Let's assume, for the sake of argument, that utilities are not interpersonally comparable. In that case, the uniquely best social welfare function is the one which maximizes the expected utility of a randomly chosen voter,

*in his own personal units of utility*. That is, the best option is the one that maximizes the average of all voters' utilities. But note that this is mathematically identical to simply

*summing*the utilities, for any given group of voters! And therefore the complaint about not being able to perform interpersonal comparisons of utility is academic, and doesn't invalidate the social choice function Smith employed for his BR calculations.

Regarding the aforementioned Twitter conversation, this also some bearing on the applicability of "labels" to the scores listed on a Score Voting ballot. That is, some folks would like to have something like:

0 - terrible, 1 - poor, 2 - mediocre, 3 - good, 4 - greatThe most obvious problem, to anyone acquainted with basic economic theory, is that this introduces error. For instance, suppose that my subjective connotation with these words is such that

u(great) - u(good) = u(good) - u(terrible)And suppose that my utilities for some set of options are such that

u(x) - u(y) = u(y) - u(z)Without labels, I would vote (assuming I'm a sincere voter)

x = 4, y = 2, z = 0But if I vote using the labels, that would be

x = 4, y = 3, z = 0This would produce a lower expected utility for me, making me worse off. Of course, if I'm a tactical voter, then it doesn't matter anyway. But why add more text than is absolutely necessary to a ballot, if it reduces the expected utility of voters? Why make the voters worse off?

There is a potential counter to this. But it appears to require that you accept the validity of interpersonal utility comparisons. Here's how it works. Suppose that Bob and Alice have the following utilities, on a universal scale.

u(x) = 8, u(y) = 6, u(z) = 0Their sincere votes (on a 0-4 scale) would look like

u(x) = 8, u(y) = 6, u(z) = 4

x = 4, y = 3, z = 0The scaling distortion makes it appear as though they have

x = 4, y = 2, z = 0

*different*opinions on y, but the

*same*opinions on z—which is precisely backwards. But suppose those labels, like "great" and "mediocre", have highly similar mappings to actual utility values for both Bob and Alice. In that case, the labels could prevent more loss than they caused—by nullifying some component of the normalization loss.

**The irony here is that our friend on Twitter supported the use of labels, but**

*opposed*the validity of interpersonal utility comparisons!The caveat is, as I said at the beginning, that a welfare function that maximizes the sum of all individuals' utilities also maximizes the expected utility of any individual voter, even if we disallow interpersonal comparisons of utility. So my aforementioned defense of labels is still equally valid, even if we don't permit interpersonal utility comparisons.

@selylidne can thank me for offering at least some evidence in support of his own argument.

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