Derivative value

Some things have derivative value. One kind of derivation is from whole to parts: a stone can have a special value by virtue of being a part of something of great significance, say a temple. Another kind of derivation is from parts to whole: a golden statue has a value deriving from the value of its atoms. Yet another kind is from friend to friend: if I do good directly to a friend of yours, I benefit you as well.

The distinction between derivative and original value is orthogonal to that between instrumental and non-instrumental value, and probably also to that between intrinsic and extrinsic value.

It is easy to create puzzles with derivative value, because derivative value is not simply additive and double counting must be avoided. Imagine a golden statue made by someone with minimal artistic skill. The maker of that statue then produced something literally worth its weight in gold, and yet they added almost no value to the world, because almost all of the value of the poorly made statue is derivative. Melting down a golder statue worth exactly its weight in gold does no harm to the world! Similarly, dissolving a ten-member committee need be no more harmful than dissolving a five-member one.

If two people are drowning, one friendless and one with ten friends, perhaps there is additional reason to save the one with ten friends, though the point is not clear. But if there is additional reason, it does not scale linearly with the number of friends. If someone had a thousand of friends, that needn’t create much more a reason to save them than if they had a hundred, I suspect.

It is tempting to initially think of derivative value as a faint shadow of original value. Sometimes this is true: the death of Alice considered as a derivative harm to her distant friends is a mere shadow of the badness of that death considered as a harm to Alice. But sometimes it’s not true: the death of Alice considered as a derivative harm to her closest friends approaches the original badness of that death considered as a harm to her. And the inartistic golden statue’s derivative value is not a whit less than the original value of its gold components.

Can we at least say that derivative value is always at most equal to the original value? Maybe, but even that is not completely clear. That Alice is loved by God makes it be the case that a harm to Alice is a harm to God. But it could be that the derivative badness to God gives us reasons to protect Alice that are stronger than those coming from the original badness to Alice, and the derivative badness here might exceed the original badness. (Recall here Anselm’s idea that sin is infinitely bad, because it offends the infinite God.) Perhaps, though, cases of love do not give rise to purely derivative value, because the derivative value is created by an interaction between the original value of the beloved and the original value of the lover. On the other hand, insofar as the inartistic golden statue’s value is purely derivative, it cannot exceed the original value of the parts.

The non-additiveness of derivative value throws a wrench in simple consequentialist systems on which we maximize the total value of everything. Perhaps, though, it is possible to talk about overall value, which is not additive in nature, so this need not be a knock-down argument against consequentialism. But it definitely seems to complicate things.

Note that similar phenomena occur for other properties than value. When one takes ten pounds of gold and makes a statue of it, one may create a ten pound object (assuming for the sake of argument that statues really exist), but one doesn’t add ten pounds to reality. We need to avoid double-counting in the case of derivative mass just as much as for derivative value.

Minor inconveniences and numerical asymmetries

As a teacher, I have many opportunities to cause minor inconveniences in the lives of my students. And subjectively it often feels like when it’s a choice between a moderate inconvenience to me and a minor inconvenience to my students, there is nothing morally wrong with the minor inconvenience to the students. Think, for example, of making online information easily accessible to students. But this neglects the asymmetry in numbers: there is one of me and many of them. The inconvenience to them needs to be multiplied by the number of students, and that can make a big difference.

I suspect that we didn’t evolve to be sensitive to such numerical asymmetries. Rather, I expect we evolved to be sensitive to more numerically balanced relationships, which may have led to a tendency to just compare the degree of inconvenience, in ways that are quite unfortunate when the asymmetry in numbers becomes very large. If I make an app that is used just once by each of 100,000 people, and my app’s takes a second longer than it could, then it should be worth spending about two working days to eliminate that delay. (Or imagine—horrors!—that I deliberately put in that delay, say in the form of a splashscreen!) If I give a talk to a hundred people and I spend a minute on an unnecessary digression, it’s rather like the case of a bore talking my ears off for an hour and a half. In fact, I rather like the idea that at the back of rooms where compulsory meetings are held there should be an electronic display calculating for each speaker the total dollar-time-value of the listeners’ time, counting up continuously. (That said, some pleasantries are necessary, in order to show respect, to relax, etc.)

