# On the transitivity of strict preference

The notion of comparing alternatives often comes up in philosophy, particularly when discussing practical reason. There are various names for this (we can talk about the reasons for choosing A over B, or how A is better than B, or how A is more desirable to B, or how A is preferred to B) but they all amount to the same thing.

The other day I was reading the SEP article on preference and was struck by this counterexample to transitivity of strict preference (I recall my friends mentioning it to me in the past, but I only thought about it critically this time around). In this quote, X≻Y represents that X is strictly preferred to Y, and X∼Y represents indifference between X and Y:

In an important type of counterexample to transitivity of strict preference, different properties of the alternatives dominate in different pairwise comparisons. Consider an agent choosing between three boxes of Christmas ornaments… Each box contains three balls, coloured red, blue and green, respectively; they are represented by the vectors ⟨R1,G1,B1⟩, ⟨R2,G2,B2⟩, and ⟨R3,G3,B3⟩. The agent strictly prefers box 1 to box 2, since they contain (to her) equally attractive blue and green balls, but the red ball of box 1 is more attractive than that of box 2. She prefers box 2 to box 3, since they are equal but for the green ball of box 2, which is more attractive than that of box 3. And finally, she prefers box 3 to box 1, since they are equal but for the blue ball of box 3, which is more attractive than that of box 1. Thus,

a. R1≻R2∼R3∼R1,
b. G1∼G2≻G3∼G1,
c. B1∼B2∼B3≻B1; and
d. ⟨R1,G1,B1⟩≻⟨R2,G2,B2⟩≻⟨R3,G3,B3⟩≻⟨R1,G1,B1⟩.

The described situation yields a preference cycle, which contradicts transitivity of strict preference.

(Note that I’ve added the labels to the listed conditions for the sake of this discussion.)

Now, I haven’t read much of the modern discussion on transitivity of preference (indeed, I didn’t even finish reading the article), so perhaps what I’m about to say is really obvious.

It seems clear to me that the above counterexample motivates the otherwise very natural distinction between (1) being better in some respect and (2) being better simply. Ultimately it has to do with why we prefer something over another. For instance, assume I prefer red balls over blue balls. Then I prefer this red ball over that blue ball simply, and I prefer this box of green and red balls over that box of green and blue balls in some respect.

I say this distinction is “very natural” because it seems necessary if we are to make sense of trade-offs, which are manifold in everyday experience. As a trivial example (which I find myself in often), imagine you need to pick one of two routes to your destination. Route A is longer but has prettier scenery and conversely route B is shorter but has uglier scenery. You have to pick one, but whatever choice you make will involve a trade-off. On account of what is this a trade-off? Well, surely it’s because shorter routes are preferable to longer ones and prettier routes are preferable to uglier ones. That is, A is better in some respect (prettiness) and B is better in some other respect (length).

This distinction resolves the above counterexample by showing us that (a)-(d) equivocate on “≻”. In (a)-(c) X≻Y means X is strictly preferred to Y simply, but in (d) it means X is strictly preferred to Y in some respect.

The SEP article immediately goes on to say the following:

These and similar examples can be used to show that actual human beings may have cyclic preferences. It does not necessarily follow, however, that the same applies to the idealizedrational agents of preference logic. Perhaps such patterns are due to irrationality or to factors, such as lack of knowledge or discrimination, that prevent actual humans from being rational.

Perhaps, but I’m inclined to think life’s more complicated than that. It seems pretty intuitive that there are various types of goods that are incommensurable. One way we might make this intuition precise is as follows: in general there seem to be two ways in which A is better than B:

1. A and B are both means to C and A is a better means.
2. B is a means to A.

(1) is where this whole business of comparing of alternatives comes in. Given our above discussion we realise that A can be a better means either in some respect or simply. Aristotle mentions something like (2) at the beginning of the Nicomachean Ethics. The guiding intuition here is that ends are preferred to means because “it is for the sake of the former that the latter are pursued” (I.1 1094a15-16).

Now, combining this with our previous discussion on basic human goods, the fact that there are multiple basic goods suggests that at least sometimes two goods will be incommensurable.