It might seem easy to average the utilities of two different people. Utilities are numbers, and averaging numbers is easy. But averaging utilities is not so easy a thing. It’s well-known among economists31 that averaging the utilities of two different people—indeed, making any interpersonal comparisons of utility whatsoever—is not meaningful given the standard structure of decision theory. It’s also well-known that with some additional technical apparatus more can be meaningfully done. Such issues are not as well-known among philosophers, however. I therefore include this explanation of the philosophical issues at stake in the mathematical structure of utilities.
There are limits to what sorts of mathematical operations can be meaningfully performed. Suppose someone wondered, “What’s the height of a tree that’s as tall as the temperature of a hot summer day?” One could reply “A hot summer day is around 95 degrees Fahrenheit, so the tree would be around 95 inches tall. It wouldn’t be a particularly tall tree, then, but more likely a sapling.” But this reply is obvious nonsense. The whole idea of directly equating some height with some temperature is absurd. Note that the numbers don’t help even though the same numbers appear when measuring heights and temperatures. There’s no equivalence to be had between 95 degrees Fahrenheit and 95 inches. Degrees Fahrenheit and inches are just very different things; the common ‘95’ doesn’t help. Of course, the selection of those units was arbitrary. One could have equally well taken degrees Celsius and centimeters and gotten a different bogus answer. But the basic problem is not the plurality of units. Then the problem would be that there were too many viable answers to the question. There are, however, no viable answers to the question. The question itself is deeply misguided.
Numbers (specifically, the natural numbers) can be put to a great many uses. Different uses capitalize on different aspects of the numbers’ numerical structure. We can use the numbers 1, 2, and 3 to describe how many coins Alice, Bob, and Carol have in their pockets. In this case, it makes sense to use the additive structure of those numbers. Just as \(1 + 2 = 3\), it makes perfect sense to say that Alice and Bob have as many coins in their pockets as Carol does in her pockets. We can also use the numbers 1, 2, and 3 to describe the order in which Alice, Bob, and Carol completed a race. But in this case it makes no sense to use the additive structure of those numbers. There’s no sense in which Alice’s race result and Bob’s race result, taken together, are equivalent to Carol’s race result. But the ordinal structure of the numbers still applies. Just as \(1< 2 < 3\), it makes perfect sense to say that Alice finished before Bob, who in turn finished before Carol.
The mathematical structure of utility functions is quite modest, too modest to allow for interpersonal comparison. Utilities have the structure only of an interval scale. Utilities have an order, and comparisons of differences between utilities can be made, but that’s it. Utilities can express that an agent prefers A to B, and utilities can express that an agent prefers A to B by twice as much as he prefers C to D. But utilities standardly express nothing more. It is, for example, not meaningful to say that A has twice as much utility as B. The reason is that utilities are standardly meant to capture preferences and nothing more. The preferences defined over simple outcomes are purely ordinal. Preferences over mixtures32 of simple outcomes get only a little more structure. An agent prefers A to B by as much as he prefers B to C just in case the agent is indifferent between the certainty of B and a 50/50 mixture of A and C. And so on. The numerical representation of an interval scale is—as Pettigrew correctly notes—unique only up to positive affine transformation (that is, adding or subtracting any number to all utilities and multiplying all utilities by any positive number leaves those utilities unchanged). All the properties that matter in a utility function will be preserved by any positive affine transformation. Suppose an agent has three options, A, B, and C. Giving them utilities of 1, 2, 3 is the same as giving them utilities of 5, 6, 7 (just add 5), is the same as giving them utilities of 10, 20, 30 (multiply by 10), is the same as giving them utilities of 15, 25, 35 (multiply by 10 and then add 5), is the same as giving them utilities of 50, 60, 70 (add 5 and then multiply by 10). All the properties a utility function has are preserved in those assignments. Those utility functions all just represent preferring C to B to A, and preferring C to B and B to A by the same amount. Those utility functions are the same utility function, just as (\(2 + 2\)) and (\(1 + 3\)) are the same number. The representations are different, but it’s the same thing being represented.
On a mathematical level it makes sense that you can’t average across different utility functions. Suppose you’re trying to take a straight average of two utility functions, giving them each equal weight. What numerical representations of the two functions should you take? If you average 1, 2, 3 and 3, 2, 1, you get 2, 2, 2. But if you average 100, 200, 300 and 3, 2, 1, you get 51.5, 101, 150.5. The latter representation of the first utility function gives it massively more of an effect than the former representation does. But any representation-dependent operation is nonsense.
On a philosophical level it makes sense that you can’t average across different utility functions. Utilities don’t measure absolute amounts of desirability. It’s not as though you can average liking X a lot with liking X a little and thereby get liking X moderately. Two agents, one of whom likes every possibility and the other of whom dislikes every possibility, can easily have the same utility function for those possibilities. Utilities show only relative desirabilities, and standings in two different hierarchies of relative desirability can’t be averaged together.
Note that—even in the worst-case scenario—formal problems with averaging across different utility functions do not mean that information about other people’s utilities has no bearing on what you should think about your own utilities. Suppose you’re uncertain about your own preferences, but have reason to believe that your preferences are similar to a friend’s preferences. Then learning that your friend prefers A to B can easily give you evidence that you prefer A to B. There are plenty of ways that the utilities of others can bear what one should think about one’s own utilities. The formal problems have a limited scope. The worry isn’t that information about other people’s utilities will be categorically unhelpful; the worry is that certain sorts of comparisons—including the sorts of comparisons that Pettigrew’s proposal requires—will not be possible.
One can, however, define non-standard utilities which express more than relative desirability according to some agent. The structural poverty of an interval scale can be enriched. And if one is to alter decision theory in the ways Pettigrew prescribes, then such enrichment is necessary.