Sadly, I rarely think this way except when I am the victim of the inconvenience. But it seems to me that in an era where more and more of us have numerically asymmetric relationships, sometimes with massive asymmetries introduced by large-scale electronic content distribution, we should think a lot more about this. We should write and talk in ways that don’t waste others’ time in numerically asymmetric situations. We should make our websites easier to navigate and our apps less frustrating. And so on. The strength of the moral reasons may be fairly small when our contributions are uncompensated and others’ participation is voluntary, but rises quite a bit when we are being paid and/or others are in some way compelled to participate.

One of my happy moments when I actually did think somewhat in this way was some years back when, after multiple speeches, I was asked to say a few words of welcome to our prospective graduate students. There were multiple speeches. I stood up, said “Welcome!”, and sat down. I am not criticizing the other speeches. But as for me, I had nothing to add to them but just a welcome from me, so I added nothing but a welcome from me. I should do this sort of thing more often.

Label independence and lotteries

Suppose we have a countably infinite fair lottery, in John Norton’s sense of label independence: in other words, probabilities are not changed by any relabeling—i.e., any permutation—of tickets. In classical probability, it’s easy to generate a contradiction from the above assumptions, given the simple assumption that there is at least one set A of tickets that has a well-defined probability (i.e., that the probability that the winning ticket is from A is well-defined) and that has the property that both A and its complement are infinite. John Norton rejects classical probability in such cases, however.

So, here’s an interesting question: How weak are the probability theory assumptions we need to generate a contradiction from a label independent countably infinite lottery? Here is a collection that works:

1. The tickets are numbered with the set N of natural numbers.

2. If A and B are easily describable subsets of the tickets that differ by an easily describable permutation of N, then they are equally probable.

3. For every easily describable set A of tickets, either A or its complement is (or both are) more likely than the empty set.

4. If A and B are disjoint and each is more likely than the empty set, then A ∪ B is more likely than A or is more likely than B.

5. Being at least as likely as is reflexive and transitive.

Here, Axioms 3 and 4 are my rather weak replacement for finite additivity (together with an implicit assumption that easily describable sets have a well-defined probability). Axiom 2 is a weak version of label independence, restricted to easily describable relabeling. Axiom 5 is obvious, and the noteworthy thing is that totality is not assumed.

What do I mean by “easily describable”? I shall assume that sets are “easily describable” provided that they can be described by a modulo 4 condition: i.e., by saying what value(s) the members of the set have to have modulo 4 (e.g., “the evens”, “the odds” and “the evens not divisible by four” are all “easily describable”). And I shall assume that a permutation of N is “easily describable” provided that it can be described by giving a formula f_i(x) using integer addition, subtraction, multiplication and division for i = 0, 1, 2, 3 that specifies what happens to an input x that is equal to i modulo 4. (E.g., the permutation that swaps the evens and the odds is given by the formulas f₂(0)=f₀(x)=x + 1 and f₃(x)=f₁(x)=x − 1.)

Proof: Let A be the set of even numbers. By (3), A or N − A is more likely than the empty set. But A and N − A differ by an easily describable permutation (swap the evens with the odds). So, by (2) they are equally likely. So they are both more likely than the empty set. Let B be the subset of A consisting of the even numbers divisible by four and let C = A − B be the even numbers not divisible by 4. Then B and C differ by an easily describable permutation (leave the odd numbers unchanged; add two to the evens divisible by four; subtract two from the evens not divisible by four). Moreover, A and B differ by an easily (but less easily!) describable permutation. (Exercise!) So, A, B and C are all equally likely by (2). So they are all more likely than the empty set. So, A = B ∪ C is more likely than either B or C (or both) by (4). But this contradicts the fact that A is equally likely as B and C.