipedia.org/wiki/Marginal_rate_of_substitution https://en.wikipedia.org/wiki/Marginal_rate_of_substitution [2]: https://en.wikipedia.org/wiki/Convex_preferences https://en.wikipedia.org/wiki/Convex_preferences

jonathanstrange 8y ago
> I'm afraid I don't understand your point. There won't always be a linear function representing the same > preferences and anyway 'ordinal' utility functions that represent the same preference order mean the same thing, so they make as much sense. I've become a bit unsure about this myself (see below). An ordinal utility function only represents the underlying preferences according to the representation condition (1) u(a)≥u(b) iff. aRb, where "R" is the underlying weak preference relation, i.e. a complete preorder relation. For finite and countably infinite domains no further conditions are needed, for uncountably infinite domains you need additional conditions (Debreu 1954). The scale of an ordinal utility function is, according to S. S. Stevens terminology, called an ordinal scale. It is characterized by admitting all strictly monotone increasing transformations.[1: 64] That means that whenever u(x) is a utility function representing R, then u'(x):=au(x) for real number a>0 also represents R in the sense of (1). Question to you: Does it not follow from this that for every nonlinear utility function representing R in the sense of (1) there is also a linear utility function representing R in the sense of (1)? For the countable and uncountably finite domain this should be easy to prove, since a linear utility function can be constructed for any such domain and any R. But I admit I don't know how to prove this for the uncountably infinite case. The way I put it, we'd have to prove that there is strictly monotonically increasing transformation of any nonlinear function into a linear function that preserves the order defined by R. I thought this was obvious but maybe I'm wrong. :/ [1]: Roberts, Fred (1979): Measurement Theory. Addison Wesley. > They use cardinal utility but they don't just assume it. The assumptions (implicit in a lot of cases, admittedly) behind Expected Utility are generally the Von Neumann-Morgenstern axioms (relating to ordinal preferences over probability distributions over outcomes) So they assume these axioms, that's what I'm saying. Not assuming them would e.g. be actually measuring preference intensities in decision makers (by direct scoring?), or eliciting preference difference comparisons of the form u(a)-u(b)>u(c)-u(d) from decision makers and having good independent reasons for them to make sense.

jonathanstrange 8y ago

> I'm afraid I don't understand your point. There won't always be a linear function representing the same > preferences and anyway 'ordinal' utility functions that represent the same preference order mean the same thing, so they make as much sense. I've become a bit unsure about this myself (see below). An ordinal utility function only represents the underlying preferences according to the representation condition (1) u(a)≥u(b) iff. aRb, where "R" is the underlying weak preference relation, i.e. a complete preorder relation. For finite and countably infinite domains no further conditions are needed, for uncountably infinite domains you need additional conditions (Debreu 1954). The scale of an ordinal utility function is, according to S. S. Stevens terminology, called an ordinal scale. It is characterized by admitting all strictly monotone increasing transformations.[1: 64] That means that whenever u(x) is a utility function representing R, then u'(x):=au(x) for real number a>0 also represents R in the sense of (1). Question to you: Does it not follow from this that for every nonlinear utility function representing R in the sense of (1) there is also a linear utility function representing R in the sense of (1)? For the countable and uncountably finite domain this should be easy to prove, since a linear utility function can be constructed for any such domain and any R. But I admit I don't know how to prove this for the uncountably infinite case. The way I put it, we'd have to prove that there is strictly monotonically increasing transformation of any nonlinear function into a linear function that preserves the order defined by R. I thought this was obvious but maybe I'm wrong. :/ [1]: Roberts, Fred (1979): Measurement Theory. Addison Wesley. > They use cardinal utility but they don't just assume it. The assumptions (implicit in a lot of cases, admittedly) behind Expected Utility are generally the Von Neumann-Morgenstern axioms (relating to ordinal preferences over probability distributions over outcomes) So they assume these axioms, that's what I'm saying. Not assuming them would e.g. be actually measuring preference intensities in decision makers (by direct scoring?), or eliciting preference difference comparisons of the form u(a)-u(b)>u(c)-u(d) from decision makers and having good independent reasons for them to make sense.