Is value additive?

When something is good, or desirable, or a reason, then this is usually because it has some good (desirable, etc.) features. The thing may also have bad features, but if the thing is good then the good features outweigh the bad features. How does this weighing work? I'd like to say that the total goodness of a thing is always the sum of the goodness of its features. This "additive" view seems to be unpopular in both ethics and economics. I'll try to defend it.

I first need to state the view more precisely.

To begin, I assume that there are ultimate bearers of value. If we're talking about personal desire, this means that there are some things an agent desires "intrinsically" or "non-derivatively". Being free from pain might be a common example. If you desire to be free from pain then this is typically not because you really desire something else, and you think being free from pain is either a means to the other thing or evidence for the other thing. You simply desire being free from pain, and that's the end of the story.

Similarly if we're talking about moral value. Perhaps other people's well-being is an ultimate bearer of value. Perhaps conforming to God's commands is an ultimate bearer of value. If so, there is no further explanation as to why these things are good (or right). The moral buck stops here. (There might be a meta-ethical explanation of why these things are good, just as there might be a psychological or metaphysical explanation of why you desire being free from pain, but that's a different kind of explanation.)

Ultimate bearers of value are not closed under conjunction or disjunction. If being rich and being famous are your ultimate goals in life, then your ultimate goals need include neither being rich or famous nor being rich and famous. To be sure, a state in which you are rich and famous will be highly desirable, but it is desirable because it has two good-making features: it is a state in which you are rich and it is a state in which you are famous.

Many bearers of value are quantities. You can be more or less rich, more or less famous, more or less in pain. You might attach different degrees of basic value to different amounts of these quantities: the richer the better, for example.

A quantity like wealth partitions (centred) logical space. The worlds within any cell in the partition agree with respect to the amount to which the quantity is instantiated. The worlds where you own exactly £1 form a cell, the worlds where you own £2 form a cell, and so on.

I'm going to assume that all ultimate bearers of values can be modelled as partitions. An all-or-nothing feature simply partitions the worlds into two cells: those where it is present and those where it is absent.

The additivity hypothesis that I'd like to defend concerns maximal conjunctions of ultimate bearers of value.

Think of a partition as a grid on logical space. Every ultimate bearer of value is a grid. Lay all these grids on top of each other. This yields another grid, typically more fine-grained then the original grids. Within any cell in the combined grid, each world is as good as every other. (Note that we could define the combined grid in terms of this equally-good relation, without knowing anything about the ultimate bearers of value.) A cell in the combined grid settles everything that matters. Let's call these cells complete propositions.

Every complete proposition is a conjunction of exactly one element from each ultimate bearer of value. Let's call these elements the components of the complete proposition. For example, if level of pain and net worth are the ultimate bearers of value, then experiencing no pain and owning £10 is a complete proposition, and its components are experiencing no pain and owning £10.

Now here's the additivity hypothesis: The value of a complete proposition is the sum of the value of its components.

This assumes that one can assign numerical value both to complete propositions and to their components: to elements of ultimate bearers of value. Such numerical representability is, of course, far from obvious. But it doesn't seem to be the reason why people reject the additivity hypothesis. Rather, people reject additivity because they think that the value of a combined thing isn't determined by the individual value of its components.

It's easy to come up with apparent examples. I like riding my bike, and I like listening to music, but I don't like doing the two at the same time. Kant allegedly held that happiness is valuable when combined with virtue, but not when combined with vice. (I've learned this from Brown (2020).)

Let's have a closer look at these examples.

Take the first case. There's a reason why I don't like riding my bike while listening to music. When I do these two things at the same time, I can't focus on either. The things that ultimately matter to me are not just riding my bike and listening to music. I also care about whether I can focus on these activities. Whenever additivity fails, we should expect some such explanation. If riding my bike and listenting to music really were the only things I cared about, then I couldn't possibly not desire doing both.

Similarly in the Kant example. Why is happiness valuable when combined with virtue but not when combined with vice? Presumably because there is another value involved, something like getting what one deserves. Kant probably thought that a virtuous person deserves happiness, but a vicious person does not. A state in which a virtuous person is happy is good because the person has what she deserves. A state in which a vicious person is happy is worse because the person doesn't have what she deserves. (She deserves to be miserable.)

So I'm not convinced by these apparent counterexamples to additivity.

I see the additivity hypothesis not as a substantive hypothesis, but as a regulative structural constraint for thinking about value. When we think systematically about value, we should think of complete value as additively determined by ultimate value. Often we start with somewhat vague ideas about complete value and ultimate value. We might initially think, for example, that all we ultimately desire is being rich and being famous. When we notice that this would render our values non-additive, I suggest that we should revise this assumption. We should realise that what we ultimately care about are, in fact, more specific properties/propositions, or that one thing we ultimately care about, besides being rich and being famous, is (say) not being both.

The additivity hypothesis is compatible with any assignment of value to complete propositions. If all else fails, we can simply assume that the grid of complete propositions is the single ultimate bearer of value. But most value systems don't look like this. Personal desirability, moral goodness, and overall reasons can be decomposed. As I said at the outset, when something is good or desirable, then this is usually because it has some good or desirable features. The additivity hypothesis says how we should think of the relation between the value of the features and the value of the complete thing: as a matter of addition.

