## Tuesday, February 27, 2024

### Incommensurability in rational choice

When I hear that two options are incommensurable, I imagine things that are very different in value. But incommensurable options could also be very close in value. Suppose an eccentric tyrant tells you that she will spare the lives of ten innocents provided that you either have a slice of delicious cake or listen to a short but beautiful song. You are thus choosing between two goods:

1. The ten lives plus a slice of delicious cake.

2. The ten lives plus a short but beautiful song.

The values of the two options are very close relatively speaking: the cake and song make hardly any difference compared to the ten lives that comprise the bulk of the value. Yet, because the cake and the song are incommensurable, when you add the same ten lives to each, the results are incommensurable.

We can make the differences between the two incommensurables arbitrarily small. Imagine that the tyrant offers you the choice between:

1. The ten lives plus a chance p of a slice of delicious cake.

2. The ten lives plus a chance p of a short but beautiful song.

Making p be as small as we like, we make the difference between the options as small as possible, but the options remain incommensurable.

Well, maybe “noncomparable” is a better term than “incommensurable”, as it is a more neutral term, without that grand sound. Then we can say that (1) and (2) are “noncomparable by a slight amount” (relative to the magnitude of the overall goods involved).

There is a common test for incommensurability. Suppose A and B are options where neither is better than the other, and we want to know if they are equal in value or incommensurable. The test is to vary one of the two options by a slight amount of value, either positive or negative. If after the tweak the two options are still such that neither is better than the other, they must be incommensurable. (Proof: If A is slightly better or worse than A, and B is equal to A, then A will be slightly better or worse than B. So if A is neither better nor worse than B, we couldn’t have had B and A equal.)

But cases of things that are noncomparable by a slight amount show that we need to be careful with the test. The test still offers a sufficient condition for incommensurability: if the fact that neither is better than the other remains after making an option better or worse, we must have incommensurability. But if the two options are noncomparable by a very, very slight amount, a merely very slight variation in one could destroy the noncomparability, and generate a false positive for incommensurability. For instance, suppose that our two options are (3) and (4) with p = 10−100. Now suppose the slight variation on (3) is that we suppose you are given a mint in addition to the goods in (3). A mint beats a 10−100 chance of a song, even if it’s incommensurable with a larger chance of a song. So the variation on (3) beats the original (4). But we still have incommensurability.

(Note: There are two concepts of incommensurability. One is purely value based, and the other is agent-centric and based on rational choice. It is the second one that I am using in this post. I am comparing not pure values, but the reasons for pursuing the values. Even if the values are strictly incommensurable, as in the case of a certainty of a mint and a 10−100 chance of a song, the former is rationally preferable at least for humans.)

#### 1 comment:

IanS said...

If you think that human ethics has limits of applicability, do you think that there is some superhuman ethics that is less limited? The post seems to raise issues for any sort of ethics that involves infinite numbers of people. Merely human or not, you have to give up strict monotonicity, make many pairs of actions incomparable, or make arbitrary choices.