[pdonis]

> It's how to take a vector of utilities, and turn it into a single utility.

Not just "how", but whether doing such a thing is even possible at all. And even that doesn't push the problem back far enough: first the utilitarian has to assume that utilities, treated as real numbers, are even measurable or well-defined at all.

[tome]

I don't think a utilitarian requires that utilities are real numbers, just that they satisfy a total ordering.

[dragonwriter]

It requires a total ordering and an aggregation function (and to be useful in the real world rather than purely abstract, a reliable and predictive measuring mechanism, but that's a different issue.) I’m pretty sure (intuitively, haven’t considered a formal argument) if both exist, then there is a representation where utilities can be represented as (a subset of) the reals.

[pdonis]

> It requires a total ordering and an aggregation function

Yes. And note that this is true even for just a single person's utilities, i.e., without even getting into the issues of interpersonal comparison. For example, a single person, just to compute their own overall utility (never mind taking into account other people's), has to be able to aggregate their utilities for different things.

> if both exist, then there is a representation where utilities can be represented as (a subset of) the reals.

Yes. In more technical language, total ordering plus an aggregation function means utilities have to be an ordered field, and for any reasonable treatment that field has to have the least upper bound property (i.e., any sequence of members of the field has to have a least upper bound that is also in the field), and the reals are the only set that satisfies those properties.

[schoen]

Can you explain the "for any reasonable treatment" part here? How is the least upper bound property (or something equivalent to it) getting used in quantitative ethical theories?

[pdonis]

Because even if we assume that we can assign numbers to a single person's utility for many different things, we cannot assume that the ratios between these utilities will be integers or rational numbers. We have to consider the possibility of utility ratios that are irrational numbers. In treatments like that of Von Neumann, where numerical utilities are assigned meaning by asking hypothetical agents to accept or decline a potentially infinite series of bets, the possibility of utility ratios that are irrational numbers arises as the least upper bound property requirement--because the final utility ratio from the infinite series of bets will be the least upper bound of the series of ratio estimates that arise from the bets.

[schoen]

Thanks!

That also raises a different question about people's introspection and ability to accurately answer questions about their utility and preferences, but that's a bit far afield from the mathematical question.