But why addition? Why not some other operation? Because the real structural constraint is separability. It has nothing to do with numbers. Separability means that if replacing some components of a complete proposition A with alternative components yields a complete proposition B that is better than A, then making the same replacement in other complete propositions A' always yields a complete proposition B' that is better than A'. Informally, swapping a component by an alternative should always affect the overall state in the same direction. If it looks like swapping a component by an alternative sometimes makes things better and sometimes worse (like swapping happiness for unhappiness in the Kant example), then we haven't identified all relevant components. It is well-known that, given some technical background assumptions, separability implies additive representability. (See e.g. Krantz et al. (1971).)

We could use a different operation. We would then have to change the scale on which we measure the component values. The additivity hypothesis goes along with the assumption that the numerical value of a component measures its additive contribution to total value.

(I haven't fully thought this through, and I've read almost nothing on the topic, so there may well be obvious problems that I've missed. One problem that I'm aware of is that the "technical background assumptions" may easily fail. But I don't know if that's a serious problem.)

Brown, Campbell. 2020. “The Significance of Value Additivity.” Erkenntnis.
Krantz, David, Duncan Luce, Patrick Suppes, and Amos Tversky. 1971. Foundations of Measurement, Vol. I: Additive and Polynomial Representations. New York Academic Press.


# on 19 November 2022, 15:18

These separable "goods" may be additive in the case of a single person, but there are other things to consider in, for example, economics and population ethics:

1. The utility to an individual of more of one particular type of "good" is not linearly additive, due to satiation and or other types of reduced marginal utility. There are many empirical examples of this and a sound theoretical foundation here as well. This is why economists use ideas like log utility.

2. In population ethics there is the idea that a life is a "good" and some hold that is linearly aditive. The question here is who is deriving the utility from this particular "good". Is is the lives deriving utility from other lives? Is it the "universe" deriving utility from these lives? Is it "God?"

Whichever of these ideas you choose may affect your analysis, but I believe except for the "God" case where you could argue there is no utility satiation with additional lives, the other cases will also be amenable to log utility arguments.

[Note by log utility in the above comments this is just a placeholder for sub linear utility functions.]

# on 19 November 2022, 19:53

Ah, thanks. I see that my explanation was a little too abstract. I'm not assuming that value is linearly additive in some ordinary quantities. The hypothesis is that "the value of a complete proposition is the sum of the value of its components". If one component is a quantity which has diminishing marginal value, then the value of this component might equal the log of the quantity.

# on 28 July 2023, 14:49

Hi Wo,

I've been thinking a little about this kind of way of arguing for additivity (thanks for the pointer!)

Here's one thing I've come up against. Suppose you've got your separable component propositions, and they're "relational", in the sense your treatment of Kant is. That is: we've got a "slot" for desert-and-happiness, with cells specifying a degree of happiness/unhappiness, and also whether the person is deserving or not.

One of the key technical assumptions you need is "restricted solvability". And roughly, that tells you that if you've got two complete propositions differing only in which component proposition they have in this slot, with different values, and then you've got some other complete proposition x whose value lies between them, then you can find a substitution for the component in the first pair that matches the value of x.

It's a sort-of continuity assumption.

Now, in something like the Kant case, continuity is going to come, intuitively, from varying degree of happiness. So suppose the pair we start with are both deserving people with different degrees of happiness. Then you'd hope to be able to match any intermediate value by finding an intermediate level of happiness for a deserving person. Seems okay.

But suppose you have a deserving person, and an undeserving person, each with different levels of happiness/unhappiness. Then it seems more substantive ethical assumption to assume that you can find some substitution for that component of overall value that will match any other realized intermediate value (picture a situation where the happiness/unhappiness of the deserving is always more important than the happiness/unhappiness of the undeserving, so the realizable values form two disconnected "islands").

So I think this assumption bears thinking about, if you're going to make the very general argument here (I was thinking of this in connection to arguments for the additivity of overall accuracy of a credence function, and that's also relational in this way---the accuracy of a given credence turns on whether the proposition which is its content is true or false. So I think that hidden in the solvability axiom is going to be a particular assumption about the way that accuracy-given-truth and accuracy-given-falsity relate).

Now, I wonder whether one can overcome this just by embedding the whole structure in a bigger one (filling in the gaps, as it were), deriving an additive representation for that, and then cutting back to the original. After all, solvability is a richness assumption, and embeddability often helps out for that in other cases.

The other technical assumptions look more innocent to me, fwiw---it seems like separability (/independence) and this restricted solvability are the two where the real action is.

# on 30 July 2023, 19:27

Thanks Robbie! Yes, restricted solvability is one of the "technical background assumption" I had in mind, and your example nicely brings out why it might fail. As you say, it does look like a typical richness/extensibility assumption in representation theorems, so maybe it's not a serious constraint. Worth thinking about some more, even though I haven't seen it mentioned by people who reject additivity.

Add a comment

Please leave these fields blank (spam trap):

No HTML please.
You can edit this comment until 30 minutes after posting.