Comment by jonathanstrange | Hacker News Reader

Comment by jonathanstrange | Hacker News Reader

jonathanstrange

I'm not convinced by the way the author argues. For example, he gives U=aln(quantity of apples)+(1-a)ln(quantity of oranges) as an example of a utility function and in the very same section claims that neoclassical economists only use ordinal scales, hence the criticism of presuming cardinal utility doesn't apply.

But does it really make sense to use a nonlinear function to represent ordinal preferences?

As a philosopher working in the so-called theory of value structure - basically, about the meaning of 'better than' - my criticism of preference modelling would be different and applies to both camps. First, if the modelling is descriptive we know that people have e.g. non-transitive preferences and completeness also doesn't hold. Second, there are also pretty good arguments against the transitivity of 'better than' from a normative perspective that at least need to be defused somehow. IMHO, the right solution points towards conditional preferences and/or lexicographic decision procedures. In a nutshell, multiple criteria interact in complicated ways and the assumption that the outcome of aggregating them is a neat complete preorder is generally not warranted. (And there is plenty of work on nontraditional decision theory by e.g. Fishburn to support alternatives.) Third, there are good arguments why some of our evaluative comparisons between items are cardinal, at least on an interval scale, whereas others are purely ordinal. I understand why this would be an undesirable point of view for an economist, though, since there is no known rationally and normatively justifiable way of aggregating ordinal and cardinal utilities into a meaningful overall assessment. From a measurement-theoretic perspective, an overall ordinal utility is the best you can get in this case. There seems to be a certain tendency of economists to dismiss certain plausible views because they are technically undesirable rather than on the basis of substantive arguments.

Another thing is that as far as I can see, economists who presume ordinal utilities do not seem to care enough about Arrow's Theorem in contexts with multiple attributes. At least they should make clear that IIA ought not hold when ordinal preferences are aggregated.

I think I agree with the crux of what you have to say, but do you have an example of people having non-transitive preferences?

I can't think of one off the top of my head, and the classical economic argument is that non-transitive preferences would create a 'happiness pump' where I get stuck in a loop trading one thing for another and feeling good about the trade every time.

These are called Spectrum Cases in the metaethical literature and have been discussed by Larry Temkin, Stuart Rachels, and others. Here is an example:

A_1: One year of intense pleasure.

A_2: Two years of slighly less intense pleasure than A_1.

A_3: Four years of slightly less intense pleasure than A_2.

. . .

A_n: 2048 years of very low pleasure (slightly less intense than A_{n-1}).

In every pairwise comparison we judge that A_{i+1} is better than A_i. However, we also judge that for some A_j in the spectrum, A_1 is better than A_j. Repeated applications of transitivity would contradict this, hence "better than" cannot be transitive.

Before you dwell on the example, note that the same construction can be made with well-being and it may be made as realistic as you desire. As the size of his book already hints, Temkin (2012) discusses a vast number of variants in many evaluative domains.

Money pumps under certainty are one argument against this. The problem is to explain what's wrong with your judgements in Spectrum Cases then. There are many other replies. John Broome suggests that "better than" is simply transitive by virtue of its meaning, but that misses Temkin's point and he's perfectly aware of this reply. You can also get away with giving up completeness, but the solution is not very convincing. It boils down to claiming that two adjacent items are incomparable somewhere in the spectrum, which is extremely implausible. In my personal opinion the only viable solution is a lexicographic approach that takes into account that two or more dimensions at play. I'll spare you the details, though.

On a side note, there is an older tradition of arguing against the transitivity of "better than" based on measurement errors or the indistinguishability of adjacent items. This started with Luce (1956). In these cases, only the transitivity of "equally good" fails but "strict better than" remains transitive, hence this can be modelled with semiorders or interval orders. The representation theorems for utility representations change, but this is well-explored and not a problem. Spectrum Cases are much worse, if you buy into them, because they show that "strict better than" is not transitive.

That's really interesting, thanks!

With the example of U given, U(2 apples + 2 oranges) > U(5 apples), which would not be true for a linear variant of U. You could transform this utility function into exp(U) = apples^a * oranges^(1-a), to make it more linear.

But I believe in general only linear transformations preserve all aspects of a utility function, when you start to look at the expected utility of a bet with probability p of outcome a and probability (1-p) of outcome b, which has expected utility E[U] = pU(a)+(1-p)U(b). I can imagine that it is harder to accept that this view of utility functions actually models human behavior.

That's exactly the problem that the author tries to explain away. Which transformations preserve the aspects of the utility functions depends on the underlying scale. (Or, better to say, the admissible transformations characterize the scale.) On an ordinal scale that merely represents preferences, any strictly monotone transformation is allowed. What you refer to is an interval scale, which is a type of cardinal utility.

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.

EU makes no sense with purely ordinal utilities either, but economists use it all the time. When they do so, they assume cardinal utilities, contrary to what the author of the original article asserts. Note also that e.g. the Principle of Diminishing Marginal Utility makes no sense with purely ordinal utilities.

> 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%93Morgenster...

[1]: https://en.wikipedia.org/wiki/Marginal_rate_of_substitution

[2]: https://en.wikipedia.org/wiki/Convex_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.

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.